Related papers: Characterizations of projective spaces and hyperqu…

For a nonsingular hypersurface $X \subset \mathbb{P}^n, n \geq 4,$ of degree $d \geq 2$, we show that the space $H^1(X, \End(T_X))$ of infinitesimal deformations of the tangent bundle $T_X$ has dimension ${n+d-1 \choose d} (d-1)$ and all…

In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.

Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the…

We study complex compact Kaehler manifolds $X$ carrying a contact structure. If $X$ is almost homogeneous and $b_2(X) \geq 2$, then $X$ is a projectivised tangent bundle (this was known in the projective case even without assumption on the…

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian…

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point…

Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $\xi$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $\chi(E\otimes F)= 0$ for $E \in \mathcal M$.…

Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual…

Let $X$ be a complex smooth projective variety such that the exterior power of the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. We prove that, up to an \'etale cover, $X$ is a Fano fiber space over an Abelian…

Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension…

We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$…

We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a…

Let $E$ be the Whitney sum of complex line bundles over a topological space $X$. Then, the projectivization $P(E)$ of $E$ is called a \emph{projective bundle} over $X$. If $X$ is a non-singular complete toric variety, so is $P(E)$. In this…

We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.

Using Hilbert schemes of points, we establish a number of results for a smooth projective variety $X$ in a sufficiently ample embedding. If $X$ is a curve or a surface, we show that the ideals of higher secant varieties are determinantally…

In this article, we prove that any smooth projective variety $X$ which is a double cover of the projective space $\mathbb{P}^n$ ($n\geq 2$) admits an Ulrich bundle. When $n=2$, we show that on any such $X$, there is an Ulrich bundle of rank…

The aim of this note is to exhibit explicit sufficient criteria ensuring bigness of globally generated, rank-$r$ vector bundles, $r \geqslant 2$, on smooth, projective varieties of even dimension $d \leqslant 4$. We also discuss connections…

We show that the moduli space of SU_X(r,L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact K\"ahler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang…

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to…