Weierstrass Sets on Finite Graphs

Alessio Borzì

doi:10.37236/10400

Abstract

We study two possible tropical analogues of Weierstrass semigroups on graphs, called rank and functional Weierstrass sets. We prove that on simple graphs, the first is contained in the second. We completely characterize the subsets of $\mathbb{N}$ arising as a functional Weierstrass set of some graph. Finally, we give a sufficient condition for a subset of $\mathbb{N}$ to be the rank Weierstrass set of some graph, allowing us to construct examples of rank Weierstrass sets that are not semigroups.

Highlights

Let X be a smooth projective algebraic curve of genus g and fix a point P ∈ X
By the Weierstrass gap theorem, the set of gaps G(P ) = N\H(P ) has cardinality exactly g. This implies that H(P ) is a numerical semigroup, that is, a cofinite additive submonoid of N
We give a sufficient condition for a subset of N to be the rank Weierstrass set of a graph

Summary

Introduction

Let X be a smooth projective algebraic curve of genus g and fix a point P ∈ X. By the Weierstrass gap theorem (see [11, III.5.3]), the set of gaps G(P ) = N\H(P ) has cardinality exactly g This implies that H(P ) is a numerical semigroup, that is, a cofinite additive submonoid of N. In [10] it was proved that for a fixed numerical semigroup S, the set of integers m that do not satisfy the above condition is finite. The previous theorem allows us to construct families of graphs in which the rank Weierstrass set is not a semigroup (see Example 31), justifying the name “Weierstrass set ”

Preliminaries

Functional Weierstrass sets

Rank Weierstrass sets