Related papers: Classes caracter\'isticas e secantes de curvas rac…

This is a detailed write-up of Koll\'ar's course at the EMS summer school in Algebraic Geometry in Eger, Hungary, August 1996. The topics include definitions and examples of rational and unirational varieties, with special attention to…

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert…

We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the…

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067,…

In this paper, extending some ideas of Fano, we study the birational geometry of the Hilbert scheme of 0-dimensional subschemes of length 2 of a rational normal scroll. This fourfold has three elementary contractions associated to the three…

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal…

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $J\colon \Cl(\mathbb R^{r,s})\toU$ be a representation of the Clifford algebra $\Cl(\mathbb R^{r,s})$…

In this paper we study the degrees of irrationality of hypersurfaces of large degree in a complex projective variety. We show that the maps computing the degrees of irrationality of these hypersurfaces factor through rational fibrations of…

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us…

In this paper, we give a full classification of the separable hypersurfaces of constant sectional curvature in the Euclidean $n$-space $\mathbb{R}^n$. In dimension $n=3$, this classification was solved by Hasanis and L\'opez [Manuscripta…

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the…

We study the link between a compact hypersurface in $\P^{n+1}$ and the set of all its tangent planes. In this context, we identify $\P^{n+1}$ to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise…

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and…

A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $\psi: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree…

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y…

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…

First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that…