AIS Schemes & Cohomology (2014)

A Brief Description of the Subject:-
The development of Algebraic Geometry in the 20th century went through a few salient phases; the first one was dominated by the Italian geometers. This phase was when algebro-geometric techniques were closely linked with topology and analytic geometry and were used in developing the theory of algebraic surfaces. But the shortcoming of this phase was that the intuitive aspects of the geometry was given more importance and there was a neglecting of the key aspects of proofs and rigour as well as analysis of special examples which were needed to support the intuition.  The period 1930-1960 was dominated by the work of Zariski, Weil intended to set right these shortcomings; this culminated in the grand synthesis in the hands of Grothendieck. An immense programme was launched by Grothendieck which introduced tools from commutative algebra into algebraic geometry which afforded a uniform language to handle geometry over characteristic p and characteristic zero. The goal was to create a geometry which synthesized the formal arithmetic aspccts of geometry as well as classical projective geometry. Some of the spectacular successes achieved by algebraic geometry both in its geometric as well as its arithmetic aspects can be directly linked to this grand synthesis initiated by the work of Grothendieck. Just to name a few, Deligne's proof of Weil conjectures, Faltings proof of the finiteness of rational points, Wiles's proof of Fermat's last theorem rely indispensably on this super structure. 
The AIS will be directed towards making young researchers in India familiar with the developments in Algebraic Geometry, with special emphasis on Grothendieck's programme. The three week programme is primarily aimed at early researchers in this field to become familiar and users of the tools and techniques in this subject. To acquire a good knowledge of modern algebraic geometry it is essential to see the “local” aspects coming  from Commutative algebra in its interplay in geometry as well as the immense machinery of cohomology. These two themes will be the main ones for the workshop. To give a flavour of the manner in which these tools have lead to fundamental theorems in algebraic geometry, the  final week will stress on some research themes.  A team of active researchers in the field has been invited for this purpose.  The basic reference for this course will be the book by Hartshorne supplemented by books by Mumford, Ueno and others. 

Schemes: Sheaves, schemes,  elementary properties, morphisms,  invertible sheaves and bundles, differentials, valuative criterion, Bertini’s theorems,  Lefschetz theorem. Derived functors, cohomology of sheaves, Cech cohomology, Cohomology of projective space, Serre duality, semicontinuity  theoremes , Zariski’s main theorem and connected theorem,  Fulton Lazarsfeld connectedness theorem.

References:
1. R. Hartshorne,  Algebraic Geometry
2. K. Ueno, Algebraic Geometry I, II, III
3. Lazarsfeld,  Positivity in Algebraic Geometry
4. Griffiths - Harris, Principles of Algebraic Geometry

Time-Table :

Tutorial Assistants:

How to reach

Kerala School of Mathematics (KSOM) is located at a distance of about 15 Kms from the city centre (train/bus stations) situated off the Kozhikode-Medical College-Kunnamangalam bus route.Prepaid auto counter is available at Railway Station itself
   (Near 1st number Platform).