Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations

TL;DR

This paper investigates the secant degree of toric surfaces using combinatorial methods, classifies delightful triangulations for low genus, and provides bounds for higher genus cases based on triangulation properties.

Contribution

It introduces the concept of delightful triangulations for computing secant degrees and classifies all such triangulations for toric surfaces with genus up to one.

Findings

Complete classification of delightful triangulations for genus g ≤ 1.

Established lower bounds for 2- and 3-secant degrees for higher genus.

Link between singularities of triangulations and secant degrees.

Abstract

The $k$-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface $X$ corresponds to a regular unimodular triangulation $D$ of the polytope defining $X$. If the secant ideal of the initial ideal with respect to $D$ coincides with the initial ideal of the secant ideal, then $D$ is said to be delightful and the $k$-secant degree of $X$ can be easily computed. All delightful triangulations of toric surfaces having sectional genus $g\leq1$ are completely classified and, for $g\geq2$, a lower bound for the $2$- and $3$-secant degree, by means of the combinatorial geometry and the singularities of non-delightful triangulations, is established.