Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

Details Density and completeness of subvarieties of moduli spaces of curves or abelian varieties Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

1996-09-13

arxiv.org/abs/alg-geom/9609008v1

Mathematics

Algebraic Geometry

AMS-LaTeX, 15 pages

Scientific paper

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V$ if $V \subset \tM_g$ (or $\cA_g$) is complete of codimension $g$.

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