By Lothar Gottsche

During this publication we learn Hilbert schemes of zero-dimensional subschemes of gentle types and a number of other similar parameter sorts of curiosity in enumerative geometry. the most target here's to explain their cohomology and Chow earrings. a few enumerative functions also are given. The Weil conjectures are used to compute the Betti numbers of a few of the types thought of, hence additionally illustrating how this strong instrument could be utilized. The e-book is largely self-contained, assuming just a uncomplicated wisdom of algebraic geometry; it truly is meant either for graduate scholars and examine mathematicians attracted to Hilbert schemes, enumertive geometry and moduli areas.

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**Sample text**

Then (G, F ) has the Lang property by the theorem of Lang. 4. (c]. 4]). Let G be a group and H, H' two commuting automorphism8 of G such that both ( G , H ) and (G,H') have the Lang property. Then Theorem (1) For all y E G we have y - i l l ( y ) e G H. 2. The Betti numbers of Hilbert schemes 42 (2) The map NH/H, : y - i l l ( y ) H yH'(y -1) definies a bijection from the set of H-classes of G H' to the set of H'-cla~aes of G H. 5. ) a partition of n. We write as above Ic~I := E c~i, and put Iv I := I~1 (obviously Iv[ = r).

By E. 2(aj-aj+l)=2 E O* s >_ aj+l. Let [xJ+l-iy s+a'-a'] r : I ----+ R / I ; xly ~ ' ) 0 if l _< i, otherwise. We can see immediately that the compatibility conditions are fulfilled and r a common eigenvector of F to the eigenvalue AJ - i # ~-a~. *

Z~], where Z 1 , . . , Z ~ are the connected components of Z and [] the class modulo G(n). 4. (~). e. Vo(k) = V0(~) = {o}. For x E V~(Ic) we put len(x) := r. For a map f : X ( k ) ~ V(k) we put Ion(f) := ~ len(f(x)). ~x(~) Gal(k, k) acts on these maps by o-(f) := crofoo--1. 3. The Betti numbers of S ["1 31 We write fl C f2, if fl(x) C f2(x) for all x 9 x ( k ) . 5. There exists a sequence of bijections commuting with the action of Gal(-k/k) such that Z1 C Z2 ~ Olen(Z,)(Z1) C Olen(Z2)(Z2). P r o o f : Let v = ( r t l , .