Topics in Birational Geometry (2023)

Convener(s)

 
Name: Omprokash Das Krishna Hanumanthu
Mailing Address: Reader, Tata Institute of Fundamental Research
Homi Bhabha Rd
Mumbai 400 005		  Associate Professor, Chennai Mathematical Institute
H1, SIPCOT IT Park, Siruseri
Kelambakkam 603103
Email: omdas at math.tifr.res.in krishna at cmi.ac.in

This program is intended for early Ph.D students interested in working in algebraic geometry. Some B.Sc and M.Sc students may also be able to attend depending on their background. There will be some preparatory lectures covering divisors, differentials, cohomology, blow ups, intersection products on surfaces and Riemann-Roch theorem on curves and surfaces. After that, the program will have lectures on the following topics: positivity properties of line bundles, rational curves on projective varieties, rationality properties (e.g. uniruled varieties, rational connectedness, etc), and the minimal model program.

Dates: 

Monday, May 22, 2023 - 09:00 to Saturday, June 3, 2023 - 18:00

Venue: 

Venue Address: 

Chennai Mathematical Institute
H1, SIPCOT IT Park, Siruseri
Kelambakkam 603103
India

Venue State: 

Venue City: 

PIN: 

603103

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

Selected Applicants: 

 

Sr. SID Full Name Gender Affiliation Position in College/ University University/ Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Deg. Date
Selected Applicants
1 45298 Ms. Ashima Bansal Female Indian institute Of Science Education And Research(IISER), Tirupati PhD Indian Institute Of Science Education And Research ,Tirupati 2020  
2 45299 Mr Shivam Vats Male Indian Indian Of Science Education And Research,Tirupati PhD Student Indian Indian Of Science Education And Research,Tirupati 2020  
3 45577 Mr Supravat Sarkar Male Princeton University PhD student Indian Statstical Institute, Bangalore 2021  
4 45589 Mr Rajat Kumar Mishra Male IIT Gandhinagar PhD IISER, Kolkata 2019  
5 45597 Mr. Pavan Hasmukhbhai Adroja Male IIT Gandhinagar PhD Student Department of Mathematics, Saurashtra University 2019  
6 45639 Mr. Biman Roy Male Indian Statistical Institute, Kolkata PhD Student Indian Statistical Institute, Kolkata 2018  
7 45939 Dr. Sourav Sen Male TIFR Mumbai Visiting Fellow Ramakrishna Mission Vivekananda University 2013 15/02/2020
8 46024 Ms Nidhi Gupta Female IISER Mohali PhD student IISER Mohali, M.Sc 2020  
9 46064 Mr Dibyendu Das Male Indian Statistical Institute Bangalore PhD Indian Statistical Institute Bangalore 2021  
10 46086 Mr Aanjaneya Rath Male NISER PHD STUDENT UTKAL University , Odisha 2019  
11 46167 Mr. Sayan Pal Male Indian Statistical Institute PhD IIT Kharagpur 2021  
12 46242 Mr. Bidhan Paul Male IIT Madras PhD student IIT Madras 2019  
13 46361 Mr. Debjit Basu Male University of Kansas PhD student IISER Mohali 2019  
14 46482 Mr. Prakash Chandra Joshi Male IISER Mohali PhD IISER Mohali 2022  
15 46504 Mr Manish Kumar Male Tata Institute of Fundamental Research Research Scholar(PhD) Indian Statistical Institute, Bangalore 2021  
16 46526 Mr. Supriya Saha Male Indian Institute of Science, Bengaluru PhD Indian Statistical Institute, Kolkata 2022  
17 46561 Mr. Himanshu Bimal Male ICTS-TIFR, Bengaluru Long term visiting student NISER 2022  
18 46612 Mr. Suchir Kaustav Male Chennai Mathematical Institute BSc Student      
19 46703 Mr Aditya Das Male IIT Palakkad PhD University Of Delhi/ Hansraj College/ M.Sc 2020  
20 46766 Mr. Anubhab Pahari Male IIT Madras PhD Scholar Jadavpur University 2020  
21 46927 Mr. Akhil Surendran Male I.I.T Gandhinagar PhD Central University of Tamilnadu 2019  
22 46944 Ms. Gargi Pravinbhai Patel Female Ramanujan Institute for Advanced Study in Mathematics PhD Gujarat University 2017  
23 47004 Mr. Buddhadev Hajra Male Tata Institute of Fundamental Research Postdoctoral Fellow Indian Institute of Technology Bombay, Mumbai (MH) 2016 20/08/2022
