Prill's problem

TL;DR

This paper solves Prill's problem by constructing a specific finite étale cover of genus 2 curves, where the preimages of points move in a pencil, advancing understanding in algebraic geometry.

Contribution

The paper provides an explicit construction of a degree 36 étale cover of genus 2 curves with special fiber properties, addressing a long-standing open problem.

Findings

Constructed a degree 36 étale cover for genus 2 curves.

Showed the preimages of points move in a pencil.

Resolved Prill's problem from the 1970s.

Abstract

We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in ACGH's "Geometry of Algebraic Curves." That is, for any curve $Y$ of genus $2$, we produce a finite \'etale degree $36$ connected cover $f: X \to Y$ where, for every point $y \in Y$, $f^{-1}(y)$ moves in a pencil.