Reduction type of smooth plane quartics

Lercier, Reynald; Liu, Qing; Lorenzo García, Elisa; Ritzenthaler, Christophe

doi:10.2140/ant.2021.15.1429

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Reduction type of smooth plane quartics

Reynald Lercier, Qing Liu, Elisa Lorenzo García and Christophe Ritzenthaler

Vol. 15 (2021), No. 6, 1429–1468

DOI: 10.2140/ant.2021.15.1429

Abstract

Let $C ∕ K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e., smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$ , in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2, 3, 5$ or $7$ . On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of work of Seshadri (Advances in Math. 26:3 (1977), 225-274). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$ . We exhibit explicit ones for ternary quartic forms under the action of ${SL}_{3, R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

Keywords

smooth plane quartic, reduction, hyperelliptic, invariants, valuation

Mathematical Subject Classification

Primary: 13A50

Secondary: 14H10, 14H25, 14L24

Milestones

Received: 27 January 2019

Revised: 17 March 2020

Accepted: 17 July 2020

Published: 16 October 2021

Authors

Reynald Lercier
IRMAR Université de Rennes Rennes France

Qing Liu
Université de Bordeaux CNRS, Bordeaux INP, IMB, UMR 5251 F-33400 Talence France

Elisa Lorenzo García
IRMAR Université de Rennes Rennes France

Christophe Ritzenthaler
IRMAR Université de Rennes Rennes France

Vol. 15, No. 6, 2021