# What do generic Torelli theorems claim?

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?

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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).

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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 '13 at 19:46
Thank you Dan for clarifying my understanding. –  Kim Mar 2 '13 at 10:42