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
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 $\overline{K}$, 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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