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Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is affine (since it is cut out by the Hasse invariant, that is a section of an ample line bundle). But what about $Y^{\operatorname{ord}}$, the ordinary locus of $Y$? So the question is the following.

Is the ordinary locus of $Y$ an affine scheme?

Thank you!

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1 Answer 1

up vote 7 down vote accepted

No. It's well known that the complement of a closed subset of codimension two or more in an affine variety is never affine. The minimal compactification has codimension g.

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