Extensions of the universal theta divisor

Abstract

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 Müller. 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.

42 pages, 5 figures. Final version. Added Section 4.1, which describes how divisor classes other than the theta divisor vary

References in corpus (2)

• Compactified universal jacobian and the double ramification cycle
• The pullback of a theta divisor to M_{g,n}

Cited by in corpus (7)

• Extending the Double Ramification Cycle using Jacobians
• Powers of the theta divisor and relations in the tautological ring
• Compactifications of the universal Jacobian over curves with marked points
• Compactified universal jacobian and the double ramification cycle
• Pullbacks of universal Brill-Noether classes via Abel-Jacobi morphisms
• Stability conditions for line bundles on nodal curves
• Orbifold Euler Characteristics of Compactified Jacobians