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

Authors :: Dmitry Korotkin
Adrien Sauvaget
Peter Zograf
Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)
Analyse, Géométrie et Modélisation (AGM - UMR 8088)
Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Source :: Mathematische Annalen, Mathematische Annalen, Springer Verlag, 2019, 375 (1-2), pp.213-246. ⟨10.1007/s00208-019-01836-1⟩
Publication Year :: 2019
Publisher :: Springer Science and Business Media LLC, 2019.
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.<br />26 pages, 1 figure, comments are welcome