Related papers: Lines, conics, and all that

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or…

Fano varieties are subvarieties of the Grassmannian whose points parametrize linear subspaces contained in a given projective variety. These expository notes give an account of results on Fano varieties of complete intersections, with a…

We classify smooth Fano weighted complete intersections of large codimension.

This note is about cycle-theoretic properties of the Fano variety of lines on a smooth cubic fivefold. The arguments are based on the fact that this Fano variety has finite-dimensional motive. We also present some results concerning Chow…

For a Fano manifold of pseudo-index at least 3 and $c_1^2-2c_2$ nef, we show irreducibility of certain spaces of curves on the Fano manifold implies the manifold is a union of rational surfaces.

The connection between these Fano 3-folds and plane quartic curves is explained.

We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.

We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.

We study the Fano scheme of $k$-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be…

Small codimensional embedded manifolds defined byequations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This…

In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P^{2m+2}, for m>0. We show that there…

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…

The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove…

Let $X^o=\mathbb P^3\setminus D$ where $D$ is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface $V$ and a plane…

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…

This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a system of partial differential equations which we describe…

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the…