AIS - Topics in Birational Geometry (2023)

Speakers and Syllabus


 

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.

 

 

 

 

References:

  1. R. Hartshorne, Algebraic Geometry.

  2. R. Lazarsfeld, Positivity in Algebraic Geometry, Volume 1.

  3. J. Kollar: Rational Curves on Algebraic Varieties.

  4. J. Kollar, S. Mori: Birational Geometry of Algebraic Varieties.

  5. O. Debarre: Higher-Dimensional Algebraic Geometry, Springer Universitext.

  6. O. Debarre: Rational curves on hypersurfaces.

  7. F. Charles, G. Mongardi, G. Pacienza: Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles.

  8. T. Graber, J. Harris, J. Starr: Families of rationally connected varieties.

  9. J.-L. Colliot-Thelene-J.-J. Sansuc: The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic groups and homogeneous spaces, 113-186, Tata Inst. Fund. Res. Stud. Math.

Names of the tutors with their affiliation :

  1. Snehajit Misra, CMI.

  2. Nabanita Ray, CMI.

  3. Jyoti Dasgupta, TIFR.

  4. Aditya Subramaniam, TIFR.

  5. Praveen Kumar Roy, CEBS.

  6. Roktim Mascharak, TIFR.

  7. Priyankur Chaudhury, TIFR.

 


Time Table

 Tentative time-table, mentioning names of the speakers and tutors with their affiliation:

Day Date Lecture 1
9:30
to
11:00
Tea
11:00

to
11.30
Lecture 2
11:30

to
13:00
Lunch
13:00

to
14:30
Tutorial 1
14:30

to
15:30
Tea
15:30

to
16:00
Tutorial 2
16:00

to
17:00
Snacks
17:00
    (name of the speaker)   (name of the speaker)          
Mon 22/5/23 KD   SB   KD+AS+RM   SB+JD+PR  
Tues 23/5/23 KD   SB   KD+AS+RM   SB+JD+PR  
Wed 24/5/23 KD   KH   KD+AS+RM   KH+JD+NR  
Thu 25/5/23 KD   KH   KD+AS+RM   KH+JD+NR  
Fri 26/5/23 SB   KH   SB+AS+RM   KH+PR+NR  
Sat 27/5/23 SB   KH   SB+AS+RM   KH+PR+NR  

 

Day Date Lecture 1
9:30
to
11:00
Tea
11:00

to
11.30
Lecture 2
11:30

to
13:00
Lunch
13:00

to
14:30
Tutorial 1
14:30

to
15:30
Tea
15:30

to
16:00
Tutorial 2
16:00

to
17:00
Snacks
17:00
    (name of the speaker)   (name of the speaker)          
Mon 29/5/23 SME   ASA   SME+NR+PC   ASA+JD+PR  
Tues 30/5/23 SME   ASA   SME+NR+PC   ASA+JD+PR  
Wed 31/5/23 SME   OD   SME+NR+PC   OD+SM+PC  
Thu 1/6/23 SME   OD   SME+NR+PC   OD+SM+PC  
Fri 2/6/23 ASA   OD   ASA+JD+PR   OD+SM+PC  
Sat 3/6/23 ASA   OD   ASA+JD+PR   OD+SM+PC  

 

Speakers: Tutors:
KD: Krishanu Dan AS: Aditya Subramaniam
SB: Surano Basu RM: Roktim Mascharak
KH: Krishna Hanumanthu PR: Praveen Kumar Roy
SME: Sukhendu Mehrotra JD: Jyoti Dasgupta
ASA: Anand Sawant NR: Nabanita Ray
OD: Omprokash Das SM: Snehajit Misra
  PC: Priyankur Chaudhury
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