24 47012 Mr. Dharm Veer Male IIT Gandhinagar Post-Doc fellow Chennai Mathematical Institute 2016  
25 47125 Mr. Sagnik Das Male University of Kansas PhD Indian Institute of Technology, Madras 2020  
26 47174 Mr. Aishik Chattopadhyay Male Ramakrishna Mission Vivekananda Educational and Research Institute PhD Pondicherry University 2018  
27 47305 Mr. Arkamouli Debnath Male TIFR Mumbai PhD Indian Statistical Institute Kolkata 2020  
28 47333 Mr. Pratik Ashok Jadhav Male Indian Institute of Science Ph.D. First year student Chennai Mathematical Institute 2022  
29 47420 Mr Mahesh Garg Male Indian Institute of Science PhD Indian Institute of Technology Delhi 2022  
30 47536 Mr. Om Sai Male Chennai Mathematical Institute B.Sc. Final year Student      
31 47577 Mr. Cyril J Jacob Male Chennai Mathematical Institute Research Scholar St. Berchman's College 2019  
32 47689 Mr. Arnab Roy Male Tata Institute of fundamental research Integrated PhD student Tata Institute of fundamental research    
33 47691 Mr Ashutosh Roy Choudhury Male Tata Institute of Fundamental Research PhD Tifr mumbai 2022  
34 47715 Dr. Bivas Khan Male Indian Institute of Science Education and Research, Pune Postdoc Fellow Indian Institute of Technology Bombay    
35 47785 Mr Manav Gaddam Male Institute of Mathematical Sciences PhD Indian Statistical Institute, Kolkata (M.Math) 2018  
36 47805 Dr. Chandranandan Gangopadhyay Male IISER Pune INSPIRE Faculty fellow Chennai Mathematical Institute 2014 15/07/2019
37 47814 Ms Yajushi Srivastava Female Indian Institute of Science Education and Research (IISER) Tirupati PhD Indian Institute of Science Education and Research (IISER) Tirupati 2022  
38 47818 Mr Mohit Upmanyu Male Tata Institute of Fundamental Research PhD Student      
39 47820 Dr. Arghya Pramanik Male ISI Kolkata Visiting Scientist University of  Hyderabad 2015 14/02/2023
40 47871 Dr. Anantadulal Paul Male ICTS Postdoc University of Pune 2015 17/01/2022
Waiting list      
1 45390 Mr. Subham De Male IIT DELHI MSc Student IIT DELHI    
2 45760 Mr. Soumitra Das Male IISER Berhampur PDRF North-Eastern Hill University 2012 09/02/2021
3 46214 Ms. Riddhi Vinay Patil Female IIT Gandhinagar PhD University of Mumbai 2021  
4 46253 Mr Anindya Mukherjee Male IISER Berhampur PHD student Jadavpur university 2021  
5 46138 Mr. Divyendu Roy Choudhuri Male Indian Institute of Science of Education and Research, Berhampur BS-MS student Indian Institute of Science Education and Research, Berhampur Awaiting Result  
6 47016 Mr. Ajay Prajapati Male Indian Institute of Science, Bangalore Ph.D. Indian Institute of Technology, Kanpur 2022  
7 47107 Mr. Saikat Manna Male Indian Institute Of Technology Gandhinagar (IIT Gandhinagar) MSc Student Indian Institute Of Technology Gandhinagar (IIT Gandhinagar) Awaiting Result  
8 47132 Mr. Soumyadeep Misra Male Indian Institute Of technology,Madras MSc Student Indian Institue of technology,Madras Awaiting Result  
9 47152 Mr. Mifron Fernandes Male Indian Institute of Technology Madras MSc Student Indian Institute of Technology Madras Awaiting Result  
10 47353 Mr. Ayush Khare Male IISER MOHALI Int PhD IISER MOHALI 2022  
11 47397 Ms. Manika Gupta Female Institute of Mathematical Sciences PhD Student Kerala School of Mathematics 2022  
12 47418 Mr Supriya Roy Male   PhD      
13 47594 Mr. Suraj Rajendra Kulkarni Male The Institute of Mathematical Sciences Integrated PhD Student IMSc, Chennai Awaiting Result  
14 47858 Mr Raman Rishi Male IIT Jodhpur MSc 2nd year student IIT JODHPUR Awaiting Result  

 

 

How to Reach: 

Click the following link for further details

https://www.cmi.ac.in/contact/how-to-get-to-cmi.php

School Short Name: 

tbg

Last Date Application: 

Friday, March 31, 2023

School Type: 

AIS

Separate faculty form: 

0