Vol. 15, No. 6, 2021

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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

Let CK 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 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 SL3,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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smooth plane quartic, reduction, hyperelliptic, invariants, valuation
Mathematical Subject Classification
Primary: 13A50
Secondary: 14H10, 14H25, 14L24
Received: 27 January 2019
Revised: 17 March 2020
Accepted: 17 July 2020
Published: 16 October 2021
Reynald Lercier
Université de Rennes
Qing Liu
Université de Bordeaux
CNRS, Bordeaux INP, IMB, UMR 5251
F-33400 Talence
Elisa Lorenzo García
Université de Rennes
Christophe Ritzenthaler
Université de Rennes