Line arrangements modeling curves of high degree: equations, syzygies and secants

TL;DR

This paper investigates the embedding properties of graph curves formed by unions of projective lines, analyzing their equations, syzygies, and secant varieties to understand their algebraic and geometric structure.

Contribution

It introduces conditions under which graph curves can be embedded as line arrangements and constructs generators for their ideals, advancing understanding of their algebraic properties.

Findings

Identified conditions for property Np in graph curve embeddings

Constructed explicit generators for the ideals of certain line arrangements

Explored secant varieties of graph curves and their higher-dimensional analogs

Abstract

We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.