14:00 to 16:00 Thursdays. First lecture: 02/05/19. No lecture on 13/06/19
Replacement lectures on Monday 13/05/19 from 14:00 to 16:00.
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 Riemann-Roch 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
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.
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. Noether-Enriques 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.
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
- Matsuki, Introduction to the Mori program, Springer
- Birkenhake, Lange, Complex Abelian Varieties, Springer
- Huybrechts, Lectures on K3 surfaces, CUP
If you require assessment for this class please contact me for recommended essay titles.