Abstract:
In this paper we determine the structure of the Chow ring of the Delaunay-Voronoi compactification $\tilde{\cal A}_3$ of the moduli space of principally polarized abelian threefolds. This compactification was introduced by Namikawa and studied by Tsushima. We use equivariant classes on level coverings of $\tilde{\cal A}_3$. We also compare this ring with the Chow ring of the moduli space of stable genus 3 curves as determined by Faber.

Abstract:
After recalling several constructions of the moduli space of curves of genus zero by different people we give our alternative construction of the moduli space. This gives a simple description of the intersection ring of this space. We give a basis for the Chow groups and an explicit duality between the Chow groups in complementary degrees. There is a recursive description of this algebra by S. Keel. Our presentation is simpler in the sense that there are fewer generators and fewer relations and our description is explicit.

Abstract:
We study the ring of characteristic classes with values in the Chow ring for principal $G$-bundles over schemes. If we consider bundles which are locally trivial in the Zariski topology, then we show, for $G$ reductive, that this ring is isomorphic to the Weyl group invariants in the algebra generated by characters of the maximal torus. For general principal bundles the same isomorphism holds after tensoring the coefficients with ${\Bbb Q}$. As a corollary, we show that any (non-torsion) topological characteristic class is algebraic when applied to Zariski locally trivial bundles over complex algebraic varieties.

Abstract:
We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.

Abstract:
We give a full description of the Chow ring of the complex Cayley plane, the simplest of the exceptional flag varieties. We describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions $Y_8$ introduced in our related preprint math.AG/0306328.

Abstract:
We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex cobordism ring of the classifying space, a topological invariant. We apply this to get torsion information on the Chow groups of varieties defined as quotients by finite groups. This generalizes Atiyah and Hirzebruch's use of such varieties to give counterexamples to the Hodge conjecture with integer coefficients.

Abstract:
Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring A\_T^*(H^d(X))\_Q and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.

Abstract:
In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.

Abstract:
Let H(d) be the (open) Hilbert scheme of rational normal curves of degree d in P^d. A presentation of the integral Chow ring of H(d) is given via equivariant Chow ring computations. Included also in the paper are algebraic computations of the integral equivariant Chow rings of the algebraic groups O(n), SO(2k+1). The results for S0(3)=PGL(2) are needed for the Hilbert scheme calculation.