Although several authors have been interested in the Hilbert scheme Hilba(IP z) parametrizing finite subschemes of length d in the projective plane ([I1, I2, F 1,.
Invent.math. 87, 343-352(1987)
///vent/ones
mathematicae 0 Springer-Verlag1987
On the homology of the Hilbert scheme of points in the plane Geir Ellingsrud 1 and Stein Arild Stromme 2 i Matematiskinstitutt,Universiteteti Oslo, Blindern,N-Oslo3, Norway 2 Matematiskinstitutt,Universiteteti Bergen,N-5014Bergen,Norway
Although several authors have been interested in the Hilbert scheme Hilba(IPz) parametrizing finite subschemes of length d in the projective plane ([I1, I2, F 1, F2, Br] among others) not much is known about the topological properties of this space. The Picard group has been calculated I-F2], and the homology groups of Hilb3(Ip2) have been computed [HI. In this paper we give a precise description of the additive structure of the homology of Hilba(Ip2), applying the results of Birula-Bialynicki [B1, B2] on the cellular decompositions defined by a torus action to the natural action of a maximal torus of SL(3) on Hilbe(IPZ). A rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the Chow groups and the homology groups are isomorphisms. In particular there is no odd homology, and the homology groups are all free. The main objective of this work is to compute their ranks: the Betti numbers of Hilba(Ipz). As a byproduct of our method we get similar results on the homology of the punctual Hilbert scheme and of the Hilbert scheme of points in the affine plane. It seems natural to generalize our results to any toric smooth surface. However, we give the results only for the rational ruled surfaces IF. with an indication of the necessary changes in the proofs. For simplicity we work over the field of complex numbers, but with an appropriate interpretation of the word "homology" our results remain valid over any base field.
w Let IP2 be the projective plane over ~E. For any positive integer d, let Hilba(lPz) denote the Hilbert scheme parametrizing finite subschemes of IP2 of length d. If A 2 denotes the complement of a line in IP 2, let Hilbe(A 1) denote the open subscheme of Hilba(IP2) corresponding to subschemes with support in A 2. Furthermore let Hilba(A 2, 0) be the closed subscheme of Hilba(A 2) parametrizing subschemes supported in the origin.
344
G. Ellingsrud and S.A. Stromme
For any complex variety X, let H,(X) be the Borel-Moore homology of X (homology with locally finite supports). By the i-th Betti number bi(X ) we shall mean the rank of the finitely generated abelian group H~(X). Let z ( X ) = zF(-1)ibi(X) be the Euler-Poincar6 characteristic of X. As usual, A,(X) is the Chow group of X, and cl: A , ( X ) ~ H , ( X ) is the cycle m a p (see [ F u ] Chap. 19.1). If m and n are non-negative integers, let P(m, n) denote the number of sequences n~bo>bl>...>b,~=O such that Ebi=m. If n>m, then P(m,n) =P(m), the number of partitions of m. Let P(m, n ) = 0 if m or n is negative. (1.1) Theorem. (i) Let X denote one of the schemes Hilbd(Ip2), Hilbd(A2), or Hilba(A2,0). Then the cycle map cl: A , ( X ) ~ H , ( X ) is an isomorphism, and in
particular the odd homology vanishes. Furthermore, both groups are free abelian groups. (ii) b2k(Hilbd(Ip2))
=
~.
~
P(p, d o -p)P(dl)P(2d z - r , r-d2)
do+dl + d 2 = d p + r = k - d l
and z(Hilba(IP2)) =
P(do) P(dx) P(d2).
~ doq-dl +d2= d
(iii) b2k(Hilbd(A Z))= P ( 2 d - k , k - d ) and
x(Hilbd(A 2))= e(d).
(iv) b2k(Hilb~(A 2, 0))=P(k, d - k ) and ~((Hilba(A 2, 0))=P(d).
Remark. The Betti numbers of Hilb3(Ip 2) were determined by Hirschowitz [H]. In Table 1 we have listed the Betti numbers of Hilbd(Ip 2) for 1 < d < 10. Table 1.
k d
0
1
1
1
1
2 3 4 5 6 7 8 9 10
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
2
3
4
5
6
7
8
9
10
3 5 6 6 6 6 6 6 6
6 10 12 13 13 13 13 13
13 21 26 28 29 29 29
24 39 49 54 56 57
47 74 94 105 110
83 131 167 189
150 232 298
257 395
440
The Betti numbers b2k(Hilbd(lp2)) are listed for l < d < 1 0 and O
