Toric ideal of slice of a polytope?

Question

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a monomial homomophism: \begin{align} \phi_A\colon k[x_1,\ldots, x_n]&\to k[y_1,\ldots, y_d],\\ x_i&\mapsto y^{a_i}:=\prod_{j=1}^{d}y_j^{a_{i,j}} \end{align} according to $I_A:=\ker \phi_A$ (suppose $k$ is algebraically closed of characteristic zero). Let $H$ be a hyperplane in $k^d$ such that $conv(A)$ (when viewed as a polytope in $\mathbb{R}^d$) has integer (or at least rational) vertices, and call $A\cap H$ the resulting point configuration, consisting of the vertices of $A\cap H$. To rule out known situations, suppose $H$ further that intersects the interior of $A$.

Question: Carrying out the construction for the toric ideal with the input point configuration $A\cap H$, is there a relation between $I_{A\cap H}, I_A$ and $H$? Or are there hypothesis that can lead to a relation between the two ideals (or the corresponding rings)?

Answer 1

There may be distinct ways of viewing this topic, but the way I am familiar with it we have that the monomial homomorphism is defined by $$\phi_{A} : k[x_{1},...,x_{n}] \rightarrow k[t_{1},...,t_{d},t_{1}^{-1},...,t_{d}^{-1}] \\ \phi(x_{i}) \mapsto \mathbf{t}^{a_{i}}:=\prod_{j=1}^{d} t_{j}^{a_{j,i}}, \forall 1 \leq i \leq n.$$ My suggestion is to try using the rational polyhedral cone $$\text{pos}_{\mathbb{Q}}((a_{1},...,a_{n})) = \left\{ \sum_{i=1}^{n} \lambda_{i}a_{i} \; | \; \lambda_{i} \in \mathbb{Q}_{\geq 0}\right\}$$ attached to the toric variety $$V(I_{A}) = \{(u_{1},...,u_{n}) \in k^{n} \; | \; F(u_{1},...,u_{n})=0, \forall F \in I_{A}\}$$ where $I_{A}=\ker(\phi_{A})$ is the toric ideal. In particular, my intuitive idea is that you construct $I_{A}$ and then pass to the toric variety $V(I_{A})$ attached to $I_{A}$ (which is in your case an affine monomial curve if you choose $a_{1}<\cdot\cdot\cdot<a_{n}$ as relatively prime positive integers, think of it in this case before you generalize!). Now, constructing the polyhedral cone from the toric variety $V(I_{A})$ will allow you to have some sort of bound to how the points of $A$, and hence of $\text{conv}(A)$ will behave. In particular, I think you will be able to define $A$ and $A \cap H$ as subsets of the polyhedral cone, and that there is an associated height of the cone for which we define the hyperplane $H$ so that $$\text{conv}\left((A \cap H)\cup \bigcup_{i=1}^{n}\chi_{H}(a_{i})\right) \subset \text{pos}_{\mathbb{Q}}((a_{1},...,a_{n}))\big|_{h}$$ where $\chi_{H}: \mathbb{N}^{d} \rightarrow \mathbb{N}^d$ is an indicator function defined in terms of the hyperplane $H$ (and a chosen orientation for a normal) which will equal $a_{i}$ when the point is on the desired side of the hyperplane (bounding the polyhedral cone to a subset with finite metric quantities) and $1$ when the point is on the undesired side of the hyperplane (the unbounded region). I am using the $\big|_{h}$ on the rational polyhedral cone to denote the restriction unto the height $h$ induced by your choice of $H$. Using some of these ideas and intuitions I recommend that you try and construct the toric varieties and rational polyhedral cones attached to $I_{A}$ and $I_{A \cap H}$ in order to understand their relationship as toric ideals.