A stratification of a moduli space of abelian varieties.

*(English)*Zbl 1052.14047
Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: BirkhĂ¤user (ISBN 3-7643-6517-X/hbk). Prog. Math. 195, 345-416 (2001).

For any fixed prime number \(p\) and any positive integer \(n\), not divisible by \(p\), let \({\mathcal A}={\mathcal A}_{g,1,n}\) denote the moduli space of principally polarized Abelian varieties of dimension \(g\) with a symplectic level-\(n\)-structure in characteristic \(p\). The paper constructs a finite stratification of \({\mathcal A}\) by locally closed quasi-affine subsets. The strata are indexed by the possible subscheme structures of the scheme of \(p\)-division points \(X[p]\) of an Abelian variety \(X\) in \({\mathcal A}\).

To be more precise: An elementary sequence is by definition a map \(\varphi: \{1,\dots, g\}\to \mathbb{Z}_{\geq 0}\) such that \(\varphi(i)\leq \varphi(i+ 1)\leq\varphi(i)+ 1\) for all \(i\). For every elementary sequence \(\varphi\) there is a locally closed subset \(S_\varphi\subset{\mathcal A}\) of dimension \(\sum^g_{i=1}\varphi(i)\) such that the disjoint union of the \(S_\varphi\) is a finite stratification of \({\mathcal A}\). The boundary of each stratum is the union of the strata meeting this boundary. The unique zero-dimensional stratum \(S_{\{0,\dots, 0\}}\) is closed and consists of the self-products of supersingular elliptic curves. There is exactly one stratum of dimension 1. It is contained in the supersingular locus and it is shown that its closure is connected. Hence all positive dimensional strata are connected. This gives another proof of the theorem that \({\mathcal A}\) is irreducible.

For the entire collection see [Zbl 0958.00023].

To be more precise: An elementary sequence is by definition a map \(\varphi: \{1,\dots, g\}\to \mathbb{Z}_{\geq 0}\) such that \(\varphi(i)\leq \varphi(i+ 1)\leq\varphi(i)+ 1\) for all \(i\). For every elementary sequence \(\varphi\) there is a locally closed subset \(S_\varphi\subset{\mathcal A}\) of dimension \(\sum^g_{i=1}\varphi(i)\) such that the disjoint union of the \(S_\varphi\) is a finite stratification of \({\mathcal A}\). The boundary of each stratum is the union of the strata meeting this boundary. The unique zero-dimensional stratum \(S_{\{0,\dots, 0\}}\) is closed and consists of the self-products of supersingular elliptic curves. There is exactly one stratum of dimension 1. It is contained in the supersingular locus and it is shown that its closure is connected. Hence all positive dimensional strata are connected. This gives another proof of the theorem that \({\mathcal A}\) is irreducible.

For the entire collection see [Zbl 0958.00023].

Reviewer: Herbert Lange (Erlangen)

##### MSC:

14K10 | Algebraic moduli of abelian varieties, classification |

14G15 | Finite ground fields in algebraic geometry |

14K05 | Algebraic theory of abelian varieties |

14K15 | Arithmetic ground fields for abelian varieties |