## Syzygies

Lecture hours/location: MW 14:00-15:15
Instructor: Manoj Kummini. Email: mkummini AT cmi DOT ac DOT in.     http://www.cmi.ac.in/~mkummini/index.html
Office: First floor.
Office Hours: MW 11:50-13:00

Important information:
• Moodle. I'll add this information and details on Moodle later; After you log in, search for `Syzygies' under `Mathematics /Jan - Apr 2012'.
Course description. A `syzygy' is a linear relation among the generators of a finitely generated module or an ideal. In this course, we will study a selection of topics from D. Eisenbud's `Geometry of Syzygies' dealing with the syzygies of the defining ideals of projective varieties (embedded in some projective space). We will begin with sheaf cohomology and local cohomology (derived functors, Cech cohomology, vanishing/non-vanishing theorems, Cohen--Macaulay and Gorenstein rings, local/Serre duality) and embedding of algebraic varieties into projective spaces. Following this, we will study free resolutions, and some numerical invariants associated to them. As specific examples, we will look at points in projective plane, rational normal curves, curves embedded by high degree and by the canonical divisor.

A somewhat long explanation: For an algebraic variety (say, a curve) we can ask whether it can be embedded as a closed subvariety of a projective space. If such embeddings exist, then the variety can be described in terms of polynomial equations. In this course, we will look at some measures of how `nice' the defining ideal of an algebraic variety (embedded in a projective space) is. My current plan is to talk about the following:

• Regular local rings and polynomial rings.
• Free resolutions, depth, projective dimension and (Castelnuovo-Mumford) regularity.
• Cohen-Macaulay and Gorenstein rings.
• Cohomology of coherent sheaves, local cohomology
• Line bundles, embeddings into projective spaces.
• Points in P^2
• Line bundles on smooth curves: Canonical bundle, Riemann-Roch theorem.
• Embedding of smooth curves: rational normal curves, curves of high degree and canonical embedding.
• If time permits: Eisenbud-Goto conjecture and the Gruson-Lasarzfeld-Peskine theorem.
Textbook:
1. D. Eisenbud, Geoemtry of syzygies, Springer. This is the main textbook.
2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer.
3. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ Press.
4. R. Hartshorne, Algebraic geometry, Springer.
5. J.-P. Serre, Faisceaux Algebriques Coherents, Ann. Math, 1955.
6. R. Hartshorne, Local cohomology, a seminar given by A. Grothendieck, Springer LNM 41.
7. S. Iyengar et. al., Twenty-four hours of local cohomology, Amer Math Soc.

Prerequisite: Some knowledge of commutative algebra at the level of Atiyah-Macdonald and algebraic geometry at the level of Hartshorne Chapter I (only!) If you are considering taking this course, talk to me this term (Aug-Dec 2011) itself. It would help me to know the background of the students before the course begins.

CMI | Manoj's Home-page | Last modified: Tue Aug 16 15:02:08 IST 2011