Related papers: Severi varieties and holomorphic nilpotent orbits

Making use of the `Veldkamp blow-up' recipe, introduced by Saniga and others (Ann. Inst. H. Poincar\' e D2 (2015) 309) for binary Segre varieties, we study geometric hyperplanes and Veldkamp lines of Segre varieties $S_k(3)$, where $S_k(3)$…

In this paper, we study the Severi varieties parametrizing integral curves of geometric genus one on polarized toric surfaces in characteristic zero and describe their irreducible components. We show that the irreducible components are in…

For a given singularity of a plane curve we consider the locus of nodal deformations of the singularity with the given number of nodes and describe possible components of the locus. As applications, we solve the local symplectic isotopy for…

We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions…

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of…

The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given…

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of…

Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a…

We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties.…

We study the problem of the irreducibility of the Hessian variety $\mathcal{H}_f$ associated with a smooth cubic hypersurface $V(f)\subset \mathbb{P}^n$. We prove that when $n\leq5$, $\mathcal{H}_f$ is normal and irreducible if and only if…

Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred…

We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy: K^2_S >= 4\chi(S).

Hypertoric varieties are hyperk\"ahler analogues of toric varieties, and are constructed as abelian hyperk\"ahler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold…

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb{V} = R^{2k} f_{*} \mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$…

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…

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An…

We complete the construction of the fundamental diagram of various partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of…