AAG Seminar: Prill and Prym problems

Abstract:  Let f: C'-->C be a (possibly branched) cover of curves. An old question of Prill asks: can the general fiber of f move in a pencil if the genus of C is at least 2? Landesman and I observed that the answer is "yes" in genus 2, using a beautiful construction of Bogomolov and Tschinkel. It turns out that in general the question is closely connected to understanding the moduli space M of covers with the same topological type as f, and the natural monodromy representation of pi_1(M) on H_1(C'). The main result (joint with Landesman and Sawin) is a computation of the Zariski-closure of the image of this monodromy representation when the genus of C is large compared to the Galois group of f. A key ingredient is a "generic Torelli theorem with coefficients."

Host: Wanlin Li