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Symbolic Powers of Toric Ideals

Giuseppe Favacchio, Graham Keiper

TL;DR

This work addresses computing symbolic powers of toric ideals, which encode geometric information about their varieties but are often difficult to compute directly. It develops two complementary frameworks: a kernel-of-linear-maps approach, where $I_A^{(t+1)}= ext{ker}(igl( ef{d.pi}igr))$ with $igl( ef{d.pi}igr)(e^{oldsymbol{eta}})=oldsymbol{eta}^{ ens t}$, and a saturation-based method, showing $I_A^{(t)}=I_A^{t}:oldsymbol{m}^{ abla}$ and $I_A^{(t)}=J^{t}:oldsymbol{m}^{ abla}$ with $J$ generated by a basis of $ ext{ker}(A)$. The contributions include explicit kernel characterizations, a fiber-decomposition viewpoint, and a practical saturation strategy that reduces generators and improves computation, underpinned by a lemma to optimize saturation. Together these results provide computable, structurally transparent tools for symbolic powers of toric ideals with potential applications in algebraic geometry and combinatorics.

Abstract

This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic powers of a toric ideal can also be expressed as saturations of regular powers with the monomial given by the product of all the variables. Finally, we conclude with a computationally significant result for computing symbolic powers of toric ideals.

Symbolic Powers of Toric Ideals

TL;DR

This work addresses computing symbolic powers of toric ideals, which encode geometric information about their varieties but are often difficult to compute directly. It develops two complementary frameworks: a kernel-of-linear-maps approach, where with , and a saturation-based method, showing and with generated by a basis of . The contributions include explicit kernel characterizations, a fiber-decomposition viewpoint, and a practical saturation strategy that reduces generators and improves computation, underpinned by a lemma to optimize saturation. Together these results provide computable, structurally transparent tools for symbolic powers of toric ideals with potential applications in algebraic geometry and combinatorics.

Abstract

This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic powers of a toric ideal can also be expressed as saturations of regular powers with the monomial given by the product of all the variables. Finally, we conclude with a computationally significant result for computing symbolic powers of toric ideals.
Paper Structure (3 sections, 10 theorems, 103 equations)

This paper contains 3 sections, 10 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Symbolic powers of toric ideals via kernel of linear maps
  3. Symbolic powers of toric ideals via saturations

Key Result

Lemma 2.3

Given $A\in\mathcal{M}_{m\times n}(\mathbb{N})$, let $f=c_{1}e^{\beta_1}+\cdots +c_{r}e^{\beta_r}\in V_{A,\sigma}$ be a $\varphi_A$-homogeneous form. Then $f\in I_A$ if and only if $c_1+\cdots+c_r=0$.

Theorems & Definitions (31)

  • Definition 2.1: Toric Ideal
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 21 more