Stable genus 2 curves and their moduli scheme
type: Item
instance of: scholarly article
title: Stable genus 2 curves and their moduli scheme
zbMATH Open document ID: 0819.14010
DOI: 10.1007/BF01444884
author: Qing Liu
published in: Mathematische Annalen
publication date: 1995-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/165039
review text: Let \(R\) be a discrete valuation ring with quotient field \(K\) and residue field \(k\). Given a projective nonsingular curve \(C\) defined over \(K\) of genus at least 1, \textit{P. Deligne} and \textit{D. Mumford} proved [Publ. Math., Inst. Hautes Étud. Sci. 36(1969), 75-110 (1970; Zbl 0181.488)] that there exists a finite extension \(K'\) of \(K\) such that \(C \times_ K K'\) is the generic fiber of a stable curve \({\mathcal C}\) defined over \(R_{K'}\), where \(R_{K'}\) denotes the integral closure of \(R\) in \(K'\). Let \(s'\) be the closed point of \(\text{Spec} (R_{K'})\) and \({\mathcal C}_{\tilde s} = {\mathcal C} \times_{R_{K'}} k(s')^{\text{alg}}\). Then \(C_{\tilde s}/k^{\text{alg}}\) is a stable curve and does not depend on the choice of \(s'\) nor of \(K'\). If \(C\) is an elliptic curve its modular invariant completely determines \(C_{\tilde s}/k^{\text{alg}}\). In this paper the author proves a similar result for a curve \(C\) of genus 2. This problem was first suggested by \textit{R. Coleman} [in: Théorie des nombres, Sémin. Paris 1985-1986, Prog. Math. 71, 1-18 (1987; Zbl 0633.14018)]. The complete solution is given in theorem 1 and involves the invariants \(J_ 2\), \(J_ 4\), \(J_ 6\), \(J_ 8\), \(J_{10}\) of \textit{J. Igusa} [cf. Ann. Math., II. Ser. 72, 612-649 (1960; Zbl 0122.390)]. If \(\text{char} (k) \neq 2\) and \(y^ 2 = P(x)\) is an equation for \(C\) then \(J_{2i}\) is a polynomial function in the coefficients of \(P(x)\) of degree \(2i\), for \(i = 1, \dots, 5\). -- One consequence of this result is the description of the moduli scheme \(\overline {\mathfrak M}_ 2/ \text{Spec} (\mathbb{Z})\) of stable curves of genus 2. Let \(\mathbb{Z} [{\mathcal I}]\) be the graded \(\mathbb{Z}\)- algebra \(\mathbb{Z} [J_ 2, \dots, J_{10}]/(J^ 2_ 4 - J_ 2 J_ 6 + 4J_ 8)\), \({\mathcal X} = \text{Proj} (\mathbb{Z} [{\mathcal I}])\) and \({\mathcal M}_ 2 : {\mathcal S} ch \to {\mathcal E} ns\) be the functor which associates to each scheme \(S\) the cl…
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 537249
zbMATH Keywords: discrete valuation ring
zbMATH Keywords: stable curve
zbMATH Keywords: modular invariant
MaRDI profile type: Publication
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OpenAlex ID: W2025376548
description: scientific article; zbMATH DE number 537249