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Zbl 0711.14025
O'Grady, Kieran G.
On the Kodaira dimension of moduli spaces of abelian surfaces. (English)
[J] Compos. Math. 72, No.2, 121-163 (1989). ISSN 0010-437X; ISSN 1570-5846/e

{\it D. Mumford} showed in Invent. Math. 42, 239-272 (1977; Zbl 0365.14012) that the moduli spaces of principally polarized abelian surfaces with a level n-structure is of general type for n big. For any prime p let ${\cal A}\sb 2(p)$ denote the moduli space of pairs (S,H), S a principally polarized abelian surface and H a rank 2 subspace of the space of p-division points of S. The main result of this paper is that ${\cal A}\sb 2(p)$ is of general type if $p\ge 17$. Since ${\cal A}\sb 2(p)$ is isomorphic to the moduli space ${\cal A}\sb{2,p\sp 2}$ of abelian surfaces with polarization of type $(1,p\sp 2)$ this is equivalent to the statement that ${\cal A}\sb{2,p\sp 2}$ is of general type for $p\ge 17.$ \par The idea of the proof is as follows: Let $\pi$ : ${\cal A}\sb 2(p)\to {\cal A}\sb 2$ denote the map (S,H)$\to S$. Define $\overline{{\cal A}\sb 2}(p)$ the natural toroidal compactification of ${\cal A}\sb 2(p)$ such that $\pi$ extends to a finite surjective map $\pi$ : $\overline{{\cal A}}\sb 2(p)\to \overline{{\frak M}}\sb 2$ onto the moduli space of stable genus 2 curves. Using Hurwitz's formula to $\pi$ an expression for the canonical class of $\overline{{\cal A}}\sb 2(p)$ is obtained. After a partial desingularization $\hat {\cal A}\sb 2(p)$ all of whose singularities are canonical an estimate for $h\sp 0(nK\sb{\hat {\cal A}\sb 2(p)})$ is obtained which implies the assertion.
[H.Lange]

MSC 2000:: *14K10 Algebraic moduli, classification (abelian varieties)
14J29 Surfaces of general type

Keywords: abelian surfaces; moduli spaces of principally polarized abelian surfaces; general type; polarization

Citations: Zbl 0365.14012

Cited in: Zbl 1074.14021 Zbl 0907.14019 Zbl 0910.14024

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