The valuative criterion for locally closed immersions
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After discussing background on valuative criteria in algebraic geometry, I'll present the little-known valuative criterion for locally closed immersions. Time permitting, I'll discuss the following applications of this criterion:
1. The moduli stack of hyperelliptic genus g curves is a locally closed substack of the the moduli stack of principally polarized abelian varieties when g is at least 2 and 2 is invertible on the base.
2. The moduli stack of degree d plane curves is a locally closed substack of moduli stack of curves for d at least 4.
Also come if you want to find out what the valuative criterion for locally closed immersions has to do with Brian Conrad's passion for libraries.