1.1. Factorization structures of dimension 2

To begin the study of factorization structures, we note that there is only one isomorphism class in dimension 1 and focus on the full understanding of the first non-trivial case, factorization structures of dimension 2. Although factorization structures were previously defined in [Reference Apostolov, Calderbank and Gauduchon4], inconsistencies between the definition, notion of isomorphism, and their classification in 2 dimensions, led to definition 1.0.1. The new definition results in the same 2-dimensional classification as in [Reference Apostolov, Calderbank and Gauduchon4]: up to isomorphism, it consists of two factorization structure, 2-dimensional Segre and Veronese factorization structures.

To restate this classification in a simplified manner and full detail, we note

Proof. Clearly,

(1.5)\begin{align} 2\geq \dim \left( \varphi(\mathfrak{h})\cap\ell^0\otimes V_2^* \right) = \dim \left( \varphi(\mathfrak{h})\cap\Sigma_{1,\ell}^0 \right) \geq 1 \end{align}

holds for any $\ell\in\mathbb{P}(V_1)$, and similarly for intersections $\varphi(\mathfrak{h})\cap\Sigma_{2,\ell}^0$ with $\ell\in\mathbb{P}(V_2)$. Note that if (1.5) were 2-dimensional in two distinct points $\ell, \bar{\ell}\in\mathbb{P}(V_1)$, then two 2-dimensional subspaces $\ell^0\otimes V_2^*$ and $\bar{\ell}^0\otimes V_2^*$, whose intersection is trivial, would lie in the 3-dimensional space $\varphi(\mathfrak{h})$. Therefore, the intersection $\varphi(\mathfrak{h}) \cap \Sigma_{1,\ell}^0$ is 2-dimensional at most at one point, hence is generically 1-dimensional, i.e., φ is a factorization structure.

We found that 2-dimensional factorization structures are merely linear inclusions $\varphi:\mathfrak{h}\to V_1^*\otimes V_2^*$, which we now classify up to isomorphism of factorization structures via the annihilator $\varphi(\mathfrak{h})^0\leq V_1\otimes V_2$.

If $\varphi(\mathfrak{h})^0$ is decomposable in $V_1\otimes V_2$, i.e., $\varphi(\mathfrak{h})^0=\gamma_1\otimes\gamma_2$ for some 1-dimensional subspaces $\gamma_j\subset V_j$, then the corresponding factorization structure, called Segre, is of the form

(1.6)\begin{align} \varphi(\mathfrak{h})=V_1^*\otimes\gamma_2^0+\gamma_1^0\otimes V_2^* \hookrightarrow V_1^* \otimes V_2^*, \end{align}

where $\gamma_j^0\subset V_j^*$ is the annihilator of γ_j (see remark 1.0.2). One easily observes that if $\tilde{\varphi}$ is another inclusion so that $\tilde{\varphi}(\mathfrak{h})^0$ is decomposable, then φ and $\tilde{\varphi}$ are isomorphic as factorization structures.

Suppose now that $\varphi(\mathfrak{h})^0$ is indecomposable. Then, any non-zero $\chi\in\varphi(\mathfrak{h})^0$, viewed as a map $\chi: V_1^* \to V_2$, is invertible, since in a basis it is represented by a 2-by-2 matrix with non-zero determinant due to the indecomposability. The composition of isomorphisms $\text{id}\otimes\chi^{-1}:V_1\otimes V_2\to V_1\otimes V_1^*$, which maps χ to the identity automorphism of $V_1^*$, and $\omega\otimes\text{id}:V_1\otimes V_1^*\to V_1^*\otimes V_1^*$, where ω is a fixed area form on V ₁, maps χ on an element of $\bigwedge^2V_1^*$. Therefore, $\langle\chi\rangle$ is mapped onto $\bigwedge^2V_1^*$ under the isomorphism $\omega\otimes\chi^{-1}$ yielding the commutative diagram

Taking in account remark 1.0.2 and dualizing (1.7) shows that the Veronese factorization structure

(1.8)\begin{align} S^2V_1\hookrightarrow V_1\otimes V_1 \end{align}

is isomorphic in the sense of factorization structures to $\varphi:\mathfrak{h}\to V_1^*\otimes V_2^*$.

We note that Segre and Veronese factorization structures are not isomorphic which can be seen from the decomposability of $\varphi(\mathfrak{h})^0$ in respective cases. This classifies 2-dimensional factorization structures.

1.2. Segre–Veronese factorization structure

We describe a large class of factorization structures, called Segre–Veronese, which generalize Segre and Veronese factorization structures discussed in §1.1.

For $i\in\{1,\ldots,m\}$ we say that the term a_i in $a_1\otimes\cdots\otimes a_m$ is in the ith slot. If a partition of m is given, $m=d_1+\cdots+d_k$, $d_j\geq1$, slots group into k groups with the jth group containing d_j slots, $j\in\{1,\ldots,k\}$. Slots belonging to the jth group are referred to as grouped j-slots. In fact, positions in such a tensor product can be labelled by pairs (j, r), where $j\in\{1,\ldots,k\}$ and $r\in\{1,\ldots,d_j\}$. For a partition of m as above and a fixed $j\in\{1,\ldots,k\}$, we define the operator

\begin{equation*}ins_j: (W_j^*)^{\otimes d_j}\otimes\bigotimes_{\substack{i=1\\i\neq j}}^k (W_i^*)^{\otimes d_i} \to \bigotimes_{i=1}^k (W_i^*)^{\otimes d_i}\end{equation*}

which acts on decomposable tensors by

\begin{align*} \left(w_j^1\otimes\cdots\otimes w_j^{d_j}\right) \otimes \bigotimes_{\substack{i=1\\i\neq j}}^k \left(w_i^1\otimes\cdots\otimes w_i^{d_i}\right) \mapsto \bigotimes_{i=1}^k \left(w_i^1\otimes\cdots\otimes w_i^{d_i}\right), \end{align*}

where W_j, $j=1,\ldots,k$, are vector spaces. Partitions $m=d_1+\cdots+d_p$ and $m=e_1+\cdots+e_q$ are considered to be the same if $\{d_1,\ldots,d_p\}=\{e_1,\ldots,e_q\}$, and distinct if they are not the same.

Definition 1.2.1. For $d_1,\ldots,d_k$ a partition of an integer $m\geq2$ and W_r, $r=1,\ldots,k$, 2-dimensional vector spaces, let $\Gamma_j \subset \bigotimes_{r=1,r\neq j}^k(W_r^*)^{\otimes d_r}$, $j\in\{1,\ldots,k\}$, be 1-dimensional subspaces such that

(1.9)\begin{align} \sum_{j=1}^{k} ins_j \left( S^{d_j}W_j^*\otimes\Gamma_j \right) \end{align}

has dimension m + 1, where $S^{d_j}W_j^*\subset(W_j^*)^{\otimes d_j}$ is viewed as the subspace of symmetric tensors. Define vector spaces $V_1,\ldots,V_m$ by

(1.10)\begin{align} V_{d_1 + \cdots + d_{j-1} + 1} = V_{d_1 + \cdots + d_{j-1} + 2} = \cdots = V_{d_1 + \cdots + d_{j-1} + d_j} = W_j, \quad j=1,\ldots,k,\nonumber\\ \end{align}

where d ₀ is defined to be zero. The standard Segre–Veronese factorization structure $\varphi: \mathfrak{h} \to V^*$ is defined to be such that $\mathfrak{h}$ is the $(m+1)$-dimensional space (1.9), $V^* = \otimes_{j=1}^m V_j^*$, where V_j is defined by (1.10), and φ is the canonical inclusion of $\mathfrak{h}$ to $V^*$, i.e., it is

(1.11)\begin{align} \sum_{j=1}^{k} ins_j \left( S^{d_j}W_j^*\otimes \Gamma_j \right) \hookrightarrow \bigotimes_{j=1}^k(W_j^*)^{\otimes d_j}. \end{align}

Factorization structures corresponding to trivial partitions,

(1.12)\begin{equation} \sum_{j=1}^m ins_j \left( W_j^* \otimes \Gamma_j \right) \hookrightarrow \bigotimes_{j=1}^m W_j^* \end{equation}

for $m=1+\cdots+1$, and

(1.13)\begin{equation} S^mW^* \hookrightarrow (W^*)^{\otimes m} \end{equation}

for m = m, are respectively called Segre and Veronese. An element of the isomorphism class of a standard Segre–Veronese factorization structure is referred to as a Segre–Veronese factorization structure.

We frequently refer to the 1-dimensional spaces $\Gamma_j$ as defining tensors of the standard Segre–Veronese factorization structure, since one does need them to define a given Segre–Veronese factorization structure, and since each $\Gamma_j$ is a linear span of a tensor.

Remark 1.2.2. Note that Segre and Veronese factorization structures recover (1.6) and (1.8) when m = 2.

To verify that (1.11) defines a factorization structure, we observe that for $i\in\{1,\ldots,k\}$ and generic $\ell\in\mathbb{P}(W_i)$, we have

(1.14)\begin{align} \varphi(\mathfrak{h}) \cap \Sigma_{d_1+\cdots+d_{i-1}+q,\ell}^0 = ins_i \left( (\ell^0)^{\otimes d_i} \otimes \Gamma_i \right), \end{align}

where $\varphi(\mathfrak{h})$ is (1.9), $q\in\{1,\ldots,d_i\}$ and d ₀ is defined to be 0. Note that there are at most finitely many $\ell\in\mathbb{P}(V_i)$ for which the dimension of the intersection in (1.14) could be strictly larger than one, and, loosely speaking, this occurs when defining tensors $\Gamma_j$, j ≠ i, decompose at the ith slot.

Determining in general which choices of $\Gamma_j$, $j=1,\ldots,k$, give rise to a factorization structure, i.e., make (1.9) an $(m+1)$-dimensional vector space, is a challenging task. Instead, in the following, we exemplify particular choices which effortlessly guarantee the correct dimension.

Example 1.2.3. We examine the standard Segre–Veronese factorization structure for k = 2. To this end, let $m=d_1+d_2$ be a partition, and $\Gamma_1\subset (W_2^*)^{\otimes d_2}$ and $\Gamma_2\subset (W_1^*)^{\otimes d_1}$ be 1-dimensional subspaces. Observe that the dimension of the image of

(1.15)\begin{align} S^{d_1}W_1^*\otimes\Gamma_1 + \Gamma_2\otimes S^{d_2}W_2^* \hookrightarrow (W_1^*)^{\otimes d_1}\otimes (W_2^*)^{\otimes d_2} \end{align}

is m + 1 if and only if $\Gamma_1\subset S^{d_2}W_2^*$ and $\Gamma_2\subset S^{d_1}W_1^*$, which completely characterizes choices of Γ₁ and Γ₂ leading to a factorization structure.

Example 1.2.4. For a partition $m=d_1+\cdots+d_k$ and 1-dimensional subspaces $ a^r \subset W_r^*$, $r=1,\ldots,k$, we define the product Segre–Veronese factorization structure as the standard Segre–Veronese factorization structure such that

(1.16)\begin{align} \Gamma_j= \bigotimes_{\substack{r=1\\r\neq j}}^k (a^r)^{\otimes d_r}, \quad j=1,\ldots,k. \end{align}

These data ensure that any two summands of (1.9) intersect in $\bigotimes_{r=1}^k(a^r)^{\otimes d_r}$, which implies that the dimension of (1.9) is m + 1. Therefore, the product Segre–Veronese factorization structure is indeed a factorization structure. The product Segre–Veronese factorization structure with partition $m=1+\cdots+1$ is called the product Segre factorization structure.

Example 1.2.5. Motivated by the example above, a natural step is to find when decomposable $\Gamma_j$ determine a factorization structure, i.e., give rise to the correct dimension of (1.9). The decomposable Segre–Veronese factorization structure is defined as the standard Segre–Veronese factorization structure such that $\Gamma_j$ are decomposable, i.e., $\Gamma_j = \bigotimes_{\substack{r=1\\r\neq j}}^k \bigotimes_{p=1}^{d_r} a_j^{r,p}$ for some 1-dimensional subspaces $a_j^{r,p} \subset W_r^*$, $j=1,\ldots,k$. In corollary 2.6.4, we show that if such decomposable $\Gamma_j$, $j=1,\ldots,k$, determine a factorization structure, then it must be that $a_j^{r,1}= \cdots= a_j^{r,d_r} =: a_j^r$, and hence

(1.17)\begin{align} \Gamma_j= \bigotimes_{\substack{r=1\\r\neq j}}^k (a^r_j)^{\otimes d_r} \end{align}

necessarily. However, it is still not plain to see which choices of $a_j^r$ lead to a factorization structure. We characterize these in §2.7.

1.3. Products and decomposable elements

In general, a complete description of 1-dimensional spaces $\Gamma_j$ determining a Segre–Veronese factorization structure is a complex task. However, we leverage the concept of a product of factorization structures to generate extensive families of hands-on examples. Specifically, we show that products of two factorization structures are parametrized by the points in the image of the Segre embedding of these two structures. As it turns out in §2.7, iterated products of Veronese factorization structures completely characterize decomposable Segre–Veronese factorization structures. We finish this subsection by presenting an example of a Segre–Veronese factorization structure whose all defining tensors are indecomposable.

Definition 1.3.1. Let $\chi:\mathfrak{g}\to W_1^*\otimes\cdots\otimes W_n^*$ and $\varphi:\mathfrak{h}\to V_1^*\otimes\cdots\otimes V_m^*$ be two factorization structures and $T\subset\chi(\mathfrak{g})$ and $S\subset\varphi(\mathfrak{h})$ any two 1-dimensional subspaces. We define the product of φ and χ to be the $(n+m)$-dimensional factorization structure given by the canonical inclusion

(1.18)\begin{align} \varphi(\mathfrak{h}) \otimes T+ S \otimes \chi(\mathfrak{g}) \hookrightarrow V_1^*\otimes\cdots\otimes V_m^*\otimes W_1^*\otimes\cdots\otimes W_n^*. \end{align}

Examples of product include the 2-dimensional Segre factorization structure viewed as a product of two 1-dimensional factorization structures, Segre–Veronese factorization structure with k = 2 from example 1.2.3 viewed as a product of two Veronese factorization structures, and the product Segre–Veronese (1.16) from example 1.2.4. In fact, the latter is a product in multiple ways as we can see in the following

Example 1.3.2. Let $I:=\{1,\ldots,k_0\}\subset\{1,\ldots,k\}$, $1\leq k_0 \lt k$, and let I^c be the complement of I. We can rewrite the product Segre–Veronese factorization structure from example 1.2.4 as

(1.19)\begin{equation} \begin{aligned} &\Bigg(\sum_{j\in I} ins_j \bigg(S^{d_j}W_j^* \otimes \bigotimes_{\substack{r\in I\\r\neq j}} (a^r)^{\otimes d_r} \bigg)\Bigg)\otimes\bigotimes_{r\in I^c}(a^r)^{\otimes d_r}+\\ &+\bigotimes_{r\in I}(a^r)^{\otimes d_r}\otimes\Bigg( \sum_{j\in I^c} ins_j \bigg( S^{d_j}W_j^* \otimes \bigotimes_{\substack{r\in I^c\\r\neq j}} (a^r)^{\otimes d_r}\bigg)\Bigg)\\ &\quad\hookrightarrow \bigotimes_{j\in I} (W_j^*)^{\otimes d_j} \otimes \bigotimes_{j\in I^c} (W_j^*)^{\otimes d_j}, \end{aligned} \end{equation}

rendering it as the product of the (product Segre–Veronese) factorization structure

(1.20)\begin{equation} \sum_{j\in I} ins_j \bigg( S^{d_j}W_j^* \otimes \bigotimes_{\substack{r\in I\\r\neq j}} (a^r)^{\otimes d_r} \bigg) \hookrightarrow \bigotimes_{j\in I} (W_j^*)^{\otimes d_j} \end{equation}

and the (product Segre–Veronese) factorization structure

(1.21)\begin{equation} \sum_{j\in I^c} ins_j \bigg( S^{d_j}W_j^* \otimes \bigotimes_{\substack{r\in I^c\\r\neq j}} (a^r)^{\otimes d_r} \bigg) \hookrightarrow \bigotimes_{j\in I} (W_j^*)^{\otimes d_j} \end{equation}

with $T=\bigotimes_{r\in I^c} (a^r)^{\otimes d_r}$ and $S=\bigotimes_{r\in I} (a^r)^{\otimes d_r}$. Clearly, such a product exists for any non-trivial $I\subset\{1,\ldots,k\}$.

Now we illustrate how products can be used to construct new examples of factorization structures. We fix the Segre–Veronese factorization structure (1.15) corresponding to the partition $d_1+d_2$ from example 1.2.3 and the Veronese factorization structure $S^{d_3}W_3^* \hookrightarrow (W_3^*)^{\otimes d_3}$. To form a product of these two factorization structures, we choose 1-dimensional spaces $\Gamma \subset S^{d_3}W_3^*$ and Γ₃ lying in the image of (1.15), and with respect to these choices, we obtain the product

(1.22)\begin{align} S^{d_1}W_1^* \otimes \Gamma_1\otimes\Gamma + \Gamma_2 \otimes S^{d_2}W_2^* \otimes \Gamma + \Gamma_3 \otimes S^{d_3}W_3^* \hookrightarrow \bigotimes_{j=1}^3 (W_j^*)^{\otimes d_j}, \end{align}

which is a factorization structure of the dimension $d_1+d_2+d_3$ and belongs again to the class of a Segre–Veronese factorization structures. We could continue further and make a product of the factorization structure (1.22) with another Veronese factorization structure $S^{d_4}W_4^* \hookrightarrow (W_4^*)^{\otimes d_4}$, or form a product of two factorization structures of type (1.15) to obtain a Segre–Veronese factorization structure corresponding to a partition of length 4, i.e., k = 4. And so on.

One could speculate that factorization structures, or at least Segre factorization structures, can be built up via products from atomic pieces. Notably, defining tensors of a product of Segre–Veronese factorization structures always decompose across slots belonging to original factors: in (1.18), defining tensors are the ones of φ tensored-from-right with T and the ones of χ tensored-from-left with S. The following example demonstrates that the building blocks of (Segre–Veronese) factorization structures need not be simple.

Example 1.3.3. We conclude this section with an example of 3-dimensional Segre factorization structure whose all defining tensors are indecomposable. Observe that only in dimension 3, the annihilator $\mathfrak{h}^0\hookrightarrow V$ of a factorization structure $\mathfrak{h}\hookrightarrow V^*$ has the right dimension for being a factorization structure. The Veronese $S^3W^*\hookrightarrow (W^*)^{\otimes 3}$ has the annihilator

(1.23)\begin{align} W\otimes\bigwedge^2W + ins_2\left( W\otimes\bigwedge^2W \right) + \bigwedge^2W\otimes W \hookrightarrow W\otimes W\otimes W, \end{align}

a factorization structure with indecomposable defining tensors.

For the later use, we study particular elements in a product of factorization structures.

Lemma 1.3.4. Let $\varphi(\mathfrak{h})\otimes T+ S\otimes\chi(\mathfrak{g}) \hookrightarrow V_1^*\otimes\cdots\otimes V_m^*\otimes W_1^*\otimes\cdots\otimes W_n^*$ be a product of factorization structures. Then

(1.24)\begin{align} I \otimes K \subset \varphi(\mathfrak{h})\otimes T+ S\otimes\chi(\mathfrak{g}) \end{align}

for some 1-dimensional subspaces $ I \subset V_1^*\otimes\cdots\otimes V_m^*$ and $ K \subset W_1^*\otimes\cdots\otimes W_n^*$ if and only if

(1.25)\begin{align} \bigg[ I= S \text{and } K \subset \chi(\mathfrak{g}) \bigg] \ \ \text{or}\ \ \bigg[ K= T \text{and } I \subset \varphi(\mathfrak{h}) \bigg] \end{align}

Proof. The ‘if’ part is obvious. To prove the ‘only if’ part of the statement, let $s\in S, t\in T, \iota\in I, \kappa\in K$ be non-zero vectors. Since any element of the product factorization structure can be written as $ \tau_1\otimes t+ s \otimes\tau_2 $ for some $\tau_1\in\varphi(\mathfrak{h})$ and $\tau_2\in\chi(\mathfrak{g})$, we need to solve

(1.26)\begin{align} \tau_1\otimes t+ s\otimes\tau_2= \iota\otimes\kappa \end{align}

for τ ₁ and τ ₂. We suppose T ≠ K and S ≠ I as the complementary situation easily gives the claim. We proceed by assuming that τ ₁ and τ ₂ solve (1.26) and analyse τ ₁ in this equation. Note that if $\tau_1=0$, then the equation reduces to the case we excluded. Now we consider two cases; τ ₁ is either in the span of S and I, or it is not. In the former case, we can write $\tau_1=as+b\iota$ for some scalars $a,b$, which transforms (1.26) into

(1.27)\begin{align} s \otimes (at + \tau_2) = \iota \otimes (\kappa - bt), \end{align}

and is true if only if I = S and $K \subset \chi(\mathfrak{g})$, hence contradicting our assumptions. In the latter case $\langle \tau_1 \rangle, S$ and I are linearly independent directions. By completing $\tau_1, s$, and ι into a basis of $V_1^* \otimes \cdots \otimes V_m^*$ and contracting (1.26) with the dual vector of τ ₁, we find that t = 0, which is a contradiction. Thus, a solution exists if and only if (1.25) holds.

1.4. Motivation for definition of factorization structures rooted in discrete geometry

The definition of 2-dimensional factorization structures was sought in the search for compactifications of ambitoric geometries [Reference Apostolov, Calderbank and Gauduchon4, Reference Apostolov, Calderbank and Gauduchon5] by formalizing the workings of hyperplane sections of the 2-dimensional Segre embedding. To provide both a presentation and motivation for this definition from a perspective rooted in discrete geometry, we turn our attention to cyclic polytopes. These were introduced by Gale [Reference Gale19] and are now a standard part of discrete geometry and combinatorics [Reference Grünbaum, Klee, Perles and Shephard21, Reference Ziegler33], however, their presentation could benefit from more context. In the rest of this subsection, we merely outline the theory of cyclic polytopes from the viewpoint assumed later in this article. For a detailed account and further motivation for studying applications of factorization structures in discrete geometry, see [Reference Brandenburg and Púček13].

The momentum curve, $t \mapsto (t,t^2,\ldots,t^m)$, is the rational normal curve, i.e., the Veronese embedding

(1.28)\begin{align} \mathbb{P}(W) &\to \mathbb{P}(S^mW^*) \end{align}

\begin{align*} \ell &\mapsto \ell^0 \otimes \cdots \otimes \ell^0, \end{align*}

in a suitable affine chart, where W is a 2-dimensional vector space, $S^mW^*$ is the $(m+1)$-dimensional space of symmetric tensors on the dual of W, and $\ell^0 \subset W^*$ denotes the annihilator of the 1-dimensional space $\ell$. The annihilators were chosen merely for convenience and consistency with the literature. A choice of finitely many points on the momentum curve determines a cyclic polytope, as well as equally many points on the rational normal curve. As any m of them are linearly independent, say parametrized by $\ell_1,\ldots,\ell_m$, they determine a hyperplane H. Its annihilator can be read from the contraction

(1.29)\begin{align} \left\langle \ell_1 \otimes \cdots \otimes \ell_m, \ell^0 \otimes \cdots \otimes \ell^0 \right\rangle, \end{align}

which is zero, and thus well-defined, if and only if $\ell\in\{\ell_1,\ldots,\ell_m\}$, where $\ell^0\otimes \cdots \otimes \ell^0 \subset S^mW^*$ is viewed as an element of $(W^*)^{\otimes m}$. Indeed, denoting the canonical inclusion of $S^mW^*$ into $(W^*)^{\otimes m}$ by φ, here called the Veronese factorization structure, the annihilator of H is the 1-dimensional space $\varphi^t \ell_1 \otimes \cdots \otimes \ell_m$.

Expressing the contraction (1.29) in coordinates, i.e., using the affine chart on $S^mW^*$ in which the rational normal curve is the momentum curve and a suitable chart on its dual, we obtain a polynomial expression

(1.30)\begin{equation} \langle (t_1,-1) \otimes \cdots \otimes (t_m,-1), (1,t) \otimes \cdots \otimes (1,t)\rangle = \prod_{j=1}^m (t_j-t). \end{equation}

The proof of Gale’s evenness condition, determining whether H defines a facet of the cyclic polytope, follows directly from (1.30) and its geometric interpretation as the contraction of a normal vector of H with a point on the momentum curve. A detailed proof would require introducing notation, which we omit for brevity. This geometric approach complements standard treatments (e.g., [Reference Gale19, Reference Grünbaum, Klee, Perles and Shephard21, Reference Ziegler33]), which primarily rely on algebraic arguments involving the expanded form of $\prod (t_j - t)$. By collecting derivatives of (1.30), one recovers the Vandermonde identities [Reference Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman6].

A detailed explanation of a generalized Gale condition and the derivation of these identities can be found in §3. The above analysis not only sheds light on the geometric characteristics of cyclic polytope theory but also encourages further investigation.

Dually, the annihilator of $\ell^0 \otimes \cdots \otimes \ell^0 \subset S^mW^*$ contains the φ ^t-image of $\sum_{j=1}^{m} \Sigma_{j,\ell}$,

\begin{align*} \nonumber \Sigma_{j,\ell} := W ^{\otimes (j-1)} \otimes \ell \otimes W^{\otimes (m-j)}, \end{align*}

or, equally, the φ ^t-image of any $\Sigma_{j,\ell}$, $j=1,\ldots,m$, since φ ^t projects onto symmetric tensors, where $W^{\otimes 0}$ is interpreted as an empty product. The ambiguity in j occurs here because the Veronese factorization structure is the simplest and most symmetric structure, and its interpretation is carried out by factorization curves. To see if $\varphi^t \Sigma_{j,\ell}$ is a hyperplane, one computes the dimension of the intersection of $\Sigma_{j,\ell}$ with $\ker \varphi^t = (\varphi \, S^mW^*)^0$, or finds the dimension of its annihilator

(1.31)\begin{align} \left( \varphi^t \Sigma_{j,\ell} \right)^0= \varphi^{-1}\left( \varphi (S^mW^*) \cap ( \Sigma_{j,\ell} )^0 \right), \end{align}

both leading to the same condition

(1.32)\begin{align} \dim \left( \varphi (S^mW^*) \cap ( \Sigma_{j,\ell})^0 \right) = 1, \end{align}

which is fulfilled for any $\ell\in\mathbb{P}(W)$. In particular, facets of the (simple) polytope dual to the cyclic polytope lie on hyperplanes $\varphi^t \Sigma_{j,\ell}$, where $\ell$ parametrize directions determined by vertices of the cyclic polytope. Furthermore, theorem 3.1.6 shows that its faces lie on subspaces of the form $\varphi^t \left( \Sigma_{i_1,\ell_1} \cap \cdots \cap \Sigma_{i_r,\ell_r} \right)$ for some $r\leq m$.

To summarize, we found that since (1.32) holds, $\varphi^t \Sigma_{j,\ell}$ is a hyperplane as well as the annihilator of $\ell^0 \otimes \cdots \otimes \ell^0$. Because $\varphi^t (\Sigma_{1,\ell_1}) \cap \cdots \cap \varphi^t (\Sigma_{m,\ell_m}) = \varphi^t (\ell_1 \otimes \cdots \otimes \ell_m)$, which can be verified in this case directly, the annihilator of the hyperplane given by $\ell_j^0 \otimes \cdots \otimes \ell_j^0$, $j=1,\ldots,m$, is $\varphi^t \ell_1 \otimes \cdots \otimes \ell_m$. When particular affine charts are used, the Gale evenness condition follows and we obtain the framework of cyclic polytopes.

From this viewpoint, a generalization of the theory becomes apparent: a general inclusion satisfying an analogue of (1.32) and general affine charts can be used to extend the cyclic polytope framework. Remarkably, even alternative affine charts within the Veronese factorization structure yield many unexpected polytope classes beyond cyclic polytopes, as explored in [Reference Brandenburg and Púček13]. This insight provides a contextual explanation of cyclic polytopes and their theory.

Wishing to preserve the clarity of computations above and maximize their use, we arrived at the definition of a factorization structure: a linear inclusion φ of an $(m+1)$-dimensional vector space into the tensor product of m 2-dimensional vector spaces such that the obvious analogue of (1.32) holds for any j and generic $\ell$. The genericity-requirement means that $\varphi^t \Sigma_{j,\ell}$ is a hyperplane only for generic $\ell$. This definition and the presentation provided apply to vector spaces over complex numbers as well. In complex case, the momentum curve can be realified, resulting in the Carathéodory curve [Reference Carathéodory16], whose polytopes are known to be cyclic [Reference Grünbaum, Klee, Perles and Shephard21, Reference Ziegler33].

Note that when $\ell_1, \ldots, \ell_m\in\mathbb{P}(W)$ vary while remaining pairwise distinct, the corresponding m 1-parametric families of hyperplanes $\varphi^t\Sigma_{j,\ell_j}$, $j=1\ldots,m$, provide coordinates on a Zariski-open subset of $\mathbb{P}(S^mW)$, since their respective annihilators $(\ell_j^0)^{\otimes m}$, $j=1,\ldots,m$, remain linearly independent. These coordinates are called separable. Thus, a point in this open subset of $\mathbb{P}(S^mW)$ is determined by a point in the m-product $\mathbb{P}(W) \times \cdots \times \mathbb{P}(W)$, showing that a projective space factors into a product of projective lines, at least locally. Hence the name factorization structures.

1.5. Factorization structures in differential geometry

Most of this section is unpublished, with references provided where applicable. It gives a brief look at factorization structures in differential geometry, thereby offering another reason to study them, and outlines applications of results from this article in studying Kähler metrics.

