Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

Krishanu Dan, NISER

4

Divisors, Sheaf cohomology

Divisors on algebraic varieties, differentials, maps to projective spaces, sheaf cohomology of projective spaces, Serre duality.

Suratno Basu, SRM-AP

4

Divisors, Sheaf cohomology

 Divisors on algebraic varieties, differentials, maps to projective spaces, sheaf cohomology of projective spaces, Serre duality.

Krishna Hanumanthu, CMI

4

Introduction to positivity

 Ample and very ample divisors, intersection theory on surfaces, blow ups, Riemann-Roch theorem for curves and surfaces.

Sukhendu Mehrotra, CMI

4

Rational curves on Varieties

Families of rational curves on varieties play an important role in the Minimal Model Program. This course will introduce the basic theory and constructions, and then turn to families of rational curves on low degree hyper-surfaces and holomorphic symplectic varieties. Time permitting, we shall discuss some applications such as the boundedness of Fano varieties.

Anand Sawant, TIFR

4

Near-rationality properties of algebraic varieties

 Notions of rationality, stable rationality, retract rationality, universal R-triviality, unirationality and rational connectedness of smooth varieties and the interrelationship between them.

Omprokash Das, TIFR

Minimal Model Program

1. Castelnouvo’s contraction theorem and MMP on smooth surfaces.

2. Mori’s idea of higher dimensional MMP.

3. Singularities of MMP.

4. Cone and Contraction Theorems (Smooth and singular version). Types of contractions in MMP.

5. Running MMP, Abundance conjecture, Canonical bundle formula.

6. Discussion of MMP in moduli of higher dimensional varieties.