Related papers: Recovering the good component of the Hilbert schem…

Van den Bergh has defined the blowup of a noncommutative surface at a point lying on a commutative divisor. We study one aspect of the construction, with an eventual aim of defining more general kinds of noncommutative blowups. Our basic…

Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. We show that if X is Frobenius split then so is the Hilbert scheme Hilb^n(X) of n points in X. In particular, we get the higher…

Let $S$ be a smooth projective surface, and $\hat{S}$ be its blow-up at a point. In this paper, we study the derived category of the Hilbert scheme of points on the blow-up $\hat{S}$. We obtain a semi-orthogonal decomposition consisting of…

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture…

The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces…

Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument…

Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define…

A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a…

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme…

Let $\mathcal{H}_{a,b}^n$ denote the component of the Hilbert scheme whose general point parameterizes an $a$-plane union a $b$-plane meeting transversely in $\mathbf{P}^n$. We show that $\mathcal{H}_{a,b}^n$ is smooth and isomorphic to…

Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define…

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…

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the…

We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of…

We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gr\"obner scheme. We introduce a Bia{\l}ynicki-Birula decomposition of the Hilbert scheme $\mathrm{Hilb}^{P}_n$ for any Hilbert polynomial $P$ such that the…

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme…

We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral…

In this paper, we give new explicit representations of the Hilbert scheme of $\mu$ points in $\PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous…

We study the component H_n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in P^n for n > 2. We show that H_n is smooth and isomorphic to the blow-up of the symmetric square of G(n-2,n)…

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective…