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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

6 votes
2 answers
1k views

Projectivity of blowups

If I have a quasiprojective variety $X$, and a subscheme $Z$, then the blowup $$f:Y = Bl(X,Z)\rightarrow X$$ is projective over $X$, since it is constructed by a relative Proj construction. Can I f …
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  • 135
4 votes
1 answer
468 views

Extending abelian schemes

Let $R$ be a regular local ring of dimension at least 2, and let $U$ be the complement of the closed point in $\mathrm{Spec} R$. Given a polarized abelian scheme over $U$, under what hypotheses can i …
anon's user avatar
  • 135
3 votes
1 answer
818 views

Smoothing subvarieties

Suppose I have a smooth complex projective variety $X$ and a singular subvariety $Z$. Can I find a general complete intersection subvariety $W$ of the same dimension as $Z$, and another smooth subvar …
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  • 135