Class timings: Tuesday 9:10 - 10:25, Friday 11:50 - 13:05, Lecture Hall 1.
Prerequisites: Chapter 1 of Hartshorne's Algebraic Geometry; basics of sheaf theory (Chapter 2, Section 1 of Hartshorne); a first course in commutative algebra (most of Atiyah-Macdonald's book). It will also be helpful (but not mandatory) to know the following topics: coherent sheaves on varieties, operations on coherent sheaves, locally free sheaves, invertible sheaves.
Course outline: The goal of the course is to cover basics of surface theory and
introduce some current research problems.
1) We will start with a short discussion of
a) sheaves of modules, divisors, and differentials on projective
varieties,
b) the theorem proving that on a nonsingular variety, the group of
divisors modulo linear equivalence is isomorphic to the group of
isomorphism classes of invertible sheaves,
c) cohomology of sheaves on projective varieties.
2) Basics of surface theory: intersection of curves on surfaces; blowing
up a point on a nonsingular surface; Riemann-Roch theorem and Hodge index
theorem.
3) Current research topics: Seshadri constants, Nagata and SHGH
conjectures, bounded negativity (changes/additions/deletions as time permits).
For (1) and (2): Hartshorne's Algebraic Geometry and Cutkosky's Introduction to Algebraic Geometry.
For (3): We will use some lecture
notes and other resources available on the web for (3).
These will be mentioned in class when we get to them.
Email: krishna [at] cmi [dot] ac [dot] in.
Academic Honesty:
Academic honesty is essential to a successful education. I will expect all
your work to be independent. Some discussion is permitted for homework but
final solutions need to be your own. Any violations will be dealt with
in a strict manner in accordance with CMI policies.