On deformations of quintic and septic hypersurfaces

An old question of Mori asks whether in dimension at least three, any smooth specialization of a hypersurface of prime degree is again a hypersurface. A positive answer to this question is only known in degrees two and three. In this paper, we settle the case of quintic hypersurfaces (in arbitrary dimension) as well as the case of septics in dimension three. Our results follow from numerical characterizations of the corresponding hypersurfaces. In the case of quintics, this extends famous work of Horikawa who analysed deformations of quintic surfaces.Comment: 23 pages, final version, to appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Positivity of the diagonal

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.Comment: 23 pages; v2: updated attributions and minor change

Effective cones of cycles on blow-ups of projective space

In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$, $n>4$, at $r$ very general points, the cone of divisors is not finitely generated as soon as $r> n+3$, whereas the cone of curves is generated by the classes of lines if $r \leq 2^n$. In fact, if $X_r^n$ is a Mori Dream Space then all the effective cones of cycles on $X_r^n$ are finitely generated.Comment: 26 pages; comments welcom

Remarks on the positivity of the cotangent bundle of a K3 surface

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell.Comment: Published versio