Vol. 15, No. 6, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 8, 1865–2122
Issue 7, 1593–1864
Issue 6, 1343–1592
Issue 5, 1077–1342
Issue 4, 821–1076
Issue 3, 569–820
Issue 2, 309–567
Issue 1, 1–308

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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 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.

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