I discuss an inductive method to describe simple higher dimensional varieties which consists of lifting extremal rays from a section of an ample vector bundle.

We give some new geometric evidence towards a strengthened conjecture of Mukai relating the Picard number and the pseudo-index of a complex Fano manifold and prove this conjecture in dimension four (joint work with Casagrande, Debarre and Druel).

After recalling the Friedman and Morgan conjecture that deformation and differentiable type should coincide for surfaces of general type, I will illustrate several counterexamples to this conjecture. I will also show that the first ones, due to Manetti, also provide counterexamples to the weaker conjecture that the canonical symplectic type should coincide with deformation type. The drawback is that all these examples are non simply connected, and some of them not so easy to describe. I will then discuss some very simple 1-connected examples which are not deformation equivalent, and report on work in progress with B. Waynrjb which seems to show that these surfaces are symplectomorphic.

This talk follows a recent joint research with C.Ciliberto, on defective varieties. I sketch a classification of irreducible projective threefolds whose k-secant variety (any k) has dimension smaller than the expected one. The classification was classically obtained by G.Scorza (and then re-obtained by several authors) only for the first case k=1. Our extension is based on Scorza's result and a recent classification of "weakly defective" surfaces.

In this talk I will comment on previous papers by R. Miranda and myself, trying to improve some of the results contained there.

I describe an approach of Chiodo to define Witten's "top Chern class", which is more direct and straightforward than Polischuk-Vaintrob.

I will explain various constructions of varieties with ample cotangent bundles.

Counting points over finite fields is a way of obtaining information on the cohomology of a variety. We apply this to the moduli space M_2 of curves of genus 2. By counting the number of genus 2 curves over finite fields we get information on the cohomology of local systems on M_2 and on vector-valued Siegel modular forms for genus 2. More precisely, we obtain convincing guesses for the `motivic' Serre polynomials of all irreducible symplectic local systems on M_2 of weight at most 14. In weight 16, a new motive appears, which we identify. We also have results for higher weights. This is joint work with Gerard van der Geer.

The compactifications of the moduli spaces of curves and theta-characteristics, introduced in the mid-eighties by Cornalba, have recently found several applications in different fields of research, from mathematical physics (generalized Witten conjecture) to projective geometry (Caporaso-Sernesi). In this talk I will describe some geometrical properties of Cornalba's compactification; in particular, I will discuss its relationship with Caporaso's compactification of the universal Picard variety.

Schottky problem, the question of characterizing Jacobians of curves among abelian varieties, is 120+ years old, and there have been numerous solutions. However, an algebraic solution - obtaining algebraic equation for the image of the Jacobian locus in the projective space under the level-two theta embedding - is only known in genus 4, and is due mostly to Schottky. In this talk we obtain intersection-theoretic formulas and explicit upper bounds for the degree of the Jacobian locus inside the projective space. Using these bounds we then explain how the Kadomtsev-Petviashvili (KP) differential equation for theta constants can be effectively rewritten as a system of algebraic equations for theta constants. As the KP solves the Schottky problem, we thus have an effective algebraic solution.

By establishing effective boundedness, some effective estimates for the cardinality of certain finite sets in complex analysis/algebraic geometry are proven. Most notably, an effective uniform version of the finiteness statement of the Shafarevich Conjecture over function fields (Theorem of Parshin-Arakelov) is proven. The proof of this particular result rests on a number of new algebraic geometric results that should be of independent interest.

I will review old and recent results on the birational geometry of quartic 3-folds. In doing this I introduce and try to give the very basics of Sarkisov program and the Maximal singularity method.

I will report on joint work with M.de Cataldo investigating the properties of some bilinear forms which are naturally associated to a projective generically finite map f:X-->Y from a nonsingular variety X. This bilinear forms generalize quite naturally the intersection form of exceptional curves of a surface resolution, and their properties can be related to theorems of Hard Lefschetz type for cupping with the first Chern class of a line bundle which is the pullback of an ample bundle on Y.

I will describe how an approach based on jet schemes can be used to attack certain problems on singularities which appear in the Minimal Model Program. The basic ingredient is a result which computes minimal log discrepancies in terms of jet schemes. As a first application, I will deduce Inversion of Adjunction for hypersurfaces in a smooth variety. Another application will be to a proof of the semicontinuity of minimal log discrepancies on a smooth variety. The talk is based on joint work with L. Ein and T. Yasuda.

We study subvarieties of a general degree $d$ hypersurface $X$ in the $n$-projective space, with a view toward the hyperbolicity of $X$. Our main theorem, which improves previous results of L. Ein and C. Voisin, has the following sharp corollary : any subvariety of $X$ is of general type, whenever $d>2n-3$ and $n>5$.

We discuss a global version of the Bryant representation formula for constant mean curvature surface in the hyperbolic space. We use the theory of stable vector bundles to give example of these surfaces with given global monodromy.

I shall describe the notion of a slightly ramified extension of local rings, and use this to prove a criterion for the smoothness of deformation spaces in mixed characteristic in terms of lifting the tangent space. From one point of view, this generalizes Cartier's proof that group schemes are smooth in characteristic zero. (joint work with T. Ekedahl.)

In the talk we prove the unirationality of the moduli space of curves of genus 14. This case was still unknown in the list of conjecturally uniruled moduli spaces M_g. The relatively elementary method of proof will be also applied to recover the known cases of genus g = 11, 12, 13.

Let $f:X\to Y$ be a semi-stable family of complex projective varieties over a curve Y of genus $q$, and smooth over the complement of a closed subset $S$ consisting of $s$ points. The variation of Hodge structures $(R^kf_*{\mathbb C})|_{Y-S}$ gives rise to a Higgs bundle $$(\bigoplus E^{k-q,q},\theta)$$. The Arakelov inequalities give an upper bound for the sum $deg(E^{k,0})+deg(E^{k-1,1})+ ... + deg(E^{k-l,l})$ where $l$ denotes the integral part of $k/2$. For families reaching this bound, the Higgs fields $$\theta^{k-q,q}:E^{k-q,q} \to E^{k-q-1,q+1}\otimes \Omega^1_Y(log S)$$ satisfy a certain maximality condition. Those imply that the family is rigid, and that the Mumford Tate group of a general fibre is determined by the monodromy representation. For families of abelian varieties, K3 surfaces and CY threefolds, the maximality of the Higgs field imply that $Y$ is a Shimura curve, and that the geometry of a general fibre of $f$ is quite special.

This is joint work with Andrew Kresch. We prove that a separated quotient Deligne-Mumford over a field, whose moduli space is quasiprojective, admits a finite flat cover by a scheme; and if the stack is smooth, the scheme can also be taken to be smooth. We derive some consequences from this: for example, we show that every smooth Deligne-Mumford stack over a field whose moduli space is affine is a quotient stack.

We generalized the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint Donat to the class of toroidal varieties with stratifications. The theory was applied in a proof of the Weak Factorization theorem which says that a birational map between complete nonsingular varieties over an algebraically closed field of characteristic zero is a composite of blow-ups and blow-downs with smooth centers.

A hundred and fifty years ago Otto Hesse introduced hessian as a function of coefficients of polynomial of several variables which vanishes if and only if the polynomial actually depends on fewer variables. This claim turned out to be wrong, and the problem of describing polynomials with vanishing hessian is still open. In this talk we describe a geometric approach to this problem and classification results in low dimensions.