As mentioned above, 2-dimensional factorization structures were originally explored in ambitoric compactifications [Reference Apostolov, Calderbank and Gauduchon4, Reference Apostolov, Calderbank and Gauduchon5], where were used to achieve a classification of extremal Kähler structures on compact toric 4-orbifolds with the second Betti number two. As a result, many geometries and their classifications were unified under the framework of ambitoric geometry (see introduction in [Reference Apostolov, Calderbank and Gauduchon5]). Importantly, the shape of ambitoric structures shows that factorization structures are not merely auxiliary computational tools but play an intrinsic role, they determine the Kähler structure.

Appendix C of [Reference Apostolov, Calderbank and Gauduchon4] shows that regular ambitoric geometries can be viewed as quotients of a 5-dimensional manifold of Sasaki type by Sasaki–Reeb vector fields. Building on this idea, Apostolov and Calderbank [Reference Apostolov and Calderbank2] extend the approach by studying weighted extremality of quotients of Sasaki-type manifolds in general dimension. Additionally, it finds two explicit families of separable geometries, which describe in terms of CR twists: twists of orthotoric geometry [Reference Apostolov, Calderbank and Gauduchon3, Reference Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman6, Reference Apostolov, Calderbank and Gauduchon7, Reference Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman9, Reference Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman10] and twists of a Kähler product of toric Riemann surfaces. In real dimension 4, as [Reference Apostolov, Calderbank, Gauduchon and Legendre8] and [Reference Apostolov and Calderbank2] show, these two families recover all ambitoric geometries.

The success of factorization structures in classifying extremal 4-orbifolds and the elegance of identifying these as natural quotients of Sasaki-type geometries motivated the author’s thesis [Reference Púček28], where the two aforementioned families of separable geometries were recognized to be associated with Veronese and product Segre factorization structures.

More generally, an m-dimensional factorization structure $\varphi:\mathfrak{h}\to V_1^* \otimes \cdots \otimes V_m^*$, $\beta \in \mathfrak{h}$, and m functions $A_1,\ldots,A_m$ of one variable determine the toric separable Kähler geometry

\begin{align*} g_\beta&= -\sum_{j=1}^m\left( \frac{\langle\partial_{x_j}\mu_\beta,\psi_j([1:x_j])\rangle} {A_j(x_j)} dx_j^2 + \frac{A_j(x_j)} {\langle\partial_{x_j}\mu_\beta,\psi_j([1:x_j])\rangle} \langle\partial_{x_j}\mu_\beta,dt\rangle^2 \right) \end{align*}

\begin{align*} \omega_\beta&= \sum_{j=1}^mdx_j\wedge\langle\partial_{x_j}\mu_\beta,dt\rangle \end{align*}

(1.33)\begin{align} J_\beta dx_j&= - \frac{A_j(x_j)}{\langle\partial_{x_j}\mu_\beta,\psi_j([1:x_j])\rangle}\langle\partial_{x_j}\mu_\beta,dt\rangle \qquad J_\beta dt=\sum_{j=1}^m\frac{\psi_j([1:x_j])\text{mod }\beta}{A_j(x_j)}dx_j, \end{align}

where dt is a 1-form valued in $\mathfrak{h} / \langle \beta \rangle$, ψ_j, $j=1,\ldots,m$, are factorization curves (§2.1) in appropriate affine charts, and $\mu_\beta = \varphi^t x / \langle x, \varphi \beta \rangle$ with $x = (1,x_1) \otimes \cdots \otimes (1,x_m)$ is the momentum map valued in the affine chart given by β. In particular, each $\partial_{x_j} \mu_\beta$ lies in the annihilator β ⁰, ensuring that the pairing $\langle \partial_{x_j} \mu_\beta, dt \rangle$ is well-defined.

While it is now possible to explicitly write down the Kähler structure for the standard Segre–Veronese factorization structure (see definition 1.2.1), doing so would require introducing additional notation related to grouped slots, which we omit for brevity. A detailed exposition will appear in future work. However, we note that the Kähler structure (1.33) is obtained as the quotient of a toric separable CR geometry whose acting torus has the Lie algebra $\mathfrak{h}$, and which is equipped with special coordinates $x_1,\ldots,x_m$, called separable (see [Reference Apostolov and Calderbank2]), in which the CR structure depends on functions of one variable. Specifically, it resembles $J_\beta dt$ from (1.33) which depends on functions $\psi_j/A_j \text{mod } \beta$, $j=1,\ldots,m,$ of one variable.

The advantages of separable Kähler and CR geometries are three-fold: a unified framework for many examples, each facet of the rational Delzant polytope or polyhedron of a separable geometry is described by $x_j = \text{\textit{const.}}$ for some j, and, unknowns in partial differential equations involving these geometries depend on functions of one variable.

Separable geometries associated with factorization structures provide a framework for vast number of toric CR and Kähler geometries which are amenable to uniform computations. Examples appearing in the literature are the aforementioned ambitoric geometries, twists of a Kähler product of toric Riemann surfaces, and twists of orthotoric geometries, which together correspond to two simplest factorization structures: Veronese and product Segre. The overflow of new separable geometries arises from the vast number of factorization structures. A classification of factorization structures would not only describe all local separable geometries in a given dimension n, but, as in ambitoric case, could also facilitate the classification of extremal structures on certain n-orbifolds.

The study of global behaviour of separable geometries includes compactifications, which are related to rational Delzant polytopes, and general boundary behaviour associated with polyhedra. In the compact case, the image of the momentum map of a toric separable Kähler geometry is in particular a compatible polytope (definition 3.1.1) with at most 2m facets, where m is the complex dimension of the geometry. Any such carries separable coordinates x_j, $j=1,\ldots,m$, in which the facets are described by $x_j = const.$ (see the last paragraph of §1.4 for separable coordinates). This enables the derivation of simple necessary and sufficient conditions for compactification, similar to those in [Reference Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman6]. A key step in compactifying is fixing an underlying rational Delzant polytope compatible with a factorization structure, constructed using Vandermonde identities (§3.3). As part of this broader framework, understanding the effects of newly introduced operations on factorization structures, namely the product (§1.3) and quotient (§2.5), on separable geometries remain to be explored.

Separable geometries provide a favourable setting where geometric PDEs are likely to be solved explicitly. In particular, this applies to the problem of finding Calabi’s extremal metrics, also famous for its connection with K-stability, which is governed by the extremality equation, a PDE seeking metrics whose scalar curvature is a Killing potential. While explicit solutions are rare and often obtained through ad hoc methods, separable geometries offer a systematic framework for recovering known solutions and discovering new ones.

The Segre–Veronese factorization structure underpins this framework. Solutions of the extremality equation for associated separable geometries are rational functions depending on tensors $\Gamma_j$, $j=1,\ldots,k$, determining the factorization structure and on finitely many parameters, whose number relates to degrees of involved factorization curves (see §2.3). A general strategy for solving the PDE is to verify, using generalized Vandermonde identities (see remark 3.3.3), which solutions of compatibility conditions satisfy the PDE. The shapes and decomposability of tensors $\Gamma_j$, $j=1,\ldots,k$ (see lemma 1.3.4 and proposition 2.6.3) are crucial in obtaining useful compatibility conditions, which necessitates the classification of Segre–Veronese factorization structures. This article achieves a partial classification by characterizing decomposable Segre–Veronese factorization structures (§2.7). Already for one of the simplest of such structures, the product Segre–Veronese factorization structure associated with a partition, known extremal metrics [Reference Apostolov and Calderbank2] are recovered for the two trivial partitions, and new solutions are obtained for any non-trivial partition. Furthermore, analysis for decomposable Segre–Veronese factorization structures corresponding to partitions $m=d_1+\cdots+d_k$ for small k indicates that the extremality equation for the associated geometries can be solved uniformly, as opposed to case-by-case approach.

2.1. Factorization curves

The defining condition of factorization structures (1.4) invites us to consider generically defined curves

(2.1)\begin{align} \mathbb{P}(V_j) &\dashrightarrow \mathbb{P}(\mathfrak{h})\nonumber\\ \ell &\mapsto \varphi^{-1} \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \end{align}

for $j=1,\ldots,m$.

For example, in the 2-dimensional Segre factorization structure $V_1^*\otimes \Gamma_1 + \Gamma_2 \otimes V_2^* \hookrightarrow V_1^* \otimes V_2^*$, we have two curves

(2.2)\begin{align} \mathbb{P}(V_1) \backslash \{\Gamma_2 \} &\to \mathbb{P}(V_1^*\otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \nonumber \\ \ell &\mapsto (V_1^*\otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \cap \ell^0 \otimes V_2^* = \ell^0 \otimes \Gamma_1 \end{align}

and

(2.3)\begin{align} \mathbb{P}(V_2) \backslash \{\Gamma_1 \} &\to \mathbb{P}(V_1^*\otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \nonumber \\ \ell &\mapsto (V_1^*\otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \cap V_1^* \otimes \ell^0 = \Gamma_2 \otimes \ell^0, \end{align}

both being (generically defined) lines in $\mathbb{P}^2$. Note that the points which are excluded from domains of these lines are exactly those where the formula (2.1) does not determine a point in a projective space. Similarly, for a general Segre factorization structure, all of its curves in the above sense are generically defined lines.

In the case of Veronese factorization structure $S^mW^* \hookrightarrow (W^*)^{\otimes m}$, its first curve reads

(2.4)\begin{align} \mathbb{P}(W) &\to \mathbb{P}(S^mW^*) \nonumber \\ \ell &\mapsto S^mW^* \cap \ell^0 \otimes (W^*)^{\otimes (m-1)} = (\ell^0)^{\otimes m}. \end{align}

This curve is defined globally, i.e., for all $\ell \in \mathbb{P}(W)$, and is known as the rational normal curve. Because the domain of any other curve is again $\mathbb{P}(W)$, and $S^mW^* \cap \Sigma_{j,\ell}^0 = (\ell^0)^{\otimes m}$ for any $\ell \in \mathbb{P}(W)$ and any $j=1,\ldots,m$, all curves coincide.

For the standard Segre–Veronese factorization structure (1.11), these curves were already found in (1.14), and, similarly to the example above, coincide in grouped slots. More concretely, for each $i=1,\ldots,k$, we have d_i identical curves

(2.5)\begin{align} \mathbb{P}(W_i) &\to \mathbb{P}\left( \sum_{j=1}^{k} ins_j \left( S^{d_j}W_j^*\otimes \Gamma_j \right) \right) \nonumber \\ \ell &\mapsto ins_i \left( (\ell^0)^{\otimes d_i} \otimes \Gamma_i \right), \end{align}

whose locus of indeterminacy is at points $\ell\in\mathbb{P}(V_j)$ for which $\dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \gt 1$. The latter happens if a defining tensor $\Gamma_j$, j ≠ i, decomposes in grouped i-slots, similarly as in 2-dimensional Segre factorization structure above.

Now we establish a way of extending these generically defined curves into projective curves. The first step is

Proposition 2.1.1. Let $\varphi: \mathfrak{h} \to V^*$ be a real/complex factorization structure of dimension m. The generically defined curve (2.1) is a regular map on an open and non-empty subset $W \subset \mathbb{P}(V_j)$ of degree at most m, i.e., it is given by homogeneous polynomials of the same degree (at most m) in homogeneous coordinates on $\mathbb{P}(V_j)$ and $\mathbb{P}(\mathfrak{h})$. More concretely, W is an open subset of the open set

\begin{equation*}U= \{\ell\in\mathbb{P}(V_j)\,|\, \dim\left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0 \right) =1\}.\end{equation*}

Proof. We describe $\varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0$ as a solution of a linear system of $2^m-1$ equations, which depend homogeneously on $\ell$, with 2^m variables. Then, we apply Cramer’s rule to show the claim for $\ell \mapsto \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0$, and thus for (2.1).

Fix a basis of $V_j^*$, $j=1,\ldots,m$, and let $c^{a_1\cdots a_m}: V^*=V_1^*\otimes\cdots\otimes V_m^* \to \mathbb{F}$, $a_j\in\{1,2\}$, be the corresponding standard coordinates, where $\mathbb{F}$ denotes the field $\mathbb{R}$ or $\mathbb{C}$ depending on the factorization structure being real or complex, respectively. For $\ell\in U$, the subspace $\Sigma_{j,\ell}^0$ in $V^*$ is then described by $2^{m-1}$ independent linear equations

(2.6)\begin{align} xc^{a_1\cdots a_{j-1}1a_{j+1}\cdots a_m}+ yc^{a_1\cdots a_{j-1}2a_{j+1}\cdots a_m} =0, a_i\in\{1,2\} \text{ for } i\neq j, \end{align}

where $\ell^0=[-y:x]$. Note, these can be viewed as equations of homogeneous polynomials of degree one in x and y with coefficients $c^{\cdots}$’s. The $(m+1)$-dimensional subspace $\varphi(\mathfrak{h})$ in $V^*$ can be described via $2^m-(m+1)$ independent linear equations, call that system (E), which do not depend on $\ell$. Finally, the subspace $\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0$, which is one dimensional for a fixed generic $\ell$, is the solution to the system of $2^{m-1}+2^m-(m+1)$ linear equations, (2.6) and (E). Clearly, this system has only $2^m-1$ independent equations, and these can be chosen as the system (E) together with another m independent linear equations from (2.6). The latter stay independent on an open subset $V\subset\mathbb{P}(V_k)$ containing $\ell$. Thus, for $\ell\in W := U\cap V$, knowing the intersection $\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0$ is equivalent to a system of $2^m-1$ independent linear equations, m of which are homogeneous of degree one in $\ell$ and the others do not depend on $\ell$. Using Cramer’s rule to solve this system (see [Reference Gorodentsev20]) shows that $\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0$ depends on $\ell$ in a homogeneous way and the degree of homogeneity is at most m which, for example, is attained in the case when $\varphi(\mathfrak{h})=S^mW^*$.

Combining proposition 2.1.1 and lemma 2.1.2 allows us to define factorization curves as extensions of (2.1).

We continue with examples of generically defined curves from above.

Example 2.1.4. Since extensions of the generically defined curves (2.1) are unique, we conclude from (2.2) and (2.3) that the 2-dimensional Segre factorization structure has two distinct factorization curves, being lines

(2.7)\begin{align} \psi_1: \mathbb{P}(V_1) &\to \mathbb{P}(V_1^* \otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \nonumber \\ \ell &\mapsto \ell^0 \otimes \Gamma_1 \end{align}

and

(2.8)\begin{align} \psi_2: \mathbb{P}(V_2) &\to \mathbb{P}(V_1^* \otimes \Gamma_1 + \Gamma_2 \otimes V_2^*) \nonumber \\ \ell &\mapsto \Gamma_2 \otimes \ell^0, \end{align}

intersecting at one point $\Gamma_2 \otimes \Gamma_1$. Note also that respective linear spans of images of (2.7) and (2.8) are $V_1^* \otimes \Gamma_1$ and $\Gamma_2 \otimes V_2^*$ (see figure 2). A general m-dimensional Segre factorization structure has m distinct factorization curves, all being lines. However, their intersections can be arbitrarily complicated. For example, for 3-dimensional Segre factorization structure from example 1.3.3, there is no intersection between any two factorization curves/lines. On the other extreme, in the product Segre factorization structure from example 1.2.4 corresponding to the partition $m = 1 + \cdots + 1$, all factorization lines intersect at the unique point $\otimes_{r=1}^m a^r$. One can form iterative products of 1-dimensional factorization structures to obtain an m-dimensional Segre factorization structure with decomposable defining tensors and with prescribed intersections of factorization lines.

As already found in (2.4), all factorization curves in a Veronese factorization structure coincide, $\psi_1 = \cdots = \psi_m$, being the rational normal curve of degree m. Such a curve has two properties we frequently use: its linear span is exactly $S^mW^*$, and any m + 1 points on the curve are linearly independent [Reference Harris23].

Example 2.1.5. The Segre–Veronese factorization structure corresponding to the partition $m=d_1+d_2$ from example 1.2.3, abbreviated here as $\varphi:\mathfrak{h} \to V^*$, has two distinct factorization curves

(2.9)\begin{align} \mathbb{P}(W_1) &\to \mathbb{P}\left( S^{d_1}W_1^* \otimes \Gamma_1 + \Gamma_2 \otimes S^{d_2}W_2^* \right) \nonumber \\ \ell &\mapsto (\ell^0)^{\otimes d_1} \otimes \Gamma_1 = \varphi(\mathfrak{h}) \cap \Sigma_{1,\ell}^0 = \cdots = \varphi(\mathfrak{h}) \cap \Sigma_{d_1,\ell}^0 \end{align}

and

(2.10)\begin{align} \mathbb{P}(W_2) &\to \mathbb{P}\left( S^{d_1}W_1^* \otimes \Gamma_1 + \Gamma_2 \otimes S^{d_2}W_2^* \right) \nonumber \\ \ell &\mapsto \Gamma_2 \otimes (\ell^0)^{\otimes d_2} = \varphi(\mathfrak{h}) \cap \Sigma_{d_1+1,\ell}^0 = \cdots = \varphi(\mathfrak{h}) \cap \Sigma_{m,\ell}^0, \end{align}

which, respectively, have degrees d ₁ and d ₂. Their respective linear spans are $S^{d_1}W_1^* \otimes \Gamma_2$ and $\Gamma_1 \otimes S^{d_2}W_2^*$, and both are rational normal curves within their linear span. They intersect if and only if both Γ₁ and Γ₂ are decomposable, in which case the intersection is the unique point $\Gamma_2 \otimes \Gamma_1$.

Example 2.1.6. To illustrate how products of factorization structures influence intersections of factorization curves, we discuss the Segre–Veronese factorization structure (1.22), abbreviated here as $\varphi:\mathfrak{h} \to V^*$, whose distinct curves are

(2.11)\begin{align} \mathcal{C}_1: \mathbb{P}(W_1) &\to \mathbb{P}(\mathfrak{h}) \nonumber \\ \ell &\mapsto (\ell^0)^{\otimes d_1} \otimes \Gamma_1 \otimes \Gamma, \end{align}

and

(2.12)\begin{align} \mathcal{C}_2: \mathbb{P}(W_2) &\to \mathbb{P}(\mathfrak{h}) \nonumber \\ \ell &\mapsto \Gamma_2 \otimes (\ell^0)^{\otimes d_2} \otimes \Gamma, \end{align}

and

(2.13)\begin{align} \mathcal{C}_3: \mathbb{P}(W_3) &\to \mathbb{P}(\mathfrak{h}) \nonumber \\ \ell &\mapsto \Gamma_3 \otimes (\ell^0)^{\otimes d_3}, \end{align}

with respective degrees $d_1, d_2$, and d ₃, and respective linear spans $S^{d_1}W_1^* \otimes \Gamma_1 \otimes \Gamma$, $\Gamma_2 \otimes S^{d_2}W_2^* \otimes \Gamma$, and $\Gamma_3 \otimes S^{d_3}W_3^*$. The following analysis of pairwise intersections of $\mathcal{C}_1, \mathcal{C}_2$, and $\mathcal{C}_3$ clarifies how to choose defining tensors $\Gamma_1, \Gamma_2$, and Γ to obtain prescribed intersection properties of the curves. Similarly to example 2.1.5, the curves $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect if and only if both Γ₁ and Γ₂ are decomposable, in which case the intersection is the unique point $\Gamma_2 \otimes \Gamma_1 \otimes \Gamma$. Curves $\mathcal{C}_1$ and $\mathcal{C}_3$ intersect if and only if Γ is decomposable and $\Gamma_3 = A \otimes \Gamma_1$ for some decomposable 1-dimensional space $A \subset S^{d_1}W_1^*$, in which case the intersection is the unique point $A \otimes \Gamma_1 \otimes \Gamma$. Finally, curves $\mathcal{C}_2$ and $\mathcal{C}_3$ intersect if and only if Γ is decomposable and $\Gamma_3 = \Gamma_2 \otimes B$ for some decomposable 1-dimensional space $B \subset S^{d_2}W_2^*$, in which case the intersection is the unique point $\Gamma_2 \otimes B \otimes \Gamma$.

Example 2.1.7. The standard Segre–Veronese factorization structure (1.11), abbreviated here as $\varphi: \mathfrak{h} \to V^*$, has k distinct factorization curves,

(2.14)\begin{align} \mathcal{C}_i: \mathbb{P}(W_i) &\to \mathbb{P}(\mathfrak{h}) \nonumber \\ \ell &\mapsto ins_i \left( (\ell^0)^{\otimes d_i} \otimes \Gamma_i \right), \quad i=1,\ldots,k, \end{align}

which relate to factorization curves $\psi_1,\ldots,\psi_m$ by $\mathcal{C}_i = \psi_{d_1+\cdots+d_{i-1}+1} = \cdots = \psi_{d_1+\cdots+d_{i-1}+d_i}$, $i=1,\ldots,k$, where d ₀ is defined to be zero. The linear span of $\mathcal{C}_i$ is $ins_i \left( S^{d_i}W_i^* \otimes \Gamma_i \right)$, $i=1,\ldots,k$, showing that the standard Segre–Veronese factorization structure is the sum of linear spans of its factorization curves. Finally, the degree of $\mathcal{C}_i$ is d_i, $i=1,\ldots,k$.

The following lemma plays an essential role in the study of factorization curves.

Its first application shows that complex factorization curves are injective (see corollary 2.1.10).

2.2. Complexification

Let $\varphi^{\mathbb{C}}:\mathfrak{h}\otimes\mathbb{C}\to V^*\otimes\mathbb{C} = V_1^* \otimes \mathbb{C} \otimes \cdots \otimes V_m^* \otimes \mathbb{C}$ be the complexification of a real factorization structure $\varphi:\mathfrak{h}\to V^*$, and denote

\begin{equation*} (V_1^*\otimes_{\mathbb{R}}\mathbb{C}) \otimes_{\mathbb{C}} \cdots \otimes_{\mathbb{C}} (V_{j-1}^*\otimes_{\mathbb{R}}\mathbb{C})\otimes_{\mathbb{C}} L \otimes_{\mathbb{C}}(V_{j+1}^*\otimes_{\mathbb{R}}\mathbb{C})\otimes_{\mathbb{C}}\cdots\otimes_{\mathbb{C}}(V_m^*\otimes_{\mathbb{R}}\mathbb{C}) \end{equation*}

by $\mathbb{C}\Sigma_{j,L}^0$ for any complex 1-dimensional subspace $L\subset V_j^*\otimes\mathbb{C}$. Such a complexification is called a complexified factorization structure.

Before we prove it, we remark on a general property which will be used multiple times and also in the proof.

Remark 2.2.2. In general, for a real/complex factorization structure, the set

(2.15)\begin{align} U_d:=\{\ell\in\mathbb{P}(V_j):\dim\left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0 \right)\geq d\} \end{align}

is the preimage of

(2.16)\begin{align} \mathcal{U}_d=\{\Lambda\in Gr(2^{m-1},V^*)\,|\, \dim\left( \varphi(\mathfrak{h})\cap\Lambda \right)\geq d\} \end{align}

under the regular map $\mathbb{P}(V_j)\to Gr(2^{m-1},V^*)$ defined by $\ell\mapsto\Sigma_{j,\ell}^0$, and hence is closed since (2.16) is a (closed) Schubert variety (see [Reference Eisenbud and Harris18, Reference Harris23, Reference Púček28]).

Proof. On an open non-empty subset of $\mathbb{P}(V_j)$, we have

(2.17)\begin{align} 1= \dim \left( \varphi \circ \psi_j(\ell) \otimes \mathbb{C} \right)&= \dim \left( (\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0)\otimes\mathbb{C} \right)= \dim \left( \varphi(\mathfrak{h})\otimes\mathbb{C}\cap\mathbb{C}\Sigma_{j,\ell\otimes\mathbb{C}}^0 \right)\nonumber\\ &= \dim \left( \varphi^{\mathbb{C}}(\mathfrak{h}\otimes\mathbb{C})\cap\mathbb{C}\Sigma_{j,\ell\otimes\mathbb{C}}^0 \right). \end{align}

If we define

(2.18)\begin{align} Q_d= \{L\in\mathbb{P}(V_j\otimes\mathbb{C})\,|\, \dim\left( \varphi^{\mathbb{C}}(\mathfrak{h}\otimes\mathbb{C})\cap\mathbb{C}\Sigma_{j,L}^0 \right) \geq d\}, \quad d=1,2, \end{align}

then (2.17) shows that for φ a real factorization structure, Q ₁ contains $\infty$-many points. Since Q ₁ is closed, we have $Q_1 = \mathbb{P}(V_j \otimes \mathbb{C})$. Furthermore,

(2.19)\begin{align} Q_1 \backslash Q_2 = \{L\in\mathbb{P}(V_j\otimes\mathbb{C})\,|\, \dim\left( \varphi^{\mathbb{C}}(\mathfrak{h}\otimes\mathbb{C})\cap\mathbb{C}\Sigma_{j,L}^0 \right) = 1\} \end{align}

is non-empty by (2.17), and open since Q ₂ is closed. Thus, $\varphi^{\mathbb{C}}$ is a complex factorization structure.

On the other hand, for $\varphi^{\mathbb{C}}$ a complex factorization structure, $Q_1 \backslash Q_2$ is open and non-empty, and thus intersects $\{\ell\otimes\mathbb{C}\in\mathbb{P}(V_j\otimes\mathbb{C})\,|\,\ell\in\mathbb{P}(V_j)\}$ in $\infty$-many points. Equalities (2.17) at these intersection points show that the closed set

(2.20)\begin{align} \{\ell\in\mathbb{P}(V_j)\,|\, \dim \left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell_k}^0 \right) \geq 1\} \end{align}

is infinite and thus is the whole $\mathbb{P}(V_j)$. They also show that the open subset where the dimension is 1 is non-empty, which shows that φ is a real factorization structure.

We remark on complexified factorization curves. Unravelling the definition of complexification shows that complexifying the curve $\psi_j: \mathbb{P}(V_j) \to \mathbb{P}(\mathfrak{h})$,

\begin{equation*}[x^1:x^2] \mapsto \psi_j([x^1:x^2])=\left[ f_1([x^1:x^2]):\cdots:f_{m+1}([x^1:x^2]) \right],\end{equation*}

means to regard its defining polynomials f_r as complex polynomials, i.e., the complexified curve $\mathcal{C}: \mathbb{P}(V_j \otimes \mathbb{C}) \to \mathbb{P}(\mathfrak{h} \otimes \mathbb{C})$ is

\begin{equation*}[z^1:z^2] \mapsto \left[ f_1([z^1:z^2]):\cdots:f_{m+1}([z^1:z^2]) \right].\end{equation*}

Using lemma 2.1.8 we conclude that $\mathcal{C}$ and the factorization curve $\psi_j^{\mathbb{C}}$, given by

\begin{equation*}\varphi^{\mathbb{C}} \circ \psi_j^{\mathbb{C}}(L) = \varphi(\mathfrak{h})\otimes\mathbb{C}\cap\mathbb{C}\Sigma_{j,L}^0,\end{equation*}

coincide, since by (2.17) they agree at $\infty$-many points.

2.3. Degree

Most claims of this subsection hold for projective curves in general, but we stay focused on factorization curves as defined in definition 2.1.3. The tautological section $\tau: \mathbb{P}(\mathfrak{h})\to \mathcal{O}_\mathfrak{h}(1) \otimes \mathfrak{h}$ assigns to each class $[z]\in\mathbb{P}(\mathfrak{h})$ the canonical inclusion of the corresponding 1-dimensional vector space $\langle z \rangle$ into $\mathfrak{h}$ viewed as an element of $\langle z \rangle^*\otimes \mathfrak{h}$, where $\mathcal{O}_\mathfrak{h}(1)$ is the the dual of the tautological line bundle. By pulling τ back via a factorization curve ψ_j, we can view the curve as a section of $\mathcal{O}_{V_j}(e_j)\otimes\mathfrak{h}$,

where $e_j\in\mathbb{Z}$ is determined via the isomorphism $(\psi_j)^*\mathcal{O}_{\mathfrak{h}}(1) \cong \mathcal{O}_{V_j}(e_j)$, using the classification of line bundles over projective spaces. On the other hand, choosing a basis for $\mathfrak{h}$ allows us to view the section $\psi_j^*\tau$ as $\dim \mathfrak{h}$ global sections of $\mathcal{O}_{V_j}(e_j)$. Such sections are homogeneous polynomials of degree e_j which together recover ψ_j.

One can consult examples 2.1.4-2.1.7 for examples of degrees. In these examples, we used a notion of degree intuitively, which, as we now see, agrees with the one from definition 2.3.1.

2.4. Decomposability

We illustrate this definition on several examples. Clearly, factorization curves of degree 1, i.e., factorization lines, are decomposable. Therefore, every factorization curve of an m-dimensional Segre factorization structure is decomposable (see example 2.1.4).

In (2.4) and example 2.1.4, we learnt that all m factorization curves of the Veronese factorization structure $S^mW^* \hookrightarrow (W^*)^{\otimes m}$ coincide, and that its degree is m. Therefore, each factorization curve $\psi_1,\ldots,\psi_m$ is decomposable, being $\psi_1(\ell) = \cdots = \psi_m(\ell) = (\ell^0)^{\otimes m}$. Note that the values of this curve are genuine decomposable tensors, which is precisely the motivation behind the definition of a decomposable factorization curve.

More generally, one can observe directly in example 2.1.7 that factorization curves of a standard Segre–Veronese factorization structure are decomposable. Note that in this case, decomposability means that the variable part of such a curve, say $\ell \mapsto ins_i \left( (\ell^0)^{\otimes d_i} \otimes \Gamma_i \right)$, is a decomposable tensors, although $\Gamma_i$ does not have to be.

To characterize decomposability via decomposable tensors as above, we make use of

The following statement confirms our intuition behind decomposability of a curve from the above examples.

