相关论文: Severi Degrees in Cogenus 3
According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials"…
In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree at least three from its inflection lines? We answer positively to the posed question for a general smooth plane quartic curve, making…
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive…
We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the…
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational…
We compute the Hodge numbers of the moduli space of semi-stable sheaves on the complex projective plane supported on quintic curves and having Euler characteristic 3. For this purpose we study the fixed-point set for a certain torus action…
We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.
We compute the degree of the projective variety of Poncelet curves of degree $n$. This variety is irreducible of dimension $2 n + 5$, and lies inside the projective space of degree $n$ plane curves. It is classically defined as the closure…
We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with…
Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected…
We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for…
Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a…
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and…
We describe the hyperplane sections of the Severi variety of curves in $E \times \mathbb{P}^1$ in a similar fashion to Caporaso-Harris' seminal work. From this description we almost get a recursive formula for the Severi degrees (we get the…
We enumerate the number of surfaces of degree $d$ in $P^3$ having a singular line of order $k$, passing through $\delta$ generic points (where $\delta$ is the dimension of moduli space of such surfaces).
We further analyze the moduli space of stable curves with level structure provided by Chiodo and Farkas in \cite{AA}. Their result builds upon Harris and Mumford analysis of the locus of singularities of the moduli space of curves and shows…
We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on…
Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra.…
We determine the class of the Hilbert scheme of points on a surface in the Grothendieck group of varieties. As a corollary we obtain its class in the Grothendieck group of motives. We give some applications to moduli spaces of sheaves on…
We study cup products in integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question, which classes can be represented by products of…