Prill’s Problem

Prill’s Problem, popularized by ACGH's "Geometry of Algebraic Curves" almost forty years ago, asks the following question: Given any curve $Y$ of genus $g \geq 2$ and a finite covering $f : X \rightarrow Y$, does $h^0(X, \mathcal{O} _X (f^{-1} (y))) = 1$ for a general point $y \in Y$? The talk will be based on an article by A. Landesman and D. Litt who found that the surprising answer to this question is negative. For any curve $Y$ of genus 2, they produce a finite étale degree 36 cover $X \rightarrow Y$ such that $h^0(X, \mathcal{O} _X( f^{-1} (y) ) \geq 2$ for any $y \in Y$.