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Matroids and semirings attached to toric singularity arrangements

Ethan Cotterill, Cristhian Garay López

Abstract

Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we establish a matroidal framework for resolving this problem for singularities determined by arrangements of toric branches; and we obtain precise quantitative results in the case of line arrangements. Our results have implications for the topology of Severi varieties of unisingular rational curves in projective space.

Matroids and semirings attached to toric singularity arrangements

Abstract

Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we establish a matroidal framework for resolving this problem for singularities determined by arrangements of toric branches; and we obtain precise quantitative results in the case of line arrangements. Our results have implications for the topology of Severi varieties of unisingular rational curves in projective space.
Paper Structure (10 sections, 13 theorems, 16 equations)

This paper contains 10 sections, 13 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Set-theoretic conventions
  3. Semirings and matroids
  4. Basic notions for idempotent semirings
  5. Basic notions for matroids
  6. Singularities with toric branches
  7. Applications to line arrangements
  8. The value semiring of an essential line arrangement
  9. Value semirings of (very) uniform essential line arrangements
  10. Conditions on rational curves imposed by $r$-tuple points

Key Result

Theorem 1.1

(Theorem semiring_generation_theorem) Whenever $\#F \geq r$, the $\pi_X$-images of $d$-forms, $d>0$ of minimal nonempty support of a toric arrangement $X\subset \mathbb{A}^n_F$ of distinguished type as above generate the semiring ${\rm S}(X)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 25 more