# NCMW - Seshadri Constants (2019)

Venue: |
NISER, Bhubaneswar, Odisha(Orissa) |

Date: |
16th, Dec 2019 to 21st, Dec 2019 |

Name |
Prof. Indranil Biswas | Dr. Krishna Hanumanthu | Dr. Ritwik Mukherjee |

Mailing Address |
Professor, School of Mathematical Sciences, TIFR. |
Associate Professor, CMI |
Reader-F, School of Mathematical Sciences, NISER |

**Speakers with their affiliations:**

Name of the speaker |
Position |
Affiliation |

D. S. Nagaraj | Professor | Indian Institute of Science Education and Research, Tirupati |

Krishna Hanumanthu | Associate Professor | Chennai Mathematical Institute, Chennai |

Omprokash Das | Reader | Tata Institute of Fundamental Research, Mumbai |

Ritwik Mukherjee | Reader-F | National Institute of Science Education and Research Bhubaneswar |

Anantadulal Paul | Research Scholar | National Institute of Science Education and Research Bhubaneswar |

Nilakantha Das | Research Scholar | National Institute of Science Education and Research Bhubaneswar |

Rahul Kumar Singh | Post-doctoral Fellow | Tata Institute of Fundamental Research, Mumbai |

Praveen Kumar Roy | Post-doctoral Fellow | Tata Institute of Fundamental Research, Mumbai |

** List of Tutors**

Name of the tutor |
Position |
Affiliation |

Snejajit Misra | Post-doctoral Fellow | Tata Institute of Fundamental Research, Mumbai |

Nabanita Ray | Post-doctoral Fellow | Tata Institute of Fundamental Research, Mumbai |

Aditya Subramaniam | Research Scholar | Chennai Mathematical Institute, Chennai |

**Syllabus:**

Name of the Speaker : |
Detailed Syllabus |

D.S. Nagaraj | Curve and surface theory. Helectured on Riemann-Roch theorem for curves and surfaces, blow ups of nonsingular surfaces at points, adjunction formula, intersection theory on surfaces, Hodge index theorem. The main reference used was relevant portions of R. Hartshorne's Algebraic geometry. |

Krishna Hanumanthu | Basics of Seshadri constants. After giving basic definitions, examples, he covered the motivation for defining them by discussing Fujita conjecture and jet separation. Then the connections to linear systems of plane curves and the Nagata and SHGH conjectures was discussed. The questions of irrationality of Seshadri constants was also discussed. The main references were the chapter on Seshadri constants in R. Lazarsfeld's Positivity in Algebraic Geometry I and some expository lecture notes. |

Omprokash Das | Positive characteristic analogues of Seshadri constants called the Frobenius Seshadri constants. Using a Frobenius analogue of jet separation, M. Musta\c{t}\u{a} and K. Schwede defined Frobenius Seshadri constants and proved many interesting properties and connections to the usual Seshadri constants. These lectures covered most of their results and the mainreference was their original paper: A Frobenius variant of Seshadri constants, Mathematische Annalen 2014. |

Ritwik Mukherjee | Gave an introduction to Enumerative Geometry of singular curves. He explained the idea of viewing Enumerative Numbers as zero sets of sections of Vector Bundles and interpreting these numbers as intersection of Cycles. The two crucial things in this subject is to show transversality and compute degenerate contributions to the intersection number. The main references were the papers by Ritwik Mukherjee and Somnath Basu on enumeration of singular curves. |

Anantadulal Paul | Continued with the setup of Ritwik Mukherjee’s lecture and explained how to enumerate curves in a surface that are tangent to a smooth divisor. As before, these numbers can also be viewed as intersection of cycles (or equivalently zeros of a section of an appropriate bundle). The main reference was Anantadulal Paul’s paper on enumerating curves with tangencies. |

Nilakantha Das | Continued with the setup of Ritwik Mukherjee’s lecture and explained how to enumerate curves in a family of surfaces. He focused on the example of enumerating singular curves in P^3, whose image lies inside a P^2. The main reference was the paper by Nilkantha Das and Ritwik Mukherjee on enumerating singular planar curves in P^3. |

Rahul Kumar Singh | Moduli space of stable maps. He explained how Kontsevich and Manin obtained a formula for the characteristic number of rational curves in P^2. He then went on to explain how one can generalize this to a family version (in particular, enumerating rational curves in P^3, whose image lies in a P^2). Finally, he spoke about generalizing this result to higher singularity (cusps). The main reference for this talk were the papers by Anantadulal, Ritwik Mukherjee and Rahul Singh on counting rational curves in a moving family of P^2 and the paper by Ritwik Mukherjee and Rahul Singh on counting cuspidal curves in a moving family of P^2. |

Praveen Kumar Roy | A research talk on some recent results on Seshadri constants for line bundles on hyperelliptic surfaces and surfaces of general type. |

**Time Table**

Day |
Date |
Lecture 1 (9.30 – 11.00) |
Tea (time) |
Lecture 2 (11.30 – 1.00) |
Lunch (time) |
Lecture 3 (02.30 – 3.30) |
Tea (time) |
Discussio n/ Tutorial (04.00 – 5.00) |
Tea & snacks (time) |

name of the speaker (abbrevia tion) | name of the speaker (abbreviati on) | name of the speaker (abbrevia tion) | name of the speaker & tutors (abbrevia tion) | ||||||

Mo n | 16/12/1 9 | DSN | DSN | RM | DSN, NR,AS | ||||

Tue s | 17/12/1 9 | DSN | DSN | RS | DSN, NR,AS | ||||

We d | 18/12/1 9 | AP | ND | AP | ND | ||||

Thu | 19/12/1 9 | KH | KH | KH, AS, SM | KH, AS, SM | ||||

Fri | 20/12/1 9 | OD | OD | PRK | OD,SM NR, | ||||

Sat | 21/12/1 9 | KH | OD | OD, NR, SM | OD,SM NR, |

**Full forms for the abbreviations of speakers and tutors :**

**Speakers:**DSN: D. S. Nagaraj

RM: Ritwik Mukherjee

RS: Rahul Singh

AP: Anantadulal Paul

ND: Nilakantha Das

KH: Krishna Hanumanthu

OD: Omprokas Das

PRK: Praveen Kumar Roy

**NR: Nabanita Ray**

Tutors:

Tutors:

SM: Snehajit Misra

AS: Aditya Subramaniam

**List of actual Participants**

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 |

1 | 28738 | Mr. Aishik Chattopadhyay | Male | Chennai Mathematical Institute | PhD | Pondicherry University | 2018 | |

2 | 29337 | Mr Arghya Pramanik | Male | IIT bombay | PHD | University Of Hyderabad | 2015 | |

3 | 29355 | Dr. Saugata Purkayastha | Male | Assam Don Bosco University | Assistant Professor | Department of Mathematics, Gauhati University | 2010 | 1 Dec 2015 |

4 | 29522 | Mr. Pradeep Das | Male | Harish-Chandra Research Institute | PhