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Thanks for the followup! Is Castelnuovo-Mumford regularity the key point in both references, or just Mumford's? (e.g. is Fujita vanishing sufficient for Lazarsfeld's argument?)
Just a morphism which is surjective on the underlying topological spaces? E.g. the "faithful" part of "faithfully flat". (this is the Stacks project definition, which I would hope is also the one in EGA) I guess surjective on closed points is another reasonable criterion, but I'm only thinking about schemes that are of finite type over a field, so these should be equivalent (right?)
Wow - great answer! I'm not so familiar with Picard and Hilbert schemes (although I want to be!), so I'm not so sure how to think about the argument that two divisors in the same component of the Picard scheme are algebraically equivalent. It seems like $H^1(X, \mathcal{O}_X)/H^1(X, \mathbb{Z})$ is a very analytic object (integer cohomology), so it's weird to me to think of it as a variety. Of course, it's a complex torus, so we can connect any two points with some complex line, but this may not be algebraic (and also is only genus 0!). Is there an elementary way to see what's going on?
Ivan - how did you get that $(\nabla_X \alpha)(Y) = \nabla_X(\alpha(Y)) - \alpha \nabla_X(Y)$? I'm considering $\nabla$ just as a vector bundle connection on the vector bundle $T^* M$, not necessarily as an affine connection on $TM$.
