Mini-Workshop: Classification of Surfaces of General Type with Small Invariants

Abstract

This mini-workshop has been organized by Fabrizio Catanese and Ciro Ciliberto. Unfortunately Catanese was unable to participate.

The classification of algebraic surfaces is a long-standing research subject in algebraic geometry, started by Castelnuovo and Enriques more than one hundred years ago, and continued by the Italian school (Severi, de Franchis, etc.) until about 1950.

In more recent times, fundamental contributions have been given by Kodaira in the 1950's and later in the 1970's by Bombieri, whose works on pluricanonical maps gave a strong impulse in studying surfaces of general type, and Mumford.

Adding important information to classical results by Noether and Castelnuovo, sharp bounds on the invariants have been given by Miyaoka and Bogomolov–Yau, allowing many authors to develop a systematic study of the “geography” of surfaces of general type.

Interesting investigations about the moduli space of surfaces of general type have been worked out in the last twenty years by Catanese, Manetti, and others.

Despite the intensive effort made in the last decades in order to make more precise our knowledge about surfaces of general type, their fine classification is still an open problem, even for small invariants. It is actually rather embarassing that, after more than one century of research on the subject, a complete classification of surfaces with geometric genus zero or one is still lacking.

This mini-workshop carried together 14 mathematicians actively working on this subject, and related arguments, with the idea of updating the state-of-the-art, exchanging information, discussing interesting open problems and stimulating collaborations. In this respect, the workshop has been very successful.

The atmosphere has been lively and very collaborative. During every talk, several questions have been posed and interesting problems pointed out. It has been especially remarkable the active presence of young participants.

During the week, 16 formal lectures have been given by the participants. This report contains extended abstracts of all the talks and also a contribution by Catanese, in collaboration with Pignatelli, about the lecture he was supposed to give.

The topics include: pluri-canonical maps for surfaces of general type (M. Mendes Lopes), canonical rings, projective embeddings and birational techniques (C. Böhning, F. Catanese, S. Papadakis, U. Persson, R. Pignatelli), irregular surfaces with low invariants (F. Polizzi, F. Zucconi), surfaces with $p_g=0$ (A. Calabri, C. Ciliberto, K. Keum, M. Mendes Lopes, C. Werner), general techniques (V. Brînzănescu, K. Konno). Ulf Persson chaired an “open problem and discussions” session, which especially concerned surfaces with $p_g=0$.

The organizers thank the Institute staff for providing a comfortable environment to the participants.