Effectively computing integral points on the moduli of smooth quartic curves

TL;DR

This paper provides an effective method to compute integral points on the moduli space of smooth quartic curves, extending Faltings' proof of the Shafarevich conjecture with explicit bounds.

Contribution

It introduces an effective approach to the Shafarevich conjecture for smooth quartic curves by extending finiteness results to del Pezzo surfaces of degree at most four.

Findings

Established an effective version of the Shafarevich conjecture for smooth quartic curves.

Extended Scholl's finiteness results to certain del Pezzo surfaces.

Provided explicit bounds for integral points on the moduli space.

Abstract

We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.