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Started by Charles Yeomans
Tue, 03 Nov 1992 12:52
Fano varieties
Author: Charles Yeomans
Date: Tue, 03 Nov 1992 12:52
Date: Tue, 03 Nov 1992 12:52
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I have seen a couple of ways to define a gadget called a Fano variety. First, let V be a variety in projective space. Then one may define F_k(V) to be the space of k-dimensional planes contained in V. Clearly, this variety is a subvariety of a Grassmann variety. Second, I have seen a Fano variety defined to be a variety for which -1 times the canonical class is ample. My question is, should these two definitions define the same object? If so, any hints as to how one might see that? Charles Yeomans cyeomans@ms.uky.edu yeomans@austin.onu.edu
Re: Fano varieties
Author: Valery A. Alexeev
Date: Tue, 03 Nov 1992 19:12
Date: Tue, 03 Nov 1992 19:12
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Answer: the only connection: Fano is the author of both gadgets. They couldn't mean the same thing: For the first definitition take any variety X and 0-planes, contained in it, you get X again. If you don't like 0-dimensional planes, then take a Veronese embedding of X and then a cone over it, or a scroll. F_1(Y) will be evidently isomorphic to X. You can take a cone again or P^2-fibration to get 2-planes and so on. I guess Fano studied lines in a 3-dimensional cubic. They are parametrized by a so called Fano surface (of this cubic). In the second definition on the other hand we have very specific varieties. V.
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