Remark 2.4.4. If S consists of the first r ₀ indices, $1\leq r_0\leq m$, then we can write

(2.22)\begin{align} \varphi\circ\psi_j(\ell)= \left( \bigotimes_{r=1}^{r_0} g_r\ell^0 \right) \otimes \Gamma_j, \end{align}

whose linear span is $((\otimes_{r=1}^{r_0} g_r) \cdot S^{r_0} V_j^*) \otimes \Gamma_j$, where ⋅ represent the action of the operator $\otimes_{r=1}^{r_0} g_r$. The general case differs from (2.22) by permutation of slots.

Decomposability is preserved under complexification.

Proof. Let ψ_i and ψ_j be equivalent factorization curves in a real factorization structure. Using equalities in (2.17), we find that for any $\ell\in\mathbb{P}(V_i)$, there exists $\ell'\in\mathbb{P}(V_j)$ such that

(2.23)\begin{align} \psi_i^{\mathbb{C}}(\ell \otimes \mathbb{C}) = \psi_i(\ell) \otimes \mathbb{C} = \psi_j(\ell') \otimes \mathbb{C} = \psi_j^{\mathbb{C}}(\ell' \otimes \mathbb{C}). \end{align}

Since factorization curves are injective, $\psi_i^{\mathbb{C}}$ and $\psi_j^{\mathbb{C}}$ coincide in $\infty$-many points, and hence lemma 2.1.8 implies they are equivalent. Furthermore, since degree is preserved in a complexification (see remark 2.3.2), we conclude that the complexification of a decomposable curve is a decomposable curve.

Suppose now $\psi_j^{\mathbb{C}}$ is decomposable and of degree d. Using theorem 2.4.3, we find

(2.24)\begin{equation} \varphi\circ\psi_j(\ell)\otimes\mathbb{C}= \varphi^{\mathbb{C}}\circ\psi_j^{\mathbb{C}}(\ell\otimes\mathbb{C})= g_1\ell^0 \otimes \cdots \otimes g_d\ell^0 \otimes \langle t \rangle \otimes \mathbb{C}, \end{equation}

up to a permutation of slots as in remark 2.4.4. The expression (2.24) clearly shows that ψ_j is equivalent with d curves, and hence decomposable.

2.5. Quotient factorization structure

To arrive at the main results of this section, we need to prove that a naturally defined quotient of a factorization structure is itself a factorization structure. Doing so requires working with seemingly weaker structures called weak factorization structures. These are employed because proving that their quotients are weak factorization structures is a more manageable task. Subsequently, through an inductive argument, it is shown that weak factorization structures are, indeed, genuine factorization structures, thereby establishing the well-behaved nature of their quotients.

Let $\varphi(\mathfrak{h})\subset V^*$ be a factorization structure of dimension m, and v a non-zero vector on a generic line $\lambda\in\mathbb{P}(V_i)$ such that $\dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{i,\lambda}^0 \right) = 1$. The inclusion of short exact sequences

defines the inclusion $\varphi_{i,v}:\mathfrak{h}_{i,\lambda}\to\hat{V}_i^*$ of the m-dimensional (quotient) vector space $\mathfrak{h}_{i,\lambda}$ into the tensor product of m − 1 2-dimensional vector spaces, which will be shown to be a factorization structure; the quotient of $\varphi:\mathfrak{h}\to V^*$ with respect to the choice $i\in\{1,\ldots,m\}$ and $\lambda\in\mathbb{P}(V_i)$.

Clearly, the inclusion $\varphi_{i,v}:\mathfrak{h}_{i,\lambda}\to\hat{V}_i^*$ is a factorization structure if and only if the intersections

(2.26)\begin{equation} \begin{aligned} \varphi_{i,v}(\mathfrak{h}_{i,v}) \cap V_1^* \otimes \cdots \otimes V_{j-1}^* \otimes \ell^0 \otimes V_{j+1}^* \otimes \cdots \otimes V_{i-1}^* \otimes V_{i+1}^* \otimes \cdots \otimes V_m^* = \\ \rho_{i,v} \left(\varphi(\mathfrak{h})\right) \cap \rho_{i,v} \Sigma_{j,\ell}^0 \end{aligned} \end{equation}

are 1-dimensional for every $j\in\{1,\ldots,m\}\backslash\{i\}$ and generic $\ell\in\mathbb{P}(V_j)$. As stated, it is difficult to prove. To make it tractable we use the aforementioned new object, weak factorization structure, defined by requiring the intersection in (1.4) to be at least 1-dimensional instead of being exactly 1-dimensional.

Definition 2.5.2. A linear inclusion $\varphi:\mathfrak{h}\to V^*$ of an $(m+1)$-dimensional vector space $\mathfrak{h}$ is called a weak factorization structure of dimension m if for every $j\in\{1,\ldots,m\}$ and generic $\ell\in\mathbb{P}(V_j)$

(2.27)\begin{align} \dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \geq 1 \end{align}

holds. Isomorphisms are defined in the same way as for factorization structures.

As the following claim proves, a weak factorization structure satisfies the defining condition (2.27) not only for generic $\ell \in \mathbb{P}(V_j)$, but in fact for all $\ell \in \mathbb{P}(V_j)$.

Proof. For m = 2, this was solved directly in §1.1. Suppose $m\geq3$. Let

(2.28)\begin{align} U_d:=\{\ell\in\mathbb{P}(V_j):\dim\left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0 \right)\geq d\} \end{align}

be the closed sets as in remark 2.2.2.

The set U ₁ is open and non-empty by definition of weak factorization structure, so $U_1=\mathbb{P}(V_k)$, and hence the condition (2.27) holds on the whole $\mathbb{P}(V_j)$ as claimed. Let

(2.29)\begin{align} U^d:=U_d\backslash U_{d+1}=\{\ell\in\mathbb{P}(V_j):|\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0|=d\}. \end{align}

The set $U^1=U_1\backslash U_2=\mathbb{P}(V_j)\backslash U_2$ is open as U ₂ is closed. Thus, if there exists $\ell\in\mathbb{P}(V_j)$ such that

(2.30)\begin{align} \dim \left(\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0\right) =1, \end{align}

i.e., $U^1\neq\emptyset$, then (2.30) holds generically in $\ell$ as claimed.

However, if the set U ¹ is empty, then $\mathbb{P}(V_j)=U_1=U_2$. Now, similarly as before, if the open set U ² is non-empty, then $\dim \left(\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0\right) =2$ generically in $\ell$. Since $\mathfrak{h}$ is a weak factorization structure this process yields the claim before d exceeds $\dim(\mathfrak{h})=m+1$.

Now we are ready to investigative quotients of (weak) factorization structures.

Lemma 2.5.4. Let $\varphi:\mathfrak{h}\to V^*$ be a weak factorization structure of dimension m. Then, for every $i\in\{1,\ldots,m\}$ there exists an open non-empty $A_i\subset\mathbb{P}(V_i)$ such that for all $\lambda\in A_i$, every non-zero $v\in\lambda$, and every $j\in\{1,\ldots,m\}\backslash\{i\}$, there is an open non-empty $U_j\subset\mathbb{P}(V_j)$ such that for all $\ell\in U_j$:

(2.31)\begin{equation} \dim \left( \rho_{i,v} \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \right) \geq 1. \end{equation}

Proof. By contradiction. Suppose there is $i\in\{1,\ldots,m\}$ such that for any open $A_i\subset\mathbb{P}(V_i)$, there is $\lambda\in A_i$, $0\neq v\in\lambda$ and j ≠ i such that for any open $U_j\subset\mathbb{P}(V_j)$, there exists $\ell$ with

(2.32)\begin{align} \varphi(\mathfrak{h})\cap \Sigma_{j,\ell}^0 \subset \ker \rho_{i,v} = \Sigma_{i,\lambda}^0. \end{align}

There are two observations to be made. Firstly, for such a fixed $i, A_i, \lambda$ and j, by suitably varying U_j we find an infinite set $S_\lambda\subset\mathbb{P}(V_j)$ such that any $\ell \in S_\lambda$ satisfies (2.32). Secondly, replacing A_i with the open set $A_i\backslash\{\lambda\}$ gives an index $j_0\neq i$, $\bar{\lambda}\in\mathbb{P}(V_i)$ s.t. $\bar{\lambda}\neq\lambda$, and the corresponding infinite set $S_{\bar{\lambda}}\subset\mathbb{P}(V_{j_0})$. Clearly, indices j and j ₀ might be different, but the freedom in choosing an open non-empty set not containing λ allows to find two infinite sets $S_\lambda, S_{\bar{\lambda}}\subset\mathbb{P}(V_j)$ corresponding to distinct $\lambda,\bar{\lambda}\in\mathbb{P}(V_i)$ such that

(2.33)\begin{align} \varphi(\mathfrak{h})\cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\lambda}^0, \ \ell \in S_\lambda, \end{align}

and

(2.34)\begin{align} \varphi(\mathfrak{h})\cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\bar{\lambda}}^0, \ \ell \in S_{\bar{\lambda}}. \end{align}

Let j be as above and $\mathcal{V}\subset\mathbb{P}(V_j)$ be the open non-empty set, where $\ell \mapsto \dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right)$ attains its minimal value d (see lemma 2.5.3). Then the closed set (see remark 2.2.2)

(2.35)\begin{align} \left\{ \ell \in \mathcal{V} \,|\, \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\lambda}^0 \right\} = \left\{ \ell \in \mathcal{V} \,|\, \dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \right) \geq d \right\} \end{align}

contains the set S_λ, and thus equals to $\mathcal{V}$. Clearly, the same argument works for $\bar{\lambda}$. Thus,

(2.36)\begin{align} \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\lambda}^0 \qquad\text{and}\qquad \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\bar{\lambda}}^0 \end{align}

for $\ell\in \mathcal{V}$, i.e., $\varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \subset \Sigma_{i,\lambda}^0 \cap \Sigma_{i,\bar{\lambda}}^0 = 0$ as $\lambda\neq\bar{\lambda}$. Therefore $\dim \left(\varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) =0$ generically, which contradicts φ being a weak factorization structure.

Corollary 2.5.5. Let $\varphi:\mathfrak{h}\to V^*$ be a factorization structure of dimension m. Then, for every $i\in\{1,\ldots,m\}$, there exists an open non-empty $A_i\subset\mathbb{P}(V_i)$ such that for all $\lambda\in A$ and any $j\in\{1,\ldots,m\}\backslash\{i\}$, there is an open non-empty $\bar{U}_j\subset\mathbb{P}(V_j)$ such that for all $\ell\in \bar{U}_j$

(2.37)\begin{equation} \dim \left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \right)=0. \end{equation}

Proof. Rank-nullity theorem together with lemma 2.5.4 give an open non-empty $U_j\subset \mathbb{P}(V_j)$ such that for $\ell \in U_j$

(2.38)\begin{align} \dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) - \dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \right) = \dim \left( \rho_{i,v} \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \right) \geq 1. \end{align}

Intersecting U_j with the open non-empty set, where $\dim \left(\varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) =1$ gives an open non-empty set $\bar{U}_j$, where the claim holds.

We define the quotient of a weak factorization structure $\varphi: \mathfrak{h} \to V^*$ with respect to $i \in \{1,\ldots,m\}$ and $v \in \lambda$, $\lambda \in \mathbb{P}(V_j)$, by (2.25) to be the linear inclusion $\varphi_{i,v}: \mathfrak{h}_{i,\lambda} \to \hat{V}_i^*$.

Proof. We need to check if the intersection (2.26) are at least 1-dimensional generically in every slot. This follows by combining lemma 2.5.4 with the set-theoretical inclusion

(2.39)\begin{equation} \rho_{i,v} \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \subset \rho_{i,v} \left(\varphi(\mathfrak{h})\right) \cap \rho_{i,v} \Sigma_{j,\ell}^0. \end{equation}

The following sufficient condition for a weak factorization structure to be a factorization structure is used to prove the main theorem of this section.

Proof. To ease the notation we proceed with $v=v_1^i$. Using respectively that $\varphi_{i,v}$ is a factorization structure, the inclusion (2.39), and the rank-nullity theorem for the contraction $\rho_{i,v}$, we find that for a generic $\ell\in\mathbb{P}(V_j)$, j ≠ i,

(2.40)\begin{align} &1 = \dim \left( \rho_{i,v} \left(\varphi(\mathfrak{h})\right) \cap \rho_{i,v} \Sigma_{j,\ell}^0 \right) \geq \dim \left( \rho_{i,v} \left( \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) \right)= \end{align}

(2.41)\begin{align} &\dim \left( \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0\right)- \end{align}

(2.42)\begin{align} &\dim \left( \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \right). \end{align}

Observe that if (2.42) were zero, then φ being a weak factorization structure implies that (2.41) is 1, and thus proving that φ satisfies the defining equation of a factorization structure for j ≠ i. To show that (2.42) is zero we consider contractions $\rho_{q,v_1^q}$ and $\rho_{q,v_2^q}$ for q ≠ j and q ≠ i. Now, as $\varphi_{q,v_1^q}$ is a factorization structure, corollary 2.5.5 implies that the right hand side of

(2.43)\begin{align} \rho_{q,v_1^q} \left( \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \right) \subset \rho_{q,v_1^q}\varphi (\mathfrak{h}) \cap \rho_{q,v_1^q}\Sigma_{j,\ell}^0 \cap \rho_{q,v_1^q}\Sigma_{i,\lambda}^0 \end{align}

is zero for appropriate generic choices of λ and $\ell$ as explained in corollary 2.5.5. Thus, for these values of $\ell$ and λ,

(2.44)\begin{align} \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \subset \ker \rho_{q,v_1^q} = \Sigma_{q,\lambda_1^q}^0. \end{align}

Repeating this process with $\varphi_{q,v_2^q}$ yields generic values of $\ell$ and λ for which

(2.45)\begin{align} \varphi (\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \cap \Sigma_{i,\lambda}^0 \subset \Sigma_{q,\lambda_1^q}^0 \cap \Sigma_{q,\lambda_2^q}^0=0 \end{align}

as required. We showed that for j ≠ i and j ≠ q, $\dim \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) = 1$ generically in $\ell$. To prove the rest one permutes the roles of $i,j$, and q.

Finally,

Finally, we are ready to describe the behaviour of factorization curves and their degrees in quotient spaces.

