Fabrizio Catanese

Fibred surfaces, varieties isogenous to a product and related moduli spaces

A fibration of an algebraic surface S over a curve B, with fibres of genus at least 2, has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group G. A variety isogenous to a (higher) product is the quotient of a product of curves of genus at least 2 by the free action of a finite group. Theorem B gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the Euler number. Theorem C classifies the groups thus occurring and shows that, after fixing the group and the Euler number, one obtains an irreducible moduli space. The result of Theorem B is extended to higher dimension in Theorem G, thus generalizing (cf. also Theorem H) results of Jost-Yau and Mok concerning varieties whose universal cover is a polydisk. Theorem A shows that fibrations where the fibre genus and the genus of the base B are at least 2 are invariants of the oriented differentiable structure. The main Theorems D and E characterize surfaces carrying constant moduli fibrations as surfaces having a Zariski open set satisfying certain topological conditions (e.g., having the right Euler number, the right fundamental group and the right fundamental group at infinity).

The maximum likelihood degree

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.

Beauville surfaces without real structures

Inspired by a construction by Arnaud Beauville of a surface of general type with K 2 = 8, p g = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2.

Embeddings of curves and surfaces

We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H 1 (2 K X ) = 0.

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with pg > 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3, and therefore pg = q = 3 and K2 = 6. Furthermore we obtain some results towards the classification of minimal surfaces with pg = q = 3. Such surfaces have 6 < Kz < 9, and we show that Kz = 6 if and only if S is the symmetric product of a curve of genus 3. We also classify the minimal surfaces with pg = q = 3 with a pencil of curves of genus 2, proving in particular that for those one has Kz = 8.

Surfaces of general type with geometric genus zero: a survey

In the last years there have been several new constructions of surfaces of general type with pg = 0, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with pg = 0, and gives an updated list of the existing surfaces, in the case where K 2 = 1;:::; 7. It also focuses on certain important aspects of this classification.

QUOTIENTS OF PRODUCTS OF CURVES, NEW SURFACES WITH pg = 0 AND THEIR FUNDAMENTAL GROUPS

We construct many new surfaces of general type with

Complex Surfaces of General Type: Some Recent Progress

q=p_g = 0

d-very-ample line bundles and embeddings of hilbert schemes of 0-cycles

whose canonical model is the quotient of the product of two curves by the action of a finite group

Pluricanonical — Gorenstein — Curves

G