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 graph curves can be embedded in projective space as line arrangements. We discuss property Np for these embeddings and are able to obtain 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.

1 Introduction

An arrangement of linear subspaces, or subspace arrangement, is the union of a finite collection of linear subspaces of projective space. In this paper we study arrangements of lines called graph curves with high degree relative to genus. We are particularly interested in the defining equations and syzygies of these subspace arrangements. We will assume an algebraically closed ground field of characteristic zero throughout.

Let G = (V, E) be a simple, connected graph with vertex set V and edge set E. Following [9], we assume that G is subtrivalent, meaning that each vertex has degree at most three. The (abstract) graph curve CG associated with G is constructed by taking the union of {Lυ | υ ∈ V}, where each Lυ is a copy of ℙ1 and lines Lu and Lυ intersect in a node if and only if there is an edge between u and υ in G. (Note that if we think of the nodes of CG as vertices and the lines Lυ as edges, then CG is the graph dual to G.) Since we are assuming that each vertex has degree less than or equal to three, CG is specified by purely combinatorial data; we may assume that on each component of CG the nodes are at 0, 1 or ∞. Note that if each vertex of G is trivalent, then each copy of ℙ1 in CG contains three nodes, and CG is stable (see [4, 9]).