Let $\psi_j^{i,\lambda}: \mathbb{P}(V_j)\to\mathbb{P}(\mathfrak{h}_{i,\lambda})$ be a factorization curve in the quotient factorization structure (2.25), thus generically given by

(2.46)\begin{align} \varphi_{i,v}\circ\psi_j^{i,\lambda}(\ell) = \varphi_{i,v}(\mathfrak{h}_{i,\lambda}) \cap \rho_{i,v}\Sigma_{j,\ell}^0. \end{align}

Corollary 2.5.5 shows that $\varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0$ does not lie in $\ker \rho_{i,v} = \Sigma_{i,\lambda}^0$ for generic $\ell\in\mathbb{P}(V_j)$, hence

(2.47)\begin{equation} \rho_{i,v} \left( \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0 \right) = \varphi_{i,v}(\mathfrak{h}_{i,\lambda}) \cap \rho_{i,v}\Sigma_{j,\ell}^0 \end{equation}

holds generically. Combining (2.46) and (2.47), we generically find

(2.48)\begin{align} \rho_{j,v} \circ \varphi \circ \psi_j (\ell) = \varphi_{i,v}\circ\psi_j^{i,\lambda}(\ell), \end{align}

which, using the commutativity of (2.25) and taking the $\varphi_{i,v}$-preimage, results in the generic equality

(2.49)\begin{align} P_{i,v} \circ \psi_j(\ell) = \psi_j^{i,\lambda}(\ell). \end{align}

Lemma 2.1.8 implies that the unique extension of $P_{i,v} \circ \psi_j$, ensured by lemma 2.1.2, and $\psi_j^{i,\lambda}$ coincide. Additionally, the injectivity of factorization curves (corollary 2.1.10) shows that the curve ψ_j intersects $\ker P_{i,v}=\psi_i(\lambda)$ at most once. In this case, we have

(2.50)\begin{align} P_{i,v} \circ \psi_j(\ell) = \psi_j^{i,\lambda}(\ell), \text{where } \ell\neq\lambda, \end{align}

otherwise the equality holds everywhere. Finally, since ψ_j and $\psi_j^{i,\lambda}$ are injective, $P_{i,v}$ restricted to $\text{Im }\psi_j \backslash \{\psi_i(\lambda)\}$ is bijective.

Theorem 2.5.11. Let φ be a complex factorization structure, i ≠ j, and fix $\lambda\in\mathbb{P}(V_i)$. Then

(2.51)\begin{align} \deg \psi_j^{i,\lambda} = \begin{cases} \deg \psi_j -1, & \text{if } \, \psi_i(\lambda) \in \operatorname{Im} \psi_j\\ \deg \psi_j, & \text{otherwise.} \end{cases} \end{align}

2.6. Decomposable curves and Segre–Veronese factorization structures

This subsection proves in theorem 2.6.2 one of the main results of this article: if all factorization curves in a factorization structure φ are decomposable, then φ is a Segre–Veronese factorization structure. Consequently, if every factorization curve is decomposable, then all factorization structures are of Segre–Veronese type. We therefore ask

Theorem 2.4.3, remark 2.4.4, and theorem 2.4.5 show that for a real/complex factorization structure $\varphi: \mathfrak{h} \to V^*$ with all curves decomposable, $\varphi(\mathfrak{h})$ contains, up to an isomorphism of factorization structures, the image of a Segre–Veronese factorization structure as a subspace, whose preimage under φ is denoted here by $\mathcal{SV}$.

In the following proof, the set of 1-dimensional spaces corresponding to a factorization curve is called a curve as well.

In example 1.2.3, we observed that defining tensors Γ₁ and Γ₂ of a standard Segre–Veronese factorization structure corresponding to a partition $m=d_1+d_2$ are symmetric. This observation generalizes as follows.

Proof. Induction on k. The case k = 1 is trivial and k = 2 was solved in (1.15) above.

Suppose the claim holds for $k\geq2$, fix $j\in\{1,\ldots,k+1\}$ and set $d_0=0$. The idea is to form d_j quotients iteratively in $(d_1+\cdots+d_{j-1}+1)$-st slot in such a way that theorem 2.5.9 is applicable, i.e., so that each quotient is a factorization structure. This contracts grouped j-slots, leaving $\Gamma_j$ behind as we will see. Note that after the first quotient, the $(d_1+\cdots+d_{j-1}+2)$-nd slot becomes $(d_1+\cdots+d_{j-1}+1)$-st slot, and so on. Clearly in each step, one can choose v and λ as in theorem 2.5.9 so that the corresponding quotient is a factorization structure. While a complete tracking of indices, v’s and λ’s is possible, it contributes no essential understanding and significantly complicates the presentation, and will therefore be bypassed. We denote the composition of all d_j quotient maps by ρ (see remark 2.5.10) and apply it on

(2.53)\begin{align} \varphi(\mathfrak{h})= \sum_{i=1}^{k+1} ins_i \left( S^{d_i}W_i^* \otimes \Gamma_i \right) \end{align}

which results in

(2.54)\begin{align} \Gamma_j + \sum_{\substack{i=1\\i\neq j}}^{k+1} ins_i \left( S^{d_i}W_i^* \otimes \rho \Gamma_i \right). \end{align}

By theorem 2.6.2,

(2.55)\begin{align} \Gamma_j \in \sum_{\substack{i=1\\i\neq j}}^{k+1} ins_i \left( S^{d_i}W_i^* \otimes \rho \Gamma_i \right) \end{align}

which together with the induction hypothesis,

(2.56)\begin{align} \rho \Gamma_i \subset \bigotimes_{\substack{b=1\\b\neq i,j}}^{k+1} S^{d_b}W_b^*, \ i\in\{1,\ldots,k+1\}\backslash\{j\}, \end{align}

give the claim.

Corollary 2.6.4. If a Segre–Veronese factorization structure of dimension $m=d_1+\ldots+d_k$ is determined by 1-dimensional spaces

(2.57)\begin{align} \Gamma_j = \bigotimes_{\substack{r=1\\r\neq j}}^k \bigotimes_{p=1}^{d_r} a_j^{r,p}, \ a_j^{r,p} \subset W_r^*, \end{align}

then $a_j^{r,1}= \cdots= a_j^{r,d_r} =: a_j^r$, i.e.,

(2.58)\begin{align} \Gamma_j= \bigotimes_{\substack{r=1\\r\neq j}}^k (a^r_j)^{\otimes d_r}. \end{align}

The following lemma shows that $d_1,\ldots,d_k$ in the partition $m = d_1 + \cdots + d_k$ corresponding to a Segre–Veronese factorization structure are invariants.

2.7. Characterization of decomposable factorization structures

This subsection proves another important result of this article. It uses products of factorization structures and the correspondence from remark 2.7.1 to characterize decomposable Segre–Veronese factorization structures in theorem 2.7.7 as iterative products of Veronese factorization structures. This structural result is the first step towards the classification of Segre–Veronese factorization structures and, in addition, it is allows to solve the extremality equation for associated separable Kähler geometries uniformly, as outlined in §1.5.

Remark 2.7.1. We explain a correspondence between isomorphism classes of decomposable Segre–Veronese factorization structures and pairs consisting of an isomorphism class of a decomposable Segre factorization structure and a set of positive integers.

Starting with a decomposable Segre–Veronese factorization structure φ of dimension m, we have a partition $m=d_1+\cdots+d_k$ and 1-dimensional subspaces

(2.59)\begin{align} \Gamma_j= \bigotimes_{\substack{r=1\\r\neq j}}^k (a_j^r)^{\otimes d_r}. \end{align}

Then, for each $r\in\{1,\ldots,k\}$ we take, in grouped r-slots, $d_r-1$ (order and choice independent) factorization structure quotients as in theorem 2.5.9 of φ to get a decomposable Segre factorization structure of dimension k

(2.60)\begin{align} \sum_{j=1}^{k} \text{ins}_j \left( W_j^* \otimes \bigotimes_{\substack{r=1\\r\neq j}}^k a_j^r \right), \end{align}

while remembering the partition $\{d_1,\ldots,d_k\}$ (see also proof of proposition 2.6.3 and remark 2.5.10 for more details on quotients). The isomorphism class of (2.60) together with $\{d_1,\ldots,d_k\}$ form the pair from the correspondence. Observe that every factorization structure isomorphic with φ corresponds to the same pair. This gives a well-defined assignment.

Conversely, starting with a decomposable Segre factorization structure, say (2.60), we use the set $\{d_1,\ldots,d_k\}$ and the Veronese embedding $W_r^*\to (W_r^*)^{\otimes d_r}$, $v\mapsto v^{\otimes d_j}$, in each slot $r\in\{1,\ldots,k\}$ to obtain an inclusion of vector spaces

(2.61)\begin{align} \sum_{j=1}^{k} \text{ins}_j \left( S^{d_j} W_j^*\otimes \Gamma_j \right) \hookrightarrow \bigotimes_{j=1}^k S^{d_j}W_j^* \hookrightarrow \bigotimes_{j=1}^k (W_j^*)^{\otimes d_j}, \end{align}

where $\Gamma_j$ are now as in (2.59). The image has dimension m + 1 as can be seen by taking consecutive factorization structure quotients, whose kernel is 1-dimensional, as above. This determines an assignment on isomorphism classes which is inverse to the one describe above.

Definition 2.7.2. A decomposable Segre factorization structure of dimension m, $m\geq2$, admits a full-product in jth slot if it equals to

(2.62)\begin{align} \text{ins}_j \left( q\otimes Q + V_j^* \otimes \Gamma_j \right), \end{align}

where Q is a decomposable Segre factorization structure of dimension m − 1 which admits a full-product in rth slot for some $r\in\{1,\ldots,m\}\backslash\{j\}$, $V_j^*$ is a 2-dimensional vector space, and $q\subset V_j^*$ and $\Gamma_j\subset Q$ are 1-dimensional subspaces. A (decomposable Segre) factorization structure of dimension 1, being a 2-dimensional vector space, admits a full-product by definition. We say that a decomposable Segre factorization structure admits a full-product if it admits a full-product in jth slot for some j.

We remark that in definition 2.7.2 we use without loss of generality the identification of a factorization structure with its image (see remark 1.0.2).

One can consult example 2.7.4 for an example of a full-product decomposable Segre–Veronese.

Example 2.7.4. We illustrate how a decomposable Segre admitting a full-product is built using inductive products of 1-dimensional factorization structures and outline the shape of defining tensors.

A 2-dimensional (decomposable) Segre factorization structure is a product of two 1-dimensional factorization structures, and hence admits full-product in both slots. Forming a product of this 2-dimensional Segre with a 1-dimensional factorization structure yields

(2.63)\begin{align} \left( V_1^* \otimes \gamma_2 + \gamma_1 \otimes V_2^* \right) \otimes \gamma + \lambda \otimes V_3^*, \end{align}

where $\lambda \subset V_1^* \otimes \gamma_2 + \gamma_1 \otimes V_2^*$ is assumed to be decomposable, and hence by lemma 1.3.4 either $\lambda=a\otimes \gamma_2$ or $\lambda=\gamma_1 \otimes b$ for some 1-dimensional $a\subset V_1^*$ or $b\subset V_2^*$. We note that regardless of the choice of λ, (2.63) admits full-products in at least two distinct slots; one being the 3rd slot and the other is the 1st or 2nd depending on the choice of λ. Note, if $\lambda=\gamma_1\otimes\gamma_2$, then the full-product exists in all slots which recovers the product Segre factorization structure. Forming yet another product

(2.64)\begin{align} \left(\left( V_1^* \otimes \gamma_2 + \gamma_1 \otimes V_2^* \right) \otimes \gamma + \lambda \otimes V_3^*\right) \otimes \delta +\pi\otimes V_4^* \end{align}

to make the pattern more visible, we see that again for any admissible choices of λ and π, (2.64) admits at least two and at most four full-products. Observe in (2.64), that three summands belong into $\Sigma_{4,\delta^0}^0$. More importantly, for any choice, π decomposes so that another three summands belong to $\Sigma_{r,\tau^0}^0$ for some $r\in\{1,2,3\}$ and $\tau\in\mathbb{P}(V_r)$.

In general, it is plain to see that m − 1 summands in a full-product (2.62) lie in $\Sigma_{j,q^0}^0$. The following lemma shows that there are another m − 1 summands with a similar property. This helps us to establish the main claim of this subsection that decomposable Segre–Veronese factorization structures are given by full-products.

Lemma 2.7.5. Let a decomposable Segre factorization structure of dimension m admit a full-product in the jth slot. Then, there exist r ≠ j, and $\lambda\in\mathbb{P}(V_r)$, such that for all $i \neq r:$

(2.65)\begin{align} \text{ins}_i(V_i^*\otimes\langle\Gamma_i\rangle) \subset \Sigma_{r,\lambda}^0. \end{align}

Proof. Induction on the dimension of a factorization structure. The base case, m = 2, is obvious. Suppose the statement holds in dimension m − 1, $m\geq3$, and write a decomposable Segre factorization structure $\varphi(\mathfrak{h})$ of dimension m which admits a full-product in the jth slot as

(2.66)\begin{align} \varphi(\mathfrak{h})= \text{ins}_j \left( q\otimes Q + V_j^* \otimes \Gamma_j \right), \end{align}

where $\Gamma_j \subset Q$, $q\subset V_j^*$ and Q is a decomposable Segre factorization structure of dimension m − 1, which admits a full-product in rth slots for some $r\in\{1,\ldots,m\}\backslash\{j\}$. In particular,

(2.67)\begin{align} Q= \text{ins}_r \left( \lambda \otimes P + V_r^*\otimes \pi \right), \end{align}

for some decomposable $\pi \subset P$ and $\lambda \subset V_r^*$, where P is a decomposable Segre factorization of dimension m − 2 which admits a full-product.

Since (2.66) is a decomposable factorization structure, in particular $\Gamma_j$ is decomposable, lemma 1.3.4 implies that either

(2.68)\begin{align} \Gamma_j = \text{ins}_r \left (\lambda \otimes \Lambda \right) \quad\text{for some } \Lambda \subset P, \end{align}

or

(2.69)\begin{align} \Gamma_j = \text{ins}_r \left( \Pi \otimes \pi \right) \quad\text{for some } \Pi \subset V_r^*. \end{align}

In (2.68) case, it is clear that

(2.70)\begin{align} \text{ins}_j\left(V_j^*\otimes\Gamma_j\right) = \text{ins}_j\left(V_j^*\otimes \text{ins}_r \left( \lambda \otimes \Lambda \right) \right) \leq \varphi(\mathfrak{h}) \end{align}

and another m − 2 summands sitting in

(2.71)\begin{align} \text{ins}_j \left( q\otimes \text{ins}_r\left(\lambda\otimes P\right)\right) \leq \varphi(\mathfrak{h}) \end{align}

lie in $\Sigma_{r,\lambda^0}^0$.

For (2.69) case, we use the induction hypothesis stating that $\text{ins}_r\left(V_r^*\otimes \pi\right) \leq Q$ and another m − 3 summands in $\text{ins}_r\left(\lambda\otimes P\right) \leq Q$ lie in $\Sigma_{i,\mu}^0$ for some $i\in\{1,\ldots,m-1\}$ and $\mu\in\mathbb{P}(V_i)$. Clearly, this gives the claim.

We note that the proof of this lemma uses less assumptions than required in the statement. However, the stronger statement suffices for proving our end-result and reveals a rigidity of decomposable Segre factorization structures as the proof is by induction. A similar situation occurred in lemma 2.5.7 when proving that every weak factorization structure is a factorization structure.

Proof. Note that if we would know that m − 1 summands in $\varphi(\mathfrak{h})$ lie in $\Sigma_{j,q^0}^0$ for some $j\in\{1,\ldots,m\}$, then

(2.72)\begin{align} \varphi(\mathfrak{h})= \text{ins}_j\left( q \otimes Q + V_j^* \otimes \Gamma_j \right) \end{align}

must be a product. Indeed, a factorization structure quotient of $\varphi(\mathfrak{h})$ in jth slot gives an m-dimensional vector space, $Q+\Gamma_j$, and theorem 2.6.2 shows $\Gamma_j \subset Q$. Now we would use the induction hypothesis for Q, saying that $(m-1)$-dimensional decomposable Segre factorization structures admit a full-product, which shows that (2.72) admits a full-product. Thus we are left to prove that there exists an index j and $q\in\mathbb{P}(V_j^*)$ such that m − 1 summands belong to $\Sigma_{j,q^0}^0$.

Fix any $j\in\{1,\ldots,m\}$, and let $\tilde{Q}$ be the image of the quotient factorization structure of $\varphi(\mathfrak{h})$ in jth slot with respect to some v and λ (see theorem 2.5.9 and remark 2.5.10). We have $\Gamma_j \subset \tilde{Q}$, and by assumptions $\tilde{Q}$ is full-product. In particular

(2.73)\begin{align} \tilde{Q}= \text{ins}_r \left( \lambda \otimes P + V_r^*\otimes \pi \right), \end{align}

and lemma 1.3.4 implies that $\Gamma_j$ decomposes either as (2.68) or (2.69). We proceed similarly to the lemma above. In the former case it is immediate that m − 1 summands in $\varphi(\mathfrak{h})$ lie in $\Sigma_{r,\lambda^0}^0$, while for the latter case we apply lemma 2.7.5 to $\tilde{Q}$ and conclude the proof.

Finally, the following theorem completely characterizes decomposable Segre–Veronese factorization structures.

Remark 2.7.9. We remark that ideas from this subsection can be directly adapted to more general factorization structures, whose defining tensors are of the form

(2.74)\begin{align} \Gamma_j = \bigotimes_{\substack{i=1\\ i\neq j}}^k \gamma_i, \end{align}

where $\gamma_i \in S^{d_i}W^*_i$.

3.1. Compatibility in general

To rephrase, a convex polytope is compatible with a factorization structure if it is full-dimensional, and is a section a pointed convex polyhedral cone whose projectivized normals lie on factorization curves.

We exemplify cones and polytopes compatible with 2-dimensional factorization structures, originally found in [Reference Apostolov, Calderbank and Gauduchon4]. To keep our cartoons uncomplicated we discuss projectivized versions of these cones/polytopes, but the reader is strongly encouraged to work out 3-dimensional polyhedral geometry according to definition 3.1.1. For more details see [Reference Brandenburg and Púček13].

Example 3.1.2. Figure 2(b) displays the images of two factorization lines/curves ψ ₁ and ψ ₂ in 2-dimensional projective space $\mathbb{P}(V_1^* \otimes \Gamma_1 + \Gamma_2 \otimes V_2^*)$, associated with 2-dimensional Segre factorization structure (example 2.1.4), together with their intersection point $\Gamma_2 \otimes \Gamma_1$ and a choice of points $a_i,b_i \in \operatorname{Im} \psi_i$, $i=1,2$.

Under the projective duality, figure 2(a) shows lines and points arrangement in $\mathbb{P}(( V_1^* \otimes \Gamma_1 + \Gamma_2 \otimes V_2^* )^*)$: the line $(\Gamma_2 \otimes \Gamma_1)^0$, dual to the point $\Gamma_2 \otimes \Gamma_1$, with two marked points $(\operatorname{Im} \psi_1)^0$ and $(\operatorname{Im} \psi_2)^0$, dual to the lines $\operatorname{Im} \psi_1$ and $\operatorname{Im} \psi_2$, and lines $(a_i)^0, (b_i)^0$, dual to $a_i,b_i$, passing through the point $(\operatorname{Im} \psi_i)^0$, $i=1,2$.

The blue region is the projectivization of a 2-dimensional polytope, whose de-projectivization is a polytope compatible with 2-dimensional Segre factorization structure, since its projectivized normals $a_i,b_i$, $i=1,2$, lie on factorization curves. The ambiguity in the definition of de-projectivization comes from the fact that these are really meant to be 3-dimensional pictures of planes, lines and of a cone section rather than their projectivizations.

Figure 2. (a) Projectivized compatible polytope and (b) Segre factorization structure.

If fact, every 2-dimensional polytope with 4 edges is compatible with a 2-dimensional Segre factorization structure. Indeed, viewing the polytope as an affine section of a cone σ, the dual cone $\sigma^\vee$ has four extremal rays lying on four 1-dimensional spaces, which are the annihilators of the planes determining facets of σ. These four 1-dimensional spaces determine two planes Π₁ and Π₂ in three possible ways. In all three cases, $\Pi_1 + \Pi_2$ is the ambient 3-dimensional space, and, after the projectivization, we obtain 2-dimensional projective space with two (intersecting) lines. To complete the argument we need to find a linear isomorphism $\Phi: \Pi_1+\Pi_2 \to V_1^* \otimes \Gamma_1 + \Gamma_2 \otimes V_2^*$ (see the definition of isomorphism in definition 1.0.1) sending the distinguished planes to the distinguished planes, which is trivial. We found that projectivized normals of σ lie on factorization curves, therefore the polytope is compatible. Said differently, a Segre factorization structure can be fit onto the vector space $\Pi_1 + \Pi_2$ so that the polytope is compatible with it.

Example 3.1.3. Figure 3(b) illustrates the quadric $\operatorname{Im} \psi$ with four marked points $a_1,\ldots,a_4$ in the 2-dimensional projective space $\mathbb{P}(S^2W^*)$ associated to the 2-dimensional Veronese factorization structure. The quadric represents the factorization curve $\psi_1 = \psi_2 =: \psi$, being the rational normal curve of degree 2, a quadric and a conic too. Dually, figure 3(a) shows lines $(a_1)^0,\ldots,(a_4)^0$ dual to points $a_1,\ldots,a_4$, which are tangent to the dual quadric $(\operatorname{Im} \psi)^*$. The orange region is the projectivization of a 2-dimensional polytope which is compatible with 2-dimensional Veronese factorization structure.

Figure 3. (a) Projectivized compatible polytope and (b) Veronese factorization structure.

In fact, every 2-dimensional polytope with 4 edges is compatible with the Veronese factorization structure. Indeed, proceeding as in example 3.1.2, we view the polytope as a section of a cone σ, which, through the extremal rays of $\sigma^\vee$, gives four lines in 3-dimensional vector space, and hence four points in $\mathbb{P}^2$. As any five points in general position determine the rational normal curve of degree 2, there is a 1-parametric family of such curves (conics) fitting these four points, and hence for any member of this family, the cone σ has its projectivized normals on factorization curves. Therefore, the polytope is compatible with a Veronese factorization structure.

A strategy for constructing compatible cones and polytopes is to start with finitely many points on factorization curves and de-projectivize them, which determines a cone $\sigma^\vee$ whose dual σ is a compatible cone by construction, provided it is full-dimensional and pointed. To clarify how to do this rigorously we continue with the following general observations, which, once restricted to our setting, provide a construction of compatible cones and polytopes.

Note that a cone $\sigma\subset\mathfrak{h}^*$ with n facets determines n points in $\mathbb{P}(\mathfrak{h})$ by projectivizing extremal rays of $\sigma^\vee$. However, not every choice of an affine chart realizes points in $\mathbb{P}(\mathfrak{h})$ as generators of extremal rays of a cone. In general, we have the following.

A finite collection of points $p_1,\ldots,p_n\in\mathbb{P}(\mathfrak{h})$ belongs to the domain of the affine chart given by $\epsilon\in \mathfrak{h}^*$ if and only if ϵ does not belong into the proper and closed set $\cup_{j=1}^n (p_j)^0$, where $(p_j)^0\subset \mathfrak{h}^*$ is the annihilator of $p_j\subset \mathfrak{h}$. The set $\cup_{j=1}^n (p_j)^0$ is a hyperplane arrangement in $\mathfrak{h}^*$ splitting it into a union of full-dimensional convex polyhedral cones which, in general, do not have n facets. Observe that for σ such a cone, bounded by $(p_{i_1})^0,\ldots,(p_{i_r})^0$, $r\leq n$, all lines $p_1,\ldots,p_n$ contain rays $p_1^+,\ldots,p_n^+$ which belong to $\sigma^\vee$, since all functionals $\alpha\in\sigma$ evaluate non-negatively on them, but the only extremal rays of $\sigma^\vee$ are $p_{i_1}^+,\ldots,p_{i_r}^+$. By construction, the projectivized normals of σ are $p_{i_1},\ldots,p_{i_r} \in \mathbb{P}(\mathfrak{h})$.

As we are also interested in compatible polytopes which are simple polytopes, we recall relevant definitions and reflect them into associated cones. An m-dimensional polytope is simple if exactly m facets are incident with each of its vertices, and simplicial if its every facet is a simplex. Observe that an m-dimensional simple polytope arises as a section of a full-dimensional and pointed cone σ whose extremal rays are intersections of exactly m facets. Dually, each facet of $\sigma^\vee$ contains exactly m extremal rays, and these are linearly independent. Equivalently, every compact slice of $\sigma^\vee$ has simplices as faces, thus $\sigma^\vee$ is a cone over a simplicial polytope. Thus, to determine if a cone is a cone over a simplicial polytope means to know how many edges lie on facets.

Proof. Say that the n extremal rays of our cone lie on 1-dimensional spaces

(3.2)\begin{align} \psi(t_i),\ i=1,\ldots,n, \end{align}

where $\psi=\psi_1=\cdots=\psi_m$ denotes the factorization curve, and $n\geq m+1$ since the cone is full-dimensional (see corollary 2.4.6). We need to show that a facet-supporting hyperplane, which is generally defined by m extremal rays and which in our case lie on 1-dimensional spaces (3.2), does not contain any other extremal rays. This is clearly true since any hyperplane intersects the degree m curve ψ in at most m points. Thus, it is a cone over a simplicial polytope.

In the rest of this subsection we describe hyperplanes and higher codimension spaces where facets and faces of a compatible cone and its dual lie. They have a particularly nice form characterized as φ ^t-images of intersections of spaces $\Sigma_{j,\ell}$, see theorem 3.1.6 below.

We start with finding the hyperplane in $\mathfrak{h}^*$ corresponding to (a projectivized normal) $\psi_j(\ell) \in \mathbb{P}(\mathfrak{h})$. Note that since (see proposition 2.1.9)

(3.3)\begin{align} \varphi\circ\psi_j(\ell)\subset \varphi(\mathfrak{h}) \cap \Sigma_{j,\ell}^0, \end{align}

we have

(3.4)\begin{align} 0= \langle \Sigma_{j,\ell}, \varphi\circ\psi_j(\ell) \rangle = \langle \varphi^t \Sigma_{j,\ell}, \psi_j(\ell) \rangle, \end{align}

and if $v\in\mathfrak{h}$ annihilates $\varphi^t \Sigma_{j,\ell}\subset\mathfrak{h}^*$, then

(3.5)\begin{align} 0= \langle \varphi^t \Sigma_{j,\ell}, v \rangle = \langle \Sigma_{j,\ell}, \varphi v \rangle, \end{align}

i.e., $\varphi(v)\in\varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0$. Therefore, $\varphi^t \Sigma_{j,\ell}\subset\mathfrak{h}^*$ is a hyperplane with the annihilator $\psi_j(\ell) \subset \mathfrak{h}$ if and only if $\dim \left( \varphi(\mathfrak{h})\cap\Sigma_{j,\ell}^0 \right) = 1$. In other words, $\varphi^t\Sigma_{j,\ell}$ is a hyperplane if and only if $\psi_j(\ell)$ does not lie on any other curve. In general, we have

Theorem 3.1.6. For $r\in\{1,\ldots,m\}$ and pairwise distinct $i_1,\ldots,i_r\in\{1,\ldots,m\}$, the space

(3.6)\begin{align} \varphi^t \left( \Sigma_{i_1,\ell_1} \cap \cdots \cap \Sigma_{i_r,\ell_r} \right) \end{align}

is of codimension r for generic choices of $\ell_j \in \mathbb{P}(V_{i_j})$, $j=1,\ldots,r$. Furthermore, for $\ell_j$, $j=1,\ldots,r$, such that (3.6) is of codimension r, if hyperplanes $\varphi^t ( \Sigma_{i_1,\ell_1} ), \cdots, \varphi^t ( \Sigma_{i_r,\ell_r} )$ are independent, then

(3.7)\begin{align} \varphi^t \left( \Sigma_{i_1,\ell_1} \cap \cdots \cap \Sigma_{i_r,\ell_r} \right) = \varphi^t ( \Sigma_{i_1,\ell_1} ) \cap \cdots \cap \varphi^t ( \Sigma_{i_r,\ell_r} ). \end{align}

In particular, as corollary 2.4.6 shows, such hyperplanes are always independent in case of Veronese factorization structure and generically independent for product Segre–Veronese factorization structure.

Proof. Set-theoretically, we always have that the space (3.6) lies in the intersection of the hyperplanes. The independence of hyperplanes implies that they intersect in a codimension r space. Thus, showing that (3.6) is of codimension r proves (3.7). To this end, we prove that its annihilator,

(3.8)\begin{align} \varphi \left( \left( \varphi^t \left( \Sigma_{i_1,\ell_1} \cap \cdots \cap \Sigma_{i_r,\ell_r} \right) \right)^0 \right) = \varphi(\mathfrak{h}) \cap \left( \Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right), \end{align}

is r-dimensional.

By combining

(3.9)\begin{align} & \dim \left(\varphi(\mathfrak{h}) +\Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) =\dim \left(\varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\nonumber\\ & \quad +\dim \left( \Sigma_{i_1,\ell_1}^0 \right) - \dim\left(\left( \varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \cap \Sigma_{i_1,\ell_1}^0\right), \end{align}

and (rank-nullity theorem)

\begin{align*} \dim\left(\rho_{i_1,\ell_{i_1}}\left(\varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\right)=&\dim\left(\varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\end{align*}

(3.10)\begin{align} &- \dim \left( \left( \varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \cap \Sigma_{i_1,\ell_1}^0 \right), \end{align}

we arrive at

(3.11)\begin{align} &\dim \left( \varphi(\mathfrak{h}) + \Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\nonumber\\ &\quad = \dim \left( \Sigma_{i_1,\ell_1}^0 \right)+ \dim \left( \rho_{i_1,\ell_{i_1}} \left( \varphi(\mathfrak{h}) + \Sigma_{i_2,\ell_2}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \right), \end{align}

where $\rho_{i_1,\ell_{i_1}}$ represents the contraction $\rho_{i_1,v}$ for some/any $v \in \ell_{i_1}$. Similar abbreviations are used in the following. Repeating the above $(r-2)$-times yields

(3.12)\begin{align} &\dim\left(\varphi(\mathfrak{h}) +\Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\nonumber\\&\quad =\dim \left( \Sigma_{i_1,\ell_1}^0 \right)+\sum_{j=1}^{r-2} \dim\left(\rho_{i_j,\ell_j} \circ \cdots \circ \rho_{i_1,\ell_1} \Sigma_{i_{j+1},\ell_{j+1}}^0\right)\nonumber\\ &\qquad +\dim\left( \rho_{i_{r-1},\ell_{r-1}} \circ \cdots \circ \rho_{i_1,\ell_1}\left( \varphi(\mathfrak{h}) + \Sigma_{r,\ell_r}^0\right)\right), \end{align}

which is valid for $\varphi(\mathfrak{h})=0$ too. Finally, inserting (3.12) with $\varphi(\mathfrak{h})=0$ and (3.12) itself into

\begin{align*} \dim\left(\varphi(\mathfrak{h}) +\Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0\right) =&\dim \left( \varphi(\mathfrak{h}) \right)+\dim\left(\Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \end{align*}

(3.13)\begin{align} &-\dim\left(\varphi(\mathfrak{h}) \cap\left( \Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \right) \end{align}

provides the final formula

(3.14)\begin{align} & \dim \left( \rho_{i_{r-1},\ell_{r-1}} \circ \cdots \circ \rho_{i_1,\ell_1} \left( \varphi(\mathfrak{h}) \right) \right) \end{align}

(3.15)\begin{align} &- \dim \left( \rho_{i_{r-1},\ell_{r-1}} \circ \cdots \circ \rho_{i_1,\ell_1} \left( \varphi(\mathfrak{h}) \right) \cap \rho_{i_{r-1},\ell_{r-1}} \circ \cdots \circ \rho_{i_1,\ell_1} \left( \Sigma_{r,\ell_r}^0 \right) \right) = \end{align}

(3.16)\begin{align} &\dim \left( \varphi(\mathfrak{h}) \right) - \dim \left( \varphi(\mathfrak{h}) \cap \left( \Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right) \right). \end{align}

Now, we use theorem 2.5.9 to find an open non-empty $A_{i_1} \subset \mathbb{P}(V_{i_1})$ where the quotient $\varphi_{i_1,v}$ of φ is a factorization structure for any non-zero $v \in \lambda$, $\lambda \in A_{i_1}$, then to find an open non-empty $A_{i_2} \subset \mathbb{P}(V_{i_2})$ where the quotient $(\varphi_{i_1,v})_{i_2,w}$ of $\varphi_{i_1,v}$ is a factorization structure for any $w \in \mu$, $\mu \in A_{i_2}$, etc. Thus, for $(\ell_1,\ldots,l_r) \in A_{i_1} \times \cdots \times A_{i_r}$, i.e., generic $\ell_j \in \mathbb{P}(V_{i_j})$, $j=1,\ldots,r$, we find that (3.14) is $m+1-(r-1)$, and (3.15) is 1. Thus,

(3.17)\begin{align} \dim\left(\varphi(\mathfrak{h}) \cap\left( \Sigma_{i_1,\ell_1}^0 + \cdots + \Sigma_{i_r,\ell_r}^0 \right)\right)=r \end{align}

as claimed.

The above suggests

Proof. Theorem 3.1.6 shows that $\varphi^t \ell_1\otimes\cdots\otimes\ell_m$ is generically 1-dimensional. Since $\varphi\circ\psi_j(\ell_j)$ has $\ell_j^0$ at the j-th slot, the computation

(3.18)\begin{equation} \left\langle \varphi^t \ell_1\otimes\cdots\otimes\ell_m, \psi_j(\ell_j) \right\rangle = \left\langle \ell_1\otimes\cdots\otimes\ell_m, \varphi \circ \psi_j(\ell_j) \right\rangle = 0 \end{equation}

gives the claim.

Note that if a cone $\sigma^\vee$ has extremal rays lying on $\psi_j(\tau_{ji})$, then extremal rays of σ lie on $\varphi^t \ell_1\otimes\cdots\otimes\ell_m$ for some $\ell_r\in\{\tau_{ji}\}_{j,i}$.

3.2. Cones compatible with the product Segre–Veronese factorization structure

This section demonstrates the construction of compatible cones through an example of a cone compatible with the product Segre–Veronese factorization structure. Furthermore, a condition for determining its facets is given which generalizes the Gale evenness condition from cyclic polytopes to this broader setting. More details can be found in [Reference Brandenburg and Púček13] where an extensive theory for the case of the Veronese factorization structure was already developed.

Recall the m-dimensional product Segre–Veronese factorization structure

(3.19)\begin{align} \mathfrak{h}:= \sum_{j=1}^{k} ins_j \left( S^{d_j}W_j^*\otimes\Gamma_j \right) \hookrightarrow{\quad \varphi \quad} \bigotimes_{j=1}^k (W_j^*)^{\otimes d_j} =: V^*, \end{align}

where

(3.20)\begin{align} \Gamma_j= \bigotimes_{\substack{r=1\\r\neq j}}^k (a^r)^{\otimes d_r} \end{align}

for some 1-dimensional subspaces $a^r\subset W_r^*$, $r=1,\ldots,k$, $d_1+\cdots+d_k = m$, and $\otimes_{r=1}^k (a^r)^{\otimes d_r}$ is called the intersection point. Its distinct factorization curves are given by

(3.21)\begin{align} \varphi\circ\psi_j(\ell)= ins_j \left( (\ell^0)^{\otimes d_j} \otimes \Gamma_j \right), \end{align}

$j=1,\ldots,k$ (see examples 2.1.4-2.1.7 for more details).

Consider points on factorization curves of the product Segre–Veronese factorization structure,

(3.22)\begin{align} \psi_j(\tau_{ji}) \in \mathbb{P}(\mathfrak{h}), \ i=1,\ldots,c_j, \ j=1,\ldots,k, \end{align}

for some c_j’s, where τ_ji’s are pairwise distinct, $\tau_{ji} \in \mathbb{P}(W_j)$. We fix a chart, and declare (3.22) in this chart as generators of the cone, which is therefore pointed. Assuming that these generators generate extremal rays of the cone, a necessary and sufficient condition for its full-dimensionality follows from corollary 2.4.6. The cone is full-dimensional if and only if there exists $j_0\in\{1,\ldots,k\}$ such that $c_{j_0} \gt d_{j_0}$ and $c_j\geq d_j$ for $j\neq j_0$. However, as noted above, not every affine chart realizes (3.22) as generators of extremal rays. Regardless, the condition for determining facets is applicable as shown below.

We express images of (3.22) in an affine chart chosen below. To do so we fix dual bases of W_r and $W^*_r$ such that the first basis vector $\underline{a}^r$ of $W_r^*$ lies on a^r (see (3.19) and (3.20) for notation). These provide coordinates t_ji for τ_ji, $t_{ji}=\tau_{ji}/\langle \tau_{ji}, \underline{a}^j \rangle$, and a basis $\epsilon_0,\epsilon_{ji}$, $i=1,\ldots,d_j$, $j=1,\ldots,k$, of $\mathfrak{h}$, uniquely characterized by

(3.23)\begin{align} \varphi\epsilon_0 &= \otimes_{r=1}^k(\underline{a}^r)^{\otimes d_r},\nonumber\\ \varphi\epsilon_{ji} &= \text{ins}_j \left( \varepsilon_{ji} \otimes \bigotimes_{\substack{r=1\\r\neq j}}^k (\underline{a}^r)^{\otimes d_r} \right), \end{align}

where ɛ_ji together with $(\underline{a}^j)^{\otimes d_j}$ denote the standard basis for symmetric tensors $S^{d_j}W_j^*$. Indeed, ϵ ₀ together with ϵ_ji, $i=1,\ldots,d_j$, form a basis of $ins_j \left( S^{d_j} W_j^* \otimes \bigotimes_{\substack{r=1 \\ r \neq j}}^k (a^r)^{\otimes d_r} \right)$ (see also examples 2.1.7 and 1.2.4).

We define the affine chart $\epsilon\in\mathfrak{h}^*$ by $\epsilon = \varphi^t \varepsilon \in \mathfrak{h}^*$, where

(3.24)\begin{align} \varepsilon:= \sum_{j=1}^k \text{ins}_j \left( (0,-1)^{\otimes d_j} \otimes (1,0)^{\otimes (m-d_j)} \right) \in V. \end{align}

The only point of any ψ_j, $j=1,\ldots,k$, which is not in this chart is the intersection point. Note that this remains true if for any $j\in\{1,\ldots,k\}$, the $(1,0)^{\otimes (m-d_j)}$-part of (3.24) is replaced by a tensor from $\otimes_{\substack{i=1\\ i\neq j}}^k \left( W_i \right)^{\otimes d_i}$ which does not belong to the annihilator of $(1,0)^{\otimes (m-d_j)} \in \otimes_{\substack{i=1\\ i\neq j}}^k \left( W_i^* \right)^{\otimes d_i}$.

In this chart and coordinates, $\psi_j([1:x])$ can be found explicitly,

(3.25)\begin{align} \frac{\varphi \circ \psi_j([1:x])}{\langle \varphi \circ \psi_j([1:x]), \varepsilon \rangle} = \text{ins}_j \left( (x,-1)^{\otimes d_j} \otimes (1,0)^{\otimes (m-d_j)} \right) \end{align}

and thus

(3.26)\begin{align} \frac{\psi_j([1:x])}{\langle \psi_j([1:x]), \epsilon \rangle} = x^{d_j}\epsilon_0 + \sum_{i=1}^{d_j} (-1)^i x^{d_j-i}\epsilon_{ji}. \end{align}

Evaluating (3.26) at $x=t_{ji}$, we obtain images of (3.22) in the chart ϵ, i.e., the vectors generating the cone $\sigma^\vee$,

(3.27)\begin{equation} \sigma^\vee = \text{cone} \left( \left(t_{ji}^{d_j}, -t_{ji}^{d_j-1},\ldots,(-1)^{d_j-1} t_{ji}, (-1)^{d_j} \right) \bigg| \, i=1,\ldots,c_i, \ j=1,\ldots,k \right). \end{equation}

To describe facet-supporting hyperplanes of $\sigma^\vee$, we note that each such is in particular a hyperplane through m linearly independent extremal rays. First, we classify hyperplanes through m linearly independent 1-dimensional spaces lying on factorization curves, which allows us to see which collections of points from (3.22) give rise to a hyperplane, and then we derive a condition for deciding which of these are facet-supporting hyperplanes.

Proof. Corollary 2.4.6 shows that there are at most $d_j+1$ linearly independent directions on ψ_j, $j=1,\ldots,k$, and note that the shape of the product Segre–Veronese factorization structure (3.19) associated to the partition $m=d_1+\cdots+d_k$ is such that its summands mutually intersect at a unique single direction (see also example 2.1.7). Therefore, for a fixed distribution of m independent directions on factorization curves, every ψ_j must contain at least $d_j-1$ of these directions. Indeed, since factorization curves ψ_j, j ≠ i, span together $(m+1-d_i)$-dimensional space, the curve ψ_i cannot contain strictly less than $d_i-1$ points. Now we cover all possible cases: there exists a curve ψ_i carrying $d_i+1$ independent directions, each curve ψ_i carries exactly d_i independent directions, and there exists a curve ψ_i carrying $d_i-1$ independent directions.

First, we start with $d_i+1$ points on ψ_i for some $i\in\{1,\ldots,k\}$, which leaves us with choosing $m-d_i-1$ independent directions on factorization curves ψ_j, j ≠ i, each now retaining exactly d_j dimensions. Then, the only way how to ensure m independent directions is to fix an index r ≠ i and $d_r-1$ independent directions on ψ_r, and d_j independent directions on ψ_j otherwise. Note that the latter distribution of points defines the hyperplane containing

(3.30)\begin{align} \sum_{\substack{j=1 \\ j\neq r}}^{k} ins_j \left( S^{d_j}W_j^*\otimes \bigotimes_{\substack{q=1\\r\neq j}}^k (1,0)^{\otimes d_q} \right) \end{align}

and $d_r-1$ points on ψ_r as above. Observe that it is the same hyperplane as the one given in (3.29).

Secondly, we consider d_j points on the curve ψ_j for each $j=1,\ldots,k$, say $\psi_j([1:x_{ji}])$, $i=1,\ldots,d_j$, $j=1,\ldots,k$, which provides m independent directions, and theorem 3.1.6 and corollary 3.1.7 show that its normal is of the form (3.28). Note that this case excludes the intersection point as one m independent directions.

Finally, it is easy to observe that having a curve ψ_j, with $d_j-1$ independent directions forces existence of a curve ψ_i, i ≠ j with $d_i+1$ independent directions, which was solved in the first case.

To proceed further, note that since a cone consists of positive combinations of its generators, a hyperplane is a (facet-)supporting hyperplane if and only if it has all the generators on its positive side. Using this we find facet-supporting hyperplanes of the cone $\sigma^\vee$ from (3.27) by separately considering the two types of hyperplanes from proposition 3.2.1. Let $\{x_{ji}\}_{i=1}^{d_j} \subset \{t_{jr}\}_{r=1}^{c_j}$, $j=1,\ldots,k$, be such that the corresponding hyperplane through $\psi_j([1:x_{ji}])$ has the normal (3.28). We compute its contraction with a general point on $\psi_j/\langle \psi_j, \epsilon \rangle$,

(3.31)\begin{equation} \begin{aligned} c\left \langle \varphi^t \otimes_{j=1}^k\otimes_{i=1}^{d_j} (1,x_{ji}), \, \frac{\psi_j([1:x])}{\langle \psi_j([1:x]), \epsilon \rangle} \right \rangle =\\ c\left\langle \otimes_{j=1}^k\otimes_{i=1}^{d_j} (1,x_{ji}), \, \text{ins}_j \left( (x,-1)^{\otimes d_j} \otimes (1,0)^{\otimes (m-d_j)} \right) \right\rangle =\\ c\prod_{i=1}^{d_j} \left\langle (1,x_{ji}), (x,-1) \right \rangle = c\prod_{i=1}^{d_j} (x-x_{ji}) = \\ c\sum_{i=0}^{d_j}(-1)^{i} x^{d_j-i} \sigma_{i}(x_{j1},\ldots,x_{jd_j}). \end{aligned} \end{equation}

The expression (3.31) is a polynomial p_j in x which vanishes at $\{x_{ji}\}_{i=1}^{d_j} \subset \{t_{jr}\}_{r=1}^{c_j}$. We conclude

In the Veronese factorization structure case, hyperplanes (3.28) are the only class of hyperplanes through m independent points (3.22), and proposition 3.2.2 recovers the Gale’s evenness condition. However, to understand if the associated compatible polytopes are cyclic requires further analysis (for details see [Reference Brandenburg and Púček13]).

Collecting our previous results together yields a condition for facet-supporting hyperplanes of $\sigma^\vee$.

The above theorem can be formulated in the spirit of proposition 3.2.2, and further worded in combinatorics, but this is beyond the scope of this article. For more details in the case of the Veronese factorization structure, see [Reference Brandenburg and Púček13].