By Lothar Göttsche
During this ebook we research Hilbert schemes of zero-dimensional subschemes of delicate types and a number of other similar parameter sorts of curiosity in enumerative geometry. the most goal 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, hence additionally illustrating how this robust device might be utilized. The e-book is largely self-contained, assuming just a uncomplicated wisdom of algebraic geometry; it's meant either for graduate scholars and study mathematicians attracted to Hilbert schemes, enumertive geometry and moduli areas.
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Extra resources for Hilbert schemes of zero-dimensional subschemes of smooth varieties
5 Supplement: Example of computation of discriminant loci 55 l ′ ′ ck tk (z0n ζ0n )a−k = 0, f (z0 , ζ0 , s, t) = (z0n ζ0n )a − s + k=1 and then we obtain l a t α −s+ a−k t α ck tk k=1 = 0, that is, s= t α l a a−k t α ck tk + k=1 . Therefore we have s = βta , where we set β := 1 + αa l ck k=1 1 αa−k . This shows the validity of (A). Next we show (B). We already proved that Xs,t is non-reduced when s = 0. Thus we consider the case s = βta , for which we shall show that the defining equation of Xs,t admits a factorization with a ′ multiple factor (z n ζ n − t/α)d for some d ≥ 2, which implies that Xs,t is nonl k , the defining equation reduced.
In the process of the deformation, Y : y n = 0 is barked off from X to become Yt : y n + txa = 0. 4. 5 (1) and (2). Note that in (2), the barked part y + tx2 = 0 is a parabola (the name “parabolic” barking comes from this). 6. 2 Barking, II (1) (2) y + tx = 0 1 m′ 1 x-axis 0 m−1 37 y + tx2 = 0 m′ x-axis 0 m−1 X0,t X0,t Fig. 5. Two examples of X0,t for parabolic barkings when (1) n = 1 and a = 1 and (2) n = 1 and a = 2 respectively. deform m X x-axis n −→ 1 1 1 gcd(n, a) 0 m−n x-axis Y X0,t Fig.
Yˇi , . . , Yl . Here Yˇi is the omission of Yi . 3 Herein, a plane curve singularity always means a reduced one. 6 Let π : M → ∆ be normally minimal such that the singular fiber X contains an irreducible component Θ0 of multiplicity 1. Let π1 : W1 → ∆ be the restriction of π to a tubular neighborhood W1 of X \Θ0 in M . Suppose that π1 : W1 → ∆ admits a splitting family Ψ1 which splits Y + := W1 ∩X into Y1+ , Y2+ , . . , Yl+ . Then π : M → ∆ admits a splitting family Ψ which splits X into X1 , X2 , .
Hilbert schemes of zero-dimensional subschemes of smooth varieties by Lothar Göttsche
Categories: Algebraic Geometry