Stable genus 2 curves and their moduli scheme
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 class of isomorphisms of proper curves over \(S\) with connected geometric fibers of genus 2. The author shows that the morphism of functors \(\varphi : {\mathcal M}_ 2 \to {\mathbf h}_{\mathcal X} = \text{Mor} (\cdot, {\mathcal X})\) induces an isomorphism between the coarse moduli scheme \({\mathfrak M}_ 2\) and the principal open subset \(D_ + (J_{10})\) of \({\mathcal X}\) (cf. \S8 of the Igusa-paper cited above). He uses theorem 1 and \(\varphi\) to show that \(\overline {\mathfrak M}_ 2\) is the normalisation of a certain blowing-up of \({\mathcal X}\) (theorem 2). The description of this result is simple over \(\mathbb{Z} [1/6]\).
scientific article; zbMATH DE number 537249
scientific article