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Description
The moduli space of abelian varieties admits a tautological ring generated by lambda classes. In contrast to the space of stable curves, this tautological ring has a remarkably simple presentation. This allows for the construction of a canonical “tautological projection” which maps the full Chow ring onto the tautological subring. Given a Chow class on the moduli space of abelian varieties, it is natural to try to compute its tautological projection. For the locus of Jacobians, an algorithm was provided by Faber.
We establish a corresponding algorithm for the locus of Prym varieties associated to G-covers, for G a finite group. The novel geometric content is a closed formula relating three different types of lambda classes on the moduli space of admissible G-covers, along with a fundamental invariance property of these lambda classes.
We have implemented this algorithm on a computer, leading to many new closed formulae for tautological projections. In the cases where the Prym cycle has codimension zero the algorithm computes the degree of the Prym-Torelli morphism, giving new intersection-theoretic proofs of results due variously to Donagi-Smith, Nagaraj-Ramanan, Bardelli-Ciliberto-Verra, Marcucci–Naranjo, and Lange–Ortega.
This is joint work in progress with Yoav Len and Sam Molcho.