Related papers: Normal bundles of rational curves on complete inte…

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$,…

We study smooth complex projective polarized varieties $(X,H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by…

We consider all genus 2 curves over Q given by an equation y^2 = f(x) with f a squarefree polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200000 isomorphism classes of curves,…

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an…

We study vector bundles on curves with rational tails and their smoothings and give a sufficient condition for the general fibre to be balanced.

A $k$-nucleus of a normal rational curve in PG$(n,F)$ is the intersection over all $k$-dimensional osculating subspaces of the curve ($k\in\{-1,0,...,n-1\}$). It is well known that for characteristic zero all nuclei are empty. In case of…

Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case…

In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results…

Let C be a smooth complex projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k>1, we find two irreducible components of the space of rational…

Let X be a subvariety of $P^n$ defined by equations of degrees $ d =(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces of dimension…

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of…

In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve.…

We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We…

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$,…

As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves,…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q+1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q+1 points with intersection…

In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace…

Let X be a projective variety which is covered by a family of rational curves of minimal degree. The classic bend-and-break argument of Mori asserts that if x and y are two general points, then there are at most finitely many curves in that…

Given $\eta=\begin{pmatrix} a&b\\c&d \end{pmatrix}\in \text{GL}_2(\mathbb{Q})$, we consider the number of rational points on the genus one curve \[H_\eta:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)+d(2x))^2.\] We prove that the set of $\eta$ for which…