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.

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