Tau functions, Prym-Tyurin classes and loci of degenerate differentials

2017/10/03 by Korotkin, Dmitri, Sauvaget, Adrien, Zograf, Peter
#14C22 #14F10 #14H15 #14H70 #30F30 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.01239

Abstract

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve.

Related

• Tau function and the Prym class
• The Q-Picard group of the moduli space of curves in positive characteristic
• Computation of λ-classes via strata of differentials
• Complete subvarieties of moduli spaces and the Prym map
• k-differentials on curves and rigid cycles in moduli space