Varieties of Nodal Surfaces, Coding Theory and Discriminants of Cubic Hypersurfaces

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.

A comprehensive survey of the theory of nodal surfaces of small degree in 3-space and of nodal K3 surfaces Gives a concise introduction to binary coding theory and describes applications to nodal surfaces This research monograph solves long standing problems and discovers new striking codes attached to singularities

Autorentext

Fabrizio Catanese is a leading algebraic geometer, member of the Accademia dei Lincei, the Goettingen Academy, and the Academia Europaea. He has been a Full Professor at the University of Pisa, Italy (1980-1996), at Goettingen University (1997-2001), and since 2002, at Bayreuth University (Germany). He has been leading several research projects, and among others, an European Project Science, a DFG Graduiertenkolleg, a DFG Schwerpunkt Program, a DFG Forschergruppe, an ERC Advanced Grant, in which about 30 students and 30 Postdocs have been participating.

Inhalt

Chapter 1. Binary codes and the components of the varieties of nodal K3 surfaces.- Chapter 2. Cubic hypersurfaces, associated discriminants and low degree nodal surfaces.- Chapter 3. Nodal Quintic surfaces.- Chapter 4. Nodal Sextic surfaces.- Chapter 5. Codes of nodal sextics with many nodes.