Global aspects of the geometry of surfaces. (English) Zbl 1247.14006

Harbourne, Brian

Ann. Univ. Paedagog. Crac. 84, Stud. Math. 9, 5-41 (2010).

The paper contains the notes of the lectures given, by the author, at the summer school of the SFB/TR 45 Bonn-Essen-Mainz, which took place March 23-27, 2009, at the Pedagogical University of Cracow. These notes contain several open problems related to the behavior of the monoid of effective divisors and the nef cone for smooth projective surfaces over an algebraically close field. The author motivates, and puts into historical context, concepts such as Mori dream space, Seshadri constants and the resurgence of homogeneous ideals in polynomial rings. Moreover, he discusses some recent work on these topics along with the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. The notes contain also several exercises with solutions.

Reviewer: Cristiano Bocci (Grosseto)

MSC:

14C20	Divisors, linear systems, invertible sheaves
14J26	Rational and ruled surfaces
13C05	Structure, classification theorems for modules and ideals in commutative rings

Keywords:

surfaces; bounded negativity; multiplier ideal; fat points; symbolic powers