Related papers: The Hilbert scheme of 11 points in A^3 is irreduci…

We show that every automorphism of the Hilbert scheme of $n$ points on a weak Fano or general type surface is natural, i.e. induced by an automorphism of the surface, unless the surface is a product of curves and $n=2$. In the exceptional…

We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3-folds and the Hom scheme…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…

We find an explicit geometric description of all coverings of the Hilbert square on a normal, complex, quasi-projective surface with finite fundamental group. We then apply this construction to show that if $\Sigma$ is an irreducible…

We prove the existence of Ulrich sheaves on the Hilbert scheme of two points on a polarized K3 surface or an abelian surface. The construction proceeds by descending Ulrich bundles on the surface to the symmetric square and lifting them to…

We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational…

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on…

Let $S$ be a smooth projective surface over $\mathbb{C}$ and $S^{[n]}$ be the Hilbert scheme of $n$ points over $S$, for any positive integer $n$. Let ${\bf a}=(n_1,\ldots,n_r)$ and ${\bf b}=(m_1,\ldots,m_s)$ be two distinct partitions of…

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the…

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…

In this paper, we show the moduli spaces of stable sheaves on K3 surfaces are irreducible symplectic manifolds, if the associated Mukai vectors are primitive. More precisely, we show that they are related to the Hilbert scheme of points. We…

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this…

We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we…

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…

The present paper provides several results on automorphisms of hyperk\"ahler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic…

We give an algebraic construction of orientation data for the Hilbert scheme of 0-dimensional subschemes of a smooth projective threefold endowed with a choice of square root of the canonical sheaf.

Let $X$ be a very general degree $d\geq 5$ hypersurface in $\mathbb{P}^3$. We compute the ample cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for various small values of $n$ (the answer is already known for large $n$). We obtain…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from…

For $\gamma \geq 7$ and $g \geq 6\gamma + 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3\gamma+1}$ over smooth non-degenerate curves of genus $\gamma$ and degree $g-2\gamma$ in…