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Gluing And Splitting of Homogeneous Toric Ideals

Philippe Gimenez, Hema Srinivasan

Abstract

We show that any two homogeneous affine semigroups can be glued by embedding them suitably in a higher dimensional space. As a consequence, we show that the sum of their homogeneous toric ideals is again a homogeneous toric ideal, and that the minimal graded free resolution of the associated semigroup ring is the tensor product of the minimal resolutions of the two smaller parts. We apply our results to toric ideals associated to graphs to show how two of them can be a splitting of a toric ideal associated to a graph or an hypergraph.

Gluing And Splitting of Homogeneous Toric Ideals

Abstract

We show that any two homogeneous affine semigroups can be glued by embedding them suitably in a higher dimensional space. As a consequence, we show that the sum of their homogeneous toric ideals is again a homogeneous toric ideal, and that the minimal graded free resolution of the associated semigroup ring is the tensor product of the minimal resolutions of the two smaller parts. We apply our results to toric ideals associated to graphs to show how two of them can be a splitting of a toric ideal associated to a graph or an hypergraph.
Paper Structure (10 sections, 7 theorems, 23 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 23 equations, 2 figures.

Table of Contents

  1. Gluing two homogeneous semigroups
  2. Splitting a homogeneous toric ideal
  3. The general homogeneous case
  4. The two dimensional case
  5. An application to non-homogeneous toric ideals
  6. Application to Gluing and splitting of Graphs
  7. Further Examples
  8. Selfgluing
  9. Constructing Gorenstein, Cohen-Macaulay or non Cohen-Macaulay toric varieties of higher dimensions
  10. Iterated gluing and splitting

Key Result

Theorem 1

Let $I_A\subset k[x_1, \ldots, x_p]$ and $I_B\subset k[y_1, y_2, \ldots y_q]$ be two homogeneous toric ideals associated to two $n\times p$ and $m\times q$ matrices $A$ and $B$ respectively. Set $R:=k[x_1, \ldots, x_{p},z, y_1, \ldots, y_q]$. For any $1\le i\le p, 1\le j\le q$, consider the ideals $

Figures (2)

  • Figure 1: The graph $G$ on the right is obtained by gluing two squares $G_1$ and $G_2$ along an edge
  • Figure 2: The graph $G$ on the left splits into two squares

Theorems & Definitions (32)

  • Theorem : Theorem \ref{['thm:A+B=C']}
  • Definition 1.1
  • Example 1.2
  • Lemma 1.3
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • proof
  • Remark 1.7
  • ...and 22 more