Extensions of the universal theta divisor

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently been studied by Grushevsky-Zakharov, Hain and M\"uller. In particular, we compute the pullback of the theta divisor studied in Alexeev's work on stable abelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.Comment: 42 pages, 5 figures. Final version. Added Section 4.1, which describes how divisor classes other than the theta divisor var

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.Comment: 25 pages, 22 figure

Perfect Matchings in Claw-free Cubic Graphs

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure

The stable set polytope and some operations on graphs

AbstractWe study some operations on graphs in relation to the stable set polytope, for instance, identification of two nodes, linking a pair of nodes by an edge and composition of graphs by subgraph identification. We show that, with appropriate conditions, the descriptions of the stable set polytopes associated with the resulting graphs can be derived from those related to the initial graphs by adding eventual clique inequalities. Thus, perfection and h-perfection of graphs are preserved