Welcome to MathOverflow

Question

4

I have a question on "Torelli theorems" in algebraic geometry. "Torelli theorems" say about how the period map of a given family of varieties behave. If I understand correctly, global Torelli theorems are the most strong ones and basically classify the varieties you have, that is, the period map is isomorphism. Local Torelli theorems state that the differential of the period map is injective. What do generic Torelli theorem claim then?

flag	asked Feb 23 at 10:40 Kim 247●3

Dan Petersen · Accepted Answer · 2013-02-23 11:56:00Z UTC

3

I would formulate it as follows: global Torelli says that the period map is injective/an immersion (but not necessarily an isomorphism!!), local Torelli says that the period map has injective differential, and generic Torelli says that the period map is generically injective (injective on a Zariski open dense set).

answered Feb 23 at 11:56

Dan Petersen
10.1k●1●21●46

2

I agree with Dan. What is often proved for generic Torelli (also called "degree one Torelli"), is only that some proper extension of the map has a regular value with a single preimage, or that some blowup of some proper extension has such a value. A related approach argues that a generic source point is determined by the image of the differential there. In all these approaches, a precise description of a dense Zariski open set on which the map must then be injective often remains unknown. (Some people say "infinitesimal Torelli" instead of "local Torelli" for "injective differential".) – roy smith Feb 23 at 19:46

Thank you Dan for clarifying my understanding. – Kim Mar 2 at 10:42

What do generic Torelli theorems claim?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.