## Math 634 Algebraic Geometry II Fall 2004

Instructor: Ali Sinan Sertöz, Department of Mathematics
Room: SA-121, Phone: 1490
Office Hours: Wednesday 10:40-12:30

Main Text:
• Compact Complex Surfaces, Barth & Hulek & Peters & Van De Ven, Springer, Second enlarged edition (2004).

Suplementary Texts:

• Complex Algebraic Surfaces, Beauville, Cambridge University Press (1996).
• Principles of Algebraic Geometry, Griffiths & Harris, Wiley Interscience (1978).
• Algebraic Surfaces, Shafarevich, American Mathematical Society, (1967).
• Algebraic Geometry, Hartshorne, Springer-Verlag, (1977).

Schedule:

 WEDNESDAY 8:40-10:30 SAZ-21 FRIDAY 10:40-12:30 SAZ-21

Topics to discuss:

• Bundles, sheaves, divisors

• Chern classes

• Curves on surfaces

• Enriques Kodaira classification

• Invariants of K3 surfaces

• Moduli for K3

• Surjectivity of the period map for K3

• Invariants for Enriques surfaces

• The period map for Enriques surfaces

Each student will choose an article from the following list and give a detailed presentation of it in class at the end of the semester and will circulate a written version of that talk incorporating all the comments and suggestions which arise during the presentation. The final grade will reflect the combined success of the talk and its written report.

Here is the list of suggested articles:

• Knutsen, Andreas Leopold; Smooth curves on projective K3 surfaces
Math Scand 90 (2002) 215-231.

• Kovacs, Sandor; The cone of curves on a K3 surface,
Math Ann 300 (1994) 681-691.

• Keum, Jong Hae; Every algebraic Kummer surface is the K3 cover of an Enriques surface
Nagoya Math J 118 (1990) 99-100.

• Morrison, David; On K3 surfaces with large Picard number,
Invent Math 75 (1984) 105-121.

• Önsiper & Sertöz; Generalized Shioda-Inose structures on K3 surfaces,
Manuscripta Math 98 (1999) 491-495.

• Saint-Donat, B; Projective models of K3 surfaces,
Amer J Math 96 (1974) 602-639.

• Sertöz; Which singular K3 surfaces cover an Enriques surface,
Proc Amer Math Soc (2004).

• Shioda & Inose; On singular K3 surfaces,
Complex Analysis and Algebraic Geometry, Edited by W. L. Baily, Jr., and T. Shioda, (1977), pp119-136.

Presentations:

Caner Koca,
Title: On K3 surfaces with large Picard number,
Date: 10-12 November, 2004

Süleyman Tek,
Title: 2-Geometries and the Hamilton-Jacobi equation, (Garcia-Godines et al, J Math Phys, 45 (2004) 725-735.)
Date:
8-10 December, 2004

Sultan Erdoğan,
Title:
Every algebraic Kummer surface is the K3 cover of an Enriques surface
Date: 15-17 December, 2004

Mesut Şahin
Title: Which singular K3 surfaces cover an Enriques surface,
Date: 22-24 December, 2004

İnan Utku Türkmen,
Title: Families of  K3 surfaces, (Mayer, Nagoya Math J, 48 (1972) 1-17.)

Date: 29-31 December, 2004

### Actual material covered in the lectures:

• Vector bundles, Picard group, divisors, sections of bundles, building new bundles and obtaining their transition functions from the original ones

• Connection, curvature, Chern-Weil theory

• Zeros of sections and relation to Chern classes, Gauss-Bonnet

• Borel & Serre, Le Theoreme de Riemann-Roch, Bull. Soc. math. France, 86 (1958), 97-136.

• Grothendieck, La theorie des classes de Chern, Bull. Soc. math. France, 86 (1958), 136-154.

• K3 surfaces

• Barth & Hulek & Peters & Van De Ven, Compact Complex Surfaces, Springer, Second enlarged edition (2004), chapter VIII, K3 surfaces and Enriques surfaces

• Miles Reid, Chapters on Surfaces,Complex algebraic geometry (Park City, UT, 1993), 3--159,
IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, RI, 1997, chapter 3, K3's.

• Surjectivity of the period map for K3 surfaces

• Namikawa, Surjectivity of period map for K3 surfaces, Katata Symposium 1982, Ed: Ueno, Progress in Math, Birkhauser, Vol 139, (1983), 379-397.

• Looijenga, A Torelli theorem for Kahler-Einstein K3 surfaces, LNM 894, Springer-Verlag, (1981), 107-112.

• Siu,  A simple proof of the surjectivity of the period map of K3 surfaces, Manuscripta Math., 35 (1981), 311-321.

• Cone of Curves on a K3

• Kovacs, The cone of curves on a K3 surface, Math Ann 300 (1994) 681-691.

• Clemens & Kollar & Mori, Higher Dimensional Complex Geometry, Asterisque, Vol 166 (1988), lecture 4, the cone of curves-the smooth case, pp22-27.

Son güncelleme: 19 Şubat 2005 Cumartesi