×

On the Kodaira dimension of moduli spaces of abelian surfaces. (English) Zbl 0711.14025

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 \({\mathcal A}_ 2(p)\) 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 \({\mathcal A}_ 2(p)\) is of general type if \(p\geq 17\). Since \({\mathcal A}_ 2(p)\) is isomorphic to the moduli space \({\mathcal A}_{2,p^ 2}\) of abelian surfaces with polarization of type \((1,p^ 2)\) this is equivalent to the statement that \({\mathcal A}_{2,p^ 2}\) is of general type for \(p\geq 17.\)
The idea of the proof is as follows: Let \(\pi\) : \({\mathcal A}_ 2(p)\to {\mathcal A}_ 2\) denote the map (S,H)\(\to S\). Define \(\overline{{\mathcal A}_ 2}(p)\) the natural toroidal compactification of \({\mathcal A}_ 2(p)\) such that \(\pi\) extends to a finite surjective map \(\pi\) : \(\overline{{\mathcal A}}_ 2(p)\to \overline{{\mathfrak M}}_ 2\) onto the moduli space of stable genus 2 curves. Using Hurwitz’s formula to \(\pi\) an expression for the canonical class of \(\overline{{\mathcal A}}_ 2(p)\) is obtained. After a partial desingularization \(\hat {\mathcal A}_ 2(p)\) all of whose singularities are canonical an estimate for \(h^ 0(nK_{\hat {\mathcal A}_ 2(p)})\) is obtained which implies the assertion.
Reviewer: H.Lange

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14J29 Surfaces of general type

Citations:

Zbl 0365.14012
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Fujiki, A. : On resolutions of cyclic quotient singularities . Publ. RIMS, Kyoto Univ.,10, 293 (1974). · Zbl 0313.32012 · doi:10.2977/prims/1195192183
[2] Harris, J. , Mumford, D. : On the Kodaira dimension of the moduli space of curves . Inv. Math., 67, 23 (1982). · Zbl 0506.14016 · doi:10.1007/BF01393371
[3] Igusa, J. : Arithmetic variety of moduli for genus two . Annals of Math., 72, 612 (1980). · Zbl 0122.39002 · doi:10.2307/1970233
[4] Mumford, D. : Hirzebruch’s proportionality theorem in the non-compact case . Inv. Math., 42, 239 (1977). · Zbl 0365.14012 · doi:10.1007/BF01389790
[5] Mumford, D. : Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry . M. Artin and J. Tate editors, Boston, Birkhauser (1983). · Zbl 0554.14008
[6] Mumford, D. , Fogarty, J. : Geometric invariant theory , Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer (1982). · Zbl 0504.14008
[7] Namikawa, Y. : Toroidal compactifications of Siegel space . Lect. Notes in Math., 812 (1980). · Zbl 0466.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.