Fabrizio Catanese - Böcker

The aim of this book is manifold, it intends to overview the wide topic of algebraic curves and surfaces (also with a view to higher dimensional varieties) from different aspects: the historical development that led to the theory of algebraic surfaces and the classification theorem of algebraic surfaces by Castelnuovo and Enriques;

The aim of this book is manifold, it intends to overview the wide topic of algebraic curves and surfaces (also with a view to higher dimensional varieties) from different aspects: the historical development that led to the theory of algebraic surfaces and the classification theorem of algebraic surfaces by Castelnuovo and Enriques;

This research book deals with some classical problems of algebraic geometry, notably the problem about the maximal number of singularities that a nodal variety can have, and the problem about the description of the components of the Severi varieties of nodal surfaces. A complete solution is found for nodal quartic surfaces in 3-space and for nodal K3 surfaces. New striking results are found also for quintic and sextic surfaces. The main focus of the book is the relation of nodal surfaces with binary coding theory, introduced by Beauville: two codes are attached to a nodal projective surface, which are invariants of these components, and in some cases determine them.The book contains a very concrete introduction to binary coding theory and new applications of Nikulin's theory of primitive embeddings of lattices. The book contains also a thorough investigation of cubic hypersurfaces and their singularities, and the associated discriminant surfaces, providing new constructions for surfaces of degree 5 and 6 with the maximal number of nodes. A surprising relation is found between the Barth 65 nodal surface and the Doro-Hall graph. The book is addressed to algebraic geometers and experts of coding theory. It is also meant to be a source of many beautiful classical constructions, due to Kummer, Togliatti and others, which should be of interest to graduate students who want to get to know classical projective geometry.

The volume consists of invited refereed research papers. The contributions cover a wide spectrum in algebraic geometry, from motives theory to numerical algebraic geometry and are mainly focused on higher dimensional varieties and Minimal Model Program and surfaces of general type.A part of the articles grew out a Conference in memory of Paolo Francia (1951-2000) held in Genova in September 2001 with about 70 participants.

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y.

This collection of surveys present an overview of recent developments in Complex Geometry. Topics range from curve and surface theory through special varieties in higher dimensions, moduli theory, Kahler geometry, and group actions to Hodge theory and characteristic p-geometry. Written by established experts this book will be a must for mathematicians working in Complex Geometry

M. Andreatta,E.Ballico,J.Wisniewski: Projective manifolds containing large linear subspaces; - F.Bardelli: Algebraic cohomology classes on some specialthreefolds; - Ch.Birkenhake,H.Lange: Norm-endomorphisms of abelian subvarieties; - C.Ciliberto,G.van der Geer: On the jacobian of ahyperplane section of a surface; - C.Ciliberto,H.Harris,M.Teixidor i Bigas: On the endomorphisms of Jac (W1d(C)) when p=1 and C has general moduli; - B. van Geemen: Projective models of Picard modular varieties; - J.Kollar,Y.Miyaoka,S.Mori: Rational curves on Fano varieties; - R. Salvati Manni: Modular forms of the fourth degree; A. Vistoli: Equivariant Grothendieck groups and equivariant Chow groups; - Trento examples; Open problems

Presents lectures given at the 3rd session of the Centro Internazionale Matematico Estivo, held in Cetraro, Italy, in July 1994. The topics covered include four manifolds, line bundles and manifolds of semi-positive curvature.

Modern approaches to the study of symplectic 4-manifolds and algebraic surfaces combine a wide range of techniques and sources of inspiration. This book presents methods for the study of moduli spaces of complex structures on algebraic surfaces, and for the investigation of symplectic topology in dimension 4 and higher.