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.

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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