Algebraic Geometry

Let X be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If X is a smooth complex variety, then the geometric invariant theory quotient X//G can be identifed with the symplectic reduction X r . Lerman introduced a construction (valid for symplectic manifolds) called symplectic cutting, which constructs a manifold X c , such that X c is the union of X r and an open subset X >0 ⊂X . Moreover, there is a natural torus action on X c such that X r is a component of the fixed locus. Using localization for equivariant cohomology, this construction can be used to study of X r . In this note, we give an algebraic version of this construction valid for projective but possibly singular varieties defined over arbitrary fields. This construction is useful for studying X r from the point of view of algebraic geometry, using the equivariant intersection theory developed by the authors. At the end of the paper we briefly give an adaptation of Lerman's proof of the Kalkman residue formula and use it to give some formulas for characteristic numbers of quotients by a torus.

For a real Enriques surface Y we prove that every homology class in H_1(Y(R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface.

Let M be a compact hyperkaehler manifold. The hyperkaehler structure equips M with a set R of complex structures parametrized by C P 1 , called "the set of induced complex structures". It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in R . A more general version of this theorem was proven by Fujiki.

We show that on real algebraic sets algebraically constructible functions coincide with the finite sums of signs of polynomials. Then we give some applications.

An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.

In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of D-modules.

We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.

For Gorenstein quotient spaces C d /G , a direct generalization of the classical McKay correspondence in dimensions d≥4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry.

We give a condition for certain divisors on the blow up of P^3 at points in general position to be ample. The result extends a theorem of G. Xu on the blow up of the projective plane.

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain twists of powers of the ideal sheaf of Y. This is applied to get rather non-trivial vanishing in case Y is a universal determinantal variety (i.e. quadratic Veronese, Segre embedding or Grassmannian of P^1's) or a curve embedded by a complete linear series of high degree. The key in the determinantal cases is to use the theory of complete objects. While the theories of complete quadrics and linear maps yield the desired results, a corresponding theory of complete skew forms seems not have been previously worked out. The desired results are obtained in an appendix.