Global aspects of the geometry of surfaces

TL;DR

This paper discusses open problems and recent developments in the geometry of smooth projective algebraic surfaces, covering topics like Mori dream spaces, Cox rings, Seshadri constants, and symbolic powers of ideals.

Contribution

It provides a comprehensive overview of current open problems and recent research trends in the geometry of algebraic surfaces, with historical context and exercises.

Findings

Discussion of open problems in surface theory

Recent results on Cox ring finite generation

Analysis of symbolic powers and homogeneous ideals

Abstract

These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the Deutsche Forschungsgemeinschaft) present a number of open problems on the theory of smooth projective algebraic surfaces, and put into historical context recent work on a range of topics, including Mori dream spaces and the finite generation of the Cox ring, Seshadri constants, and the resurgence of homogeneous ideals and the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. These notes include many exercises, with solutions.