Some properties of the decompositions of algebraic varieties determined by actions of a torus

Complete symmetric varieties

Several authors have been interested in the Hilbert scheme SPq:=Hilb"(S) parametrizing finite subschemes of length n on a smooth projective surface S. In EF 1] and I-F 2] Fogarty shows that S tnj is smooth and there exists a natural birational morphism co n: St"J~S("). Here S (n) is the n th symmetric power of S. So S t"j is a natural desingularisation of S (n). He then computes the Picard group of S tnl. In I-I 1] the highest dimensional fiber Hilb"(spec(k[I-x, y]])) of co n is studied, and in I-I 2] it is shown, that the expectional locus of con is an irreducible divisor. Fujiki has shown in I-Fj] that S t2J is a symplectic variety, if S is a K 3 surface, thereby disproving a conjecture of Bogomolov. In [B] Beauville generalizes this result to arbitrary n. Several authors have worked on the eohomology of St*J: The ease n___ 2 is trivial by the results of IF 2]. The homology groups of Hilb3(P2) have been computed in I-H]. Finally in [E-S] the homology groups of Hilbn(P2), Hilbn(A2), Hilb"(spee(k[[x,y]])), and Hilb~(27m) are computed. Here P2 is the projective plane, A 2 is the attine plane and Zm is the mth Hirzebruch surface. In this paper we will compute the Betti numbers ofS t"j for an arbitrary smooth surface S using the Weil conjectures. We want to state our main result: Let F+ be a finite field with q elements, l~a its algebraic closure. Let X be a smooth projective variety over C or over 1~.. In the first case let b~(X) be the i th Betti number of X. In the second ease let b+(X) ~ the rank of the i th l-adic cohomology group Hi(X, Ql) of X. In both eases let p(X, z) be the Poinear6 polynomial ~ b+(X)z + of X and e(X): = ~ (-1)/b,~X) the i t Euler number of X. We put ~X, z): = p ( X , z ) .

The Betti numbers of the Hilbert scheme of points on a smooth projective surface

Geir Ellingsrud 1 and Stein Arild Stromme 2 i Matematisk institutt, Universitetet i Oslo, Blindern, N-Oslo 3, Norway 2 Matematisk institutt, Universitetet i Bergen, N-5014 Bergen, Norway 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, 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(Ip 2) 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 de- fined 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.

/pdf/on-the-homology-of-the-hilbert-scheme-of-points-in-the-plane-1nxx4so15l.pdf

On the homology of the hilbert scheme of points in the plane

The product of two Schubert cohomology classes on a Grassmannian ${\rm Gr}_k (\mathbb{c}^n)$ has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (eg, the Littlewood-Richardson rule or the more symmetric puzzle rule from A Knutson, T Tao, and C Woodward [KTW]) Recently, W~Graham showed in [G], nonconstructively, that a similar positivity statement holds for {\em $T$-equivariant} cohomology (where the coefficients are polynomials) We give the first manifestly positive formula for these coefficients in terms of puzzles using an ``equivariant puzzle piece'' The proof of the formula is mostly combinatorial but requires no prior combinatorics and only a modicum of equivariant cohomology (which we include) As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinary-cohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the ``most equivariant'' case This formula is closely related to the one in A Molev and B Sagan [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G] We include a cohomological interpretation of their problem and a puzzle formulation for it

Puzzles and (equivariant) cohomology of Grassmannians

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Some properties of the decompositions of algebraic varieties determined by actions of a torus

Citations

Complete symmetric varieties

The Betti numbers of the Hilbert scheme of points on a smooth projective surface

On the homology of the hilbert scheme of points in the plane

Puzzles and (equivariant) cohomology of Grassmannians

Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field

Related Papers (5)

Some theorems on actions of algebraic groups

Equivariant cohomology, Koszul duality, and the localization theorem

Lectures on Hilbert Schemes of Points on Surfaces

The Yang-Mills equations over Riemann surfaces

Linear algebraic groups