Related papers: Chern classes of the tangent bundle on the Hilbert…

We give formulas for the extension groups between tautological sheaves and more general between tautological objects twisted by a determinant line bundle on the Hilbert scheme of points on a smooth quasi-projective surface. We do this using…

We develop a formula for tautological integrals over geometric subsets of the Hilbert scheme of points on complex manifolds. As an illustration of the theory, we derive a new iterated residue formula for the number of nodal curves in…

We use semi-orthogonal decompositions to construct autoequivalences of Hilbert schemes of points on Enriques surfaces and of Calabi-Yau varieties which cover them. While doing this, we show that the derived category of a surface whose…

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points in the complex plane. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum…

We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the…

We define a subcategory of the category of diffeological spaces, which contains smooth manifolds, the diffeomorphism subgroups and its coadjoint orbits. In these spaces we construct a tangent bundle, vector fields and a de Rham cohomology.

We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on the complex projective plane. The top non-vanishing Chern classes of the cohomology of these complexes yield actions of the…

We compute the cohomology of the right generalised projective Stiefel manifolds and use it to find bounds on the rank of the complementary bundle for certain vector bundles. Further the cohomology computations are also used to find bounds…

Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct…

We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly…

We revisit Ellingsrud and Str{\o} mme's cellular decomposition of the Hilbert scheme of points in the projective plane. We study the product of cohomology classes defined by the closures of cells, deriving necessary conditions for the…

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $S$ be a smooth projective surface over $k$ with $k$-rational point $x$. For $n \geq 2$, let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we…

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on…

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms…

We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric…

Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an…

The quotient of a finite-dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety. Under appropriate assumptions, the McKay correspondence relates the geometry of nice resolutions of…

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old…

We introduce and study the Chern filtration on the cohomology of the moduli of bundles on curves. This can be viewed as a natural cohomological invariant defined via tautological classes that interpolates between additive Betti numbers and…