Saturday, 24 June 2006

ag.algebraic geometry - Family of Enriques surfaces and GRR, Part 2

Q1: It is the other way round. For a smooth family the differential TYtofastTT is surjective and the relative tangent is the kernel, so you have an exact sequence



0toTftoTYtofastTTto0.



In this way the tangent to f actually restricts to the tangent of the fibers.



Q2: I don't think that the classes ci(Tf) are determined by the fibers alone; they depend on the family. It does not even make sense to say that ci(Tf) are determined by the fibers since these classes live in H2i(Y) anyway, so you have to know at least the total space.



But since Tf restricts to the tangent of the fibers, you know, by naturality of the Chern classes, that if icolonEtoY is the inclusion of a fiber iastci(Tf)=ci(TE).



And these you can compute using the fact that E is Enriques. Namely 2c1(TE)=0 since twice the canonical is trivial and c2(TE)=chi(E)=12.



Q3: Surely it is not injective in the top degree, for trivial dimensional reasons. I do not see any reason why it should be in other degrees.



Q4: As is written in the article, this follows from fastc2=12. This is more or less clear in cohomology. In this case f is the integration along fibers, and since c2(TE) is 12 times the fundamental class of E for all fibers E (see Q2), that integral is 12.



To translate this in the Chow language, I think the folllowing will do. Let D be a cycle representing c2(Tf). Since Y is smooth, we can compute the intersection number DcdotE=c2(Tf)capE=c2(TE)capE=12. So D intersectts the generic fiber in 12 points, and the morphism DtoT has degree 12. In follows that fastD=12[T], which is what you want.

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