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Prill's problem

Aaron Landesman, Daniel Litt

TL;DR

The paper resolves Prill's problem in genus $2$ by constructing a finite étale cover $f:X\to Y$ of degree $36$ for any genus $2$ curve $Y$ over $\mathbb{C}$ such that $f$ is Prill exceptional, i.e., $h^0(X,\mathscr O_X(f^{-1}(y)))\ge 2$ for all $y\in Y$. It achieves this by connecting Prill's condition to the infinitesimal variation of Hodge structure on $H^1(X,\mathbb{Q})$ and identifying an isotrivial isogeny factor in $\mathrm{Pic}^0_{\mathscr X/\mathscr M}$, following a Bogomolov–Tschinkel framework. The main technical step shows that $f_*\omega_X$ is not generically globally generated, which implies the pencil condition on the fibers; the degree-$36$ construction arises from a three-step isogeny composition, with isotriviality ensuring independence from the complex structure on $Y$. A companion paper extends these ideas to obtain a Prill exceptional cover for a general genus-2 curve using a more intricate argument.

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 étale degree $36$ connected cover $f: X \to Y$ where, for every point $y \in Y$, $f^{-1}(y)$ moves in a pencil.

Prill's problem

TL;DR

The paper resolves Prill's problem in genus by constructing a finite étale cover of degree for any genus curve over such that is Prill exceptional, i.e., for all . It achieves this by connecting Prill's condition to the infinitesimal variation of Hodge structure on and identifying an isotrivial isogeny factor in , following a Bogomolov–Tschinkel framework. The main technical step shows that is not generically globally generated, which implies the pencil condition on the fibers; the degree- construction arises from a three-step isogeny composition, with isotriviality ensuring independence from the complex structure on . A companion paper extends these ideas to obtain a Prill exceptional cover for a general genus-2 curve using a more intricate argument.

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 of genus , we produce a finite étale degree connected cover where, for every point , moves in a pencil.
Paper Structure (3 sections, 4 theorems, 8 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The Proof of \ref{['theorem:prill-example']}

Key Result

Theorem 1.2

If $Y$ is any smooth proper connected curve of genus $2$ over the complex numbers, there is a finite étale cover $f: X \to Y$ which is Prill exceptional.

Figures (1)

  • Figure 1: A diagram depicting the relevant curves in the proof of \ref{['proposition:bt']}.

Theorems & Definitions (12)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Remark 2.5
  • ...and 2 more