Rational curves and bounds on the Picard number of Fano manifolds

We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.Comment: Modified the proof of the main theorem and eliminated a statement about the cone of curves. To appear in Geometriae Dedicat

Cohomological characterizations of projective spaces and hyperquadrics

We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface.Comment: Added Lemma 2.8 and slightly changed proof of Lemma 6.2 to make them apply for torsion-free sheaves and not only to vector bundle

On Fano schemes of complete intersections

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.Comment: Added lacking references, corrected acknowledgments, minor editorial change

INCUBATORS OF PUBLIC SPACES (2): TOOLS FOR SELF-ORGANISATION IN URBAN REGENERATION

The paper introduces a specific aspect of the Incubators of Public Space European Project: the method and system that intends linking agreed visions for the positive change of an urban area and the combination of individual self-interest driven actions on that area through an interactive and web-based tool. The Project aims to support individuals’, groups’, organisations’, and enterprises’ own ability to evolve towards grassroots interventions in urban regeneration process