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Tropical computations for toric intersection theory in Macaulay2

Alessio Borzì

Abstract

We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.

Tropical computations for toric intersection theory in Macaulay2

Abstract

We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.
Paper Structure (8 sections, 4 theorems, 13 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Toric intersection theory
  4. Tropical computations
  5. Applications
  6. Wonderful compactifications
  7. The moduli space $\overline{M}_{0,n}$
  8. Characteristic polynomials

Key Result

Theorem 3.1

Let $Y$ be a subvariety of the algebraic torus $T^n$ and let $\overline{Y}$ be its closure in a toric variety $X_\Sigma$ such that $|\Sigma| = \operatorname{trop}(Y)$ and $\Sigma$ is simplicial. Let $\Sigma'$ be a completion of the fan $\Sigma$ and let $i:X_{\Sigma} \rightarrow X_{\Sigma'}$ be the i where $m(\sigma)$ is the multiplicity of $\sigma$ in $\operatorname{trop}(Y)$.

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure :
  • Figure :

Theorems & Definitions (7)

  • example 1
  • Theorem 3.1
  • Proposition 4.1
  • Theorem 4.2: maclagan2015tropical
  • example 2
  • Theorem 4.3: Huh-Katz huhkatz
  • example 3