Schedule
14:00 to 16:00 Thursdays. First lecture: 02/05/19. No lecture on
13/06/19 and
20/06/19
Replacement lectures on Monday 13/05/19 from 14:00 to 16:00.
Overview
Enriques' classification of complex algebraic surfaces is a beautiful piece of classical algebraic geometry.
We will begin by introducing the theory of algebraic surfaces; reviewing intersection theory on surfaces,
the RiemannRoch theorem and Picard group. We then move toward understanding the classification, via a
number of landmark results. We then give a survey of the principal classes of surfaces which appear in
the classification.
We will assume some familiarity with the basic notions in algebraic geometry; although the Picard group, amplitude,
and intersection multiplicity of curves will all be covered (though quite briefly). A basic familiarity with complex manifolds
will be assumed. Some tolerance for the language of schemes has advantages, but is not required.
Lecture Notes
Notes will be added here throughout the course.
 Lecture 1 Foundations: Schemes vs. Complex manifolds, GAGA. Invariants of complex surfaces, Hodge diamond. Curves and Divisors on surfaces. Slides.
 Lecture 2 Amplitude. First examples. Blowing up and Castelnuovo's criterion Slides.

Lecture 3 Anatomy of birational maps. Ruled surfaces. NoetherEnriques theorem. Slides.

Lecture 4 Castelnuovo's rationality criterion. Slides.

Lecture 5 Stein factorization, Kodaira dimension zero surfaces.

Lecture 6 Smoothness of blow down, Enriques' theorem, minimal rational surfaces.

Lecture 7 K3 surfaces and Torelli theorems.

Lecture 8 Higher Kodiara dimension. Examples of surfaces of general type.
References
Main references for the course.
 Beauville, Complex Algebraic Surfaces, LMS Student Texts
 Barth, Hulek, Peters, Van de Ven, Compact complex surfaces, Springer
 Reid, Chapters on algebraic surfaces, IAS/Park city lecture notes series, AMS
 Vakil, Complex algebraic surfaces, Lecture notes
General Algebraic Geometry references
 Hartshorne, Algebraic Geometry, Springer
 Lazarsfeld, Positivity in algebraic geometry: I, Springer
 Huybrechts, Complex Geometry, Springer
Additional references:
 Matsuki, Introduction to the Mori program, Springer
 Birkenhake, Lange, Complex Abelian Varieties, Springer
 Huybrechts, Lectures on K3 surfaces, CUP
Assessment
If you require assessment for this class please contact me for recommended essay titles.