Graph Curves

Graph curves are a useful construction for considering various problems in algebraic geometry. In this paper, we explain some of the results about graph curves from the seminal paper by Bayer and Eisenbud [3]. We provide background in simplicial cohomology and the canonical series of a curve for the beginning reader. In our study of algebraic curves, it is helpful to consider degenerations – objects that are simpler to characterize, while still having a lot of explanatory power. In this paper, we will investigate graph curves, in a sense the simplest degeneration one could use. Definition 0.1. A graph curve is a connected union of projective lines, each line meeting three others in ordinary double points. The lines and their intersections can be recorded in a dual graph G = (V,E), where each vertex of V corresponds to a line, and each edge e connecting v1 and v2 represents a node where the corresponding lines intersect. C(G) is used to denote the curve corresponding to a graph G. There are many ways to relax the definition of a graph curve, allowing some more general statements while giving up some of the simplicity and symmetry of the geometry. For instance, instead of each line intersecting three others, lines may be allowed to intersect one or two lines instead, as in [4]. Additionally, instead of allowing only lines in the curve, we may allow rational curves corresponding to the vertices, as in [1] and [2]. Example 0.2 (Bayer-Eisenbud, Introduction). Consider a set of four distinct lines in the plane. For simplicity take the zero sets of the polynomials X,Y, Z, and (X + Y + Z), which (like all other lines in P2) intersect. The corresponding graph is K4, the complete graph on four vertices, since we have four lines that intersect each of the others. This graph is trivalent. The polynomial defining this curve is trivially P (X,Y, Z) = XY Z(X + Y + Z).