# School on Algebraic Surfaces (2015) - Speakers and Syllabus

Time Table

Note: The number is bracket following the initials of a speaker is the serial number of the lecture, and the prefix ‘t’ denotes a tutorial session.

 Week I 20-VII-15 21-VII-15 22-VII-15 23-VI-15 24-VI-15 25-VII-15 9:30-11:00 KHP(1) YP(2) KHP(2) VMM(1) VMM(2) NN(3) 11:30-13:00 NN(1) NN(2) YP(3) AM(1) YP(4) AM(2) 14:30-15:30 YP(1) YP(t1) KHP(t2) VMM(t1) VMM(t2) — 16:00-17:00 KHP(t1) NN(t1) NN(t2) AM(t1) YP(t2) — Week II 27-VII-15 28-VII-15 29-VII-15 30-VII-15 31-VII-15 01-VIII-15 09:30-11:00 VMM(3) VMM(4) RS(1) RS(2) RS(3) KHP(4) 11:30-13:00 AM(3) AM(4) KHP(3) NN(4) RS(4) Val 14:30-15:30 VMM(t3) AM(t3) RS(t1) RS(t2) RS(t3) 16:00-17:00 AM(t2) KHP(t3) KHP(t4) NN(t3) KHP(t5)

Speakers

• AM: Alok Maharana
• KHP: Kapil Hari Parajape
• NN: Nitin Nitsure
• RS: Ramesh Sreekantan
• VMM: Vivek Mohan Mallick
• YP: Yashonidhi Pandey

Lectures

The planned content of the lectures is as follows. There may be some modifications in this as the School progresses.

1. KHP(1): Introduction to the course.

2. NN(1): Recollection of the theory of varieties and schemes.

3. YP(1) Recollection of the theory of algebraic curves and plane curves.

4. NN(2): The exponential sequence, line bundles and analytic Picard variety. Hodge Theory for surfaces.

5. KHP(2) Intersection Theory of Curves on a Surface and blow-ups.Ample divisors, Hodge Index Theorem.

6. YP(2): Differential Forms. Adjunction Formulas. De Rham Theorem.

7. YP(3): Vector bundles on curves, ruled surfaces, elementary transformations.

8. AM(1): Projective and Affine plane,Rational surfaces.

9. VMM(1): Analytic theory of Elliptic curves and Abelian surfaces.

10. VMM(2): Ample divisors, Vanishing theorems, Hodge Index Theorem. Cone of Effective curves.

11. NN(3): Tsen’s theorem for conic bundles on a curve, Brauer groups.Rationalty Questions.

12. YP(4): Hilbert and Quot Schemes.

13. AM(2): Affine surfaces and log surfaces.

14. VMM(3): Rational double points.

15. RS(1): Elliptic surfaces and Kodaira-Neron models.

16. AM(3): Topology of Projective Algebraic Surfaces. Lefschetz Pencils. Noether’s Formula.

17. VMM(4): Singular Surfaces. Resolution of Singularities of a surface.

18. AM(4): Fundamental Groups of Surfaces.

19. NN(4): Picard and Albanese Schemes.

20. RS(2): K3 surfaces. Moduli etc.

21. KHP(3): Classification of Surfaces. Surfaces of General Type.

22. RS(3): Hilbert modular surfaces.

23. RS(4): K-Theory and Cycles on an algebraic surface. (Some mention of arithmetic surfaces.)

24. KH(4): Wrapping up. Open Problems.

25. Val: Feedback. Valedictory session.

The study of Algebraic Surfaces forms a core component of Algebraic Geometry. The workshop is meant to introduce the study of Algebraic Surfaces to students who wish to pursue research in this area or to utilise this under-standing in other areas of the mathematical sciences.

Lectures will be broadly in two streams

• General Theory
• Special Surfaces.

The former will provide the algebro-geometric tools which are used to study surfaces (and higher dimensional objects in algebraic geometry). The latter will introduce various important surfaces and study their properties in detail.

A short list of topics to be covered is as follows:

• Construction of Picard and Albanese schemes
• Grothendieck-Riemann-Roch formula and Noether’s Formula
• The Cone of effective divisors
• Rational and ruled surfaces
• Bundles of Conics
• Elliptic and Abelian surfaces
• The Kummer Surface and K3 surfaces
• Singularities on an Algberaic Surface.

Detailed Break-Up of Lecture Programme: The total hours of instruction including lectures and tutorials will be roughly equally divided amongst all the speakers listed below. This means an estimated 8 contact hours for each of them.

General Theory portion will be co-ordinated by Prof. Nitsure (TIFR).

• The lecturers will be Dr. Vivek Mallick (IISER Pune) and Dr. Yashonidhi Pandey (IISER Mohali).
• The Special surfaces portion will be co-ordinated by Prof. Kapil Paranjape (IISER Mohali). The lecturers will be Dr. Ramesh Srikantan (ISI Bangalore Centre), Dr. Alok Maharana (IISER Mohali).

Detailed Syllabus: Topics under General Theory that lead up to lead up to the following ”big” theorems.

• Construction of Pic and Albanese

• GRR and Noether’s theorem.

• Vanishing theorems.

• Lefschetz theorem on hyperplane sections.

These are the topics in detail.

(1) Recalling basic notions of schemes, algebraic and differential topology etc. (2 Lectures no proofs)

(2) Recall of basic theory of curves and Riemann surfaces. (2 Lectures no proofs)

(3) The exponential sequence, line bundles and analytic Picard variety. Analytic/Topological intersections and Hodge Index Theorem for surfaces. (1 Lecture)

(4) Divisors and intersection products done algebraically. (1 Lecture)

(5) Differential forms, Kahler Differentials, Adjunction formula. (1 Lecture)

(6) The cone of effective curves and the algebraic Hodge Index theorem. (1 Lecture.)

(7) Linear systems of curves on a surface. Degenerations. Lefschetz pencils. (1 Lecture).

(8) Curves with negative self-intersection and blow-down. (1 Lecture).

Topics under Special Surfaces are to study the various examples and lead up to the classification.

(1) Rational surfaces. Blowing up. Curves in the projective plane.

(2) Projective bundle of a vector bundle, ruled surfaces. Joins. Blow-ups of ruled surfaces. Horizontal curves on ruled surfaces.

(3) Tsen’s theorem for conic bundles on a curve.

(4) Analytic theory of Elliptic curves and Abelian surfaces.

(5) The Kummer surface.

(6) K3 surfaces (plane doubled along sextic, quartic). Deformations of the same.

(7) Elliptic surfaces and Kodaira-Neron models.

(8) Rational Singularities and other examples of singular surfaces.

(9) Affine surfaces and log surfaces. Degenerations of surfaces