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arXiv · 1803.05816

Reduction type of smooth quartics

Reynald Lercier·Qing Liu·Elisa Lorenzo García·Christophe Ritzenthaler
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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 (Seshadri 1977). 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 $\textrm{SL}_{3,R}$ 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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Reynald Lercier, Qing Liu, Elisa Lorenzo García, Christophe Ritzenthaler. 2018-03-15. Reduction type of smooth quartics. https://doi.org/10.2140/ant.2021.15.1429

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