Daniel Naie

RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of . For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion . As applications we show various instances in which X is determined by . We also formulate a basic question about the birational invariance of ẽ(X, Y).

NUMERICAL CAMPEDELLI SURFACES CANNOT HAVE THE SYMMETRIC GROUP AS THE ALGEBRAIC FUNDAMENTAL GROUP

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order [les ]9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.

Twisted Kodaira-Spencer classes and the geometry of surfaces of general type

We study the cohomology groups

Surfaces d'Enriques et une construction de surfaces de type général avecp g =0

H^1(X,\Theta_X(-mK_X))

Jumping numbers of a unibranch curve on a smooth surface

, for

The irregularity of cyclic multiple planes after Zariski

m\geq1

Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces

, where

Enriques diagrams and log-canonical thresholds of curves on smooth surfaces

X

Special rank two vector bundles over Enriques surfaces

is a smooth minimal complex surface of general type,

Log-canonical threshold for curves on a smooth surface

\Theta_X