The punctual Hilbert scheme

Let R =, . xj]] be the field of formal power series in d variables over a 

If T = ( i)i o is a sequence of non-negative integers, of which only finitely many do not vanish, we put T = The initial degree do of T is the smallest i such that i, ( - ). 

As Z is finite dimensional, there exists an to with Z,0 = 0. For such an io we have I 3 m °. There is an isomorphism 

In a similar way one can prove Let X be a smooth projective variety over an algebraically closed field k. Let x X be a point and Z C X subscheme of length n with supp(Z) = x. Let Iz,x be the stalk of the ideal of Z at X. Then we have 

Remark 1.3.3. As every ideal of colength n in R contains m , we can regard it as an ideal in R/m . Thus the Hilbert scheme Hilb (fZ/mn) also parametrizes the ideals of colength n in R. We also see that the reduced schemes (Hilbn(ii/m ))re,j are naturally isomorphic for k n. We will therefore denote these schemes also by Hilb (R)rei. Hilbn(R)red is the closed subscheme with the reduced induced structure of the Grassmannian Grass(n, R jmn) of n dimensional quotients of R/mn whose geometric points are the ideals of colength n of k[[xi. z j]]/mn. 

Definition 1.3.4. Let T — ( ), o be a sequence of non-negative integers with T = n. Let Zt C Hilbn(f )red be the locally closed subscheme (with the reduced induced structure) parametrizing ideals I C R with Hilbert function T. Let Gt C Zt be the closed subscheme (with the reduced induced structure) parametrizing homogeneous ideals I C R with Hilbert function T. Let 

---

In section 2.2 we consider the punctual Hilbert schemes Hilbn(fc[[x, y]]). We give a cell decomposition of the strata and so determine their Betti numbers. I have published most of the results of this section in a different form in [Gottsche (3)]. They have afterwards been used in [Iarrobino-Yameogo (1)] to study the structure of the cohomology ring of the Gt- We also recall the results of [Ellingsrud-Str0mme (1),(2)] on a cell decomposition of Hilb"(fc[[x,j/]]) and P. ... 

Another proof of the irreducibility of the punctual Hilbert scheme of a smooth surface, preprint 1992. 

Algebraic families on an algebraic surface II, the Picard scheme of the punctual Hilbert scheme, Amercan Journal of Math. 95 (1973), 660-687. 

Betti numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables, Manuscripta Math. 66 (1990), 253-259. 

In chapter 1 we recall some fundamental facts, that will be used in the rest of the book. First in section 1.1, we give the definition and the most important properties of Xfnl then in section 1.2 we explain the Weil conjectures in the form in which we are later going to use them in order to compute Betti numbers of Hilbert schemes, and finally in section 1.3 we introduce the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety. We hope that the non-expert reader will find in particular sections 1.1 and 1.2 useful as a quick reference. 

An intersection number for the punctual Hilbert scheme of a surface, preprint, alg-... 

C17] [fog 2] J. FOGARTHY.- The Picard scheme of the punctual Hilbert Scheme. American Journal of Mathematics (1973). 

---