Related papers: Recovering the good component of the Hilbert schem…
Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $\omega$ and a projection onto $\mathbb{C}$. We provide a full…
We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not…
We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of…
This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and…
We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…
An example is given of a compact absolute retract that is not a Hilbert cube manifold but whose second symmetric porduct is the Hilbert cube. A factor theorem is given for nth symmetric product of the cartesian product of any absolute…
X(5) is a paradigm for the structure at the critical point of a particular first-order phase transition for which the intrinsic energy surface has two degenerate minima separated by a low barrier. For a finite system, we show that the…
Given a complex smooth algebraic variety X, we compute the generating function of the stringy motives of its symmetric powers as a function of motive of X. In dimension two we recover the Goettsche formulas for Hilbert schemes. We use the…
An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of…
This paper is concerned with blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system. We first derive the precise blow-up scenario for strong solutions to the system. Then, we present several new blow-up…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this…
We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second…
We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of…
For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf…
By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\Hilb^n(X)$ can be identified, for all $n\geq 1$, with moduli spaces of Gieseker stable vector…
For a smooth projective variety $X$ with exceptional structure sheaf, and $\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X)…
At a critical point of a second order phase transition the intrinsic energy surface is flat and there is no stable minimum value of the deformation. However, for a finite system, we show that there is an effective deformation which can…
In the present paper, we give a full description of the jet schemes of the polynomial ideal $\left( x_1\ldots x_n \right) \in k[x_1, \ldots, x_n]$ over a field of zero characteristic. We use this description to answer questions about…
Let $p(t)$ be an admissible Hilbert polynomial in $\PP^n$ of degree $d$. The Hilbert scheme $\hilb^n_p(t)$ can be realized as a closed subscheme of a suitable Grassmannian $ \mathbb G$, hence it could be globally defined by homogeneous…