Download e-book for kindle: Hilbert schemes of zero-dimensional subschemes of smooth by Lothar Göttsche

Algebraic Geometry

By Lothar Göttsche

During this ebook we research Hilbert schemes of zero-dimensional subschemes of tender kinds and several other similar parameter different types of curiosity in enumerative geometry. the most objective this is to explain their cohomology and Chow jewelry. a few enumerative purposes also are given. The Weil conjectures are used to compute the Betti numbers of some of the kinds thought of, therefore additionally illustrating how this strong device might be utilized. The booklet is largely self-contained, assuming just a easy wisdom of algebraic geometry; it really is meant either for graduate scholars and study mathematicians drawn to Hilbert schemes, enumertive geometry and moduli areas.

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7 Let (x, H ) ∈ Z . Then X H has an ordinary double point at X if and only if pr2 : Z → (P N )∗ is a immersion at the point (x, H ). 18. Taking a local chart C N ⊂ P N containing x, and taking normalised affine representatives for the homogeneous forms of degree 1 on P N in the neighbourhood of H , we can replace P N in the statement by C N , and (P N )∗ by the set K of affine forms 1≤i≤N αi xi + β on C N . Then, in the neighbourhood of (x, H ), Z ⊂ X × K is defined by Z = {(y, k) ∈ X × K | k(y) = 0, dk(y) = 0}, where k is viewed as the restriction to X of the affine function k on C N .

We locally choose a complex hypersurface Y ⊂ X passing through 0, of equation t = 0, where t is a holomorphic function with non-zero differential at 0. We apply the induction hypothesis to Y , which gives holomorphic coordinates z 1 , . , z n−1 on Y satisfying the conclusion of the lemma. Extending these functions to holomorphic functions on X , we obtain n−1 f − f (0) − z i2 = 0 i=1 on Y . Then we have n−1 f − f (0) − z i2 = tg, i=1 where g is holomorphic and vanishes at 0, since d f (0) = 0 = g(0)dt.

When the Hessian is non-degenerate, the local inversion theorem shows that in a neighbourhood of 0, the set χ −1 (0) of critical points of f is reduced to {0}. 8 We proceed by induction on n. If n > 0, then clearly we can find a hypersurface Y ⊂ X passing through 0, defined in the neighbourhood of 0, and smooth at 0, such that f |Y admits 0 as a non-degenerate 22 1 The Lefschetz Theorem on Hyperplane Sections critical point. Indeed, for this last condition to be satisfied, it suffices that the non-degenerate quadratic form Hess0 f remain non-degenerate on the hyperplane TY,0 ⊂ TX,0 .

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