82 research outputs found
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
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