Related papers: Smooth components on special iterated Hilbert sche…

Let $(S,p)$ be a smooth pointed surface. In the first part of this paper we study motivic invariants of punctual nested Hilbert schemes attached to $(S,p)$ using the Hilbert-Samuel stratification. We compute two infinite families of motivic…

We continue the study of maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s > 3, we extend the two ranges where W is a…

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert…

The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and…

If $X$ is a smooth scheme of characteristic zero or a complex analytic manifold, and $S$ is a locally split infinitesimal thickening of $X$, we compute explicitly the derived self-intersection of $X$ in $S$.

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their…

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the…

We discuss the parabolic Hilbert scheme of points on smooth surface, which is an interesting generalization of Hilbert scheme of points on surface. Some of the studies by Ellingsrud-Stromme, Goettsche, Cheah, Nakajima and Grojnowski can be…

We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works…

To produce cartographic maps, simplification is typically used to reduce complexity of the map to a legible level. With schematic maps, however, this simplification is pushed far beyond the legibility threshold and is instead constrained by…

We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which…

In this paper we determine the irreducible components of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g. We show that these Hilbert schemes are connected, in spite of having…

For infinitely many $d$, Hassett showed that special cubic fourfolds of discriminant $d$ are related to polarized K3 surfaces of degree $d$ via their Hodge structures. For half of the $d$, each associated K3 surface $(S,L)$ canonically…

For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant…

The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of…

We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be…

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant…

We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this…

We construct a linearly normal smooth rational surface S of degree 11 and sectional genus 8 in the projective fivespace. Surfaces satisfying these numerical invariants are special, in the sense that $h^1(\mathscr{O}_S(1))>0$. Our…