Rational curves on general projective hypersurfaces

Let $k$ be an integer such that $1\leq k\leq n-5$, and $X_{2n-2-k}\subset \mathbf P^n$ a general projective hypersurface of degree $d=2n-2-k$. In this paper we prove that the only $k$-dimensional subvariety $Y$ of $X_{2n-2-k}$ having geometric genus zero is the one covered by the lines. As an immediate corollary we obtain that, for $n>5$, the general $X_{2n-3}\subset \mathbf P^n$, contains no rational curves of degree $\delta >1$.Comment: Final version to appear in the Journal of Algebraic Geometry. Exposition improved, according to referee's suggestions. 26 pages, Late

Subvarieties of general type on a general projective hypersurface

We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf P^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface $X_{d}\subset {\mathbf P}^n$, for $n\geq 6$ and $d\geq 2n-2$, is of general type.Comment: Title changed, introduction completely rewritten, exposition improved. To appear in the Transactions of the A.M.S. 18 pages, Late

On the uniformity of the Iitaka fibration

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration of smooth projective varieties of positive Kodaira dimension, provided that the base of the Iitaka fibration is not uniruled, the variation of the fibration is maximal, and the generic fiber has a good minimal model.Comment: 18 page

Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

We study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that any ample linear system on a projective holomorphic symplectic variety of K3[n]-type contains a uniruled divisor. As an application we provide a generalization of the Beauville-Voisin result on the Chow group of 0-cycles on such varieties.Comment: The results are corrected and completed in arXiv:1907.10970, which replaces the pape

On the log minimal model program for irreducible symplectic varieties

We show the termination of any log-minimal model program for a pair $(X,\Delta)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $\Delta$

Stability of coisotropic fibrations on holomorphic symplectic manifolds

We investigate the stability of fibers of coisotropic fibrations on holomorphic symplectic manifolds and generalize Voisin's result on Lagrangian subvarieties to this framework. We present applications to the moduli space of holomorphic symplectic manifolds which are deformations equivalent to Hilbert schemes of points on a $K3$ surface or to generalized Kummer manifolds.Comment: Minor changes in the Introduction and in the body of the paper to improve the presentation of the result

Rational connectedness modulo the Non-nef locus

It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair (X,\D) such that -(K_X+\D) is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair (X,\D) such that -(K_X+\D) is big is rationally connected modulo the non-nef locus of -(K_X+\D). This result is a consequence of a more general structure theorem for arbitrary pairs (X,\D) with -(K_X+\D) pseff

On the logarithmic Kobayashi conjecture

We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of $(\mathbb{P}^n, X)$ is of log-general type, give a new proof of the algebraic hyperbolicity of $(\mathbb{P}^n, X)$, and exclude the existence of maximal rank families of entire curves in the complement of the universal degree $d$ hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.Comment: 17 page

The Green Conjecture for Exceptional Curves on a K3 Surface

We use the Brill-Noether theory to prove the Green conjecture for exceptional curves on K3 surfaces. Such curves count among the few ones having Clifford dimension at least three. We obtain our result by adopting an infinitesimal approach due to Pareschi, and using a degenerate version of a theorem due to Hirschowitz-Ramanan-Voisin

Remarks about bubbles

We make some remarks about bubbling on, not necessarily proper, champs de Deligne-Mumford, i.e. compactification of the space of mappings from a given (wholly scheme like) curve, so, in particular, on quasi-projective projective varieties. Under hypothesis on both the interior and the boundary such as Remark \ref{rmk:interior} below, this implies an optimal logarithmic variant of Mori's Bend-and-Break. The main technical remark is \ref{thm:logMM}, while our final remark, the cone theorem, \ref{rmk:cone}, is a variant.Comment: 13 pages, 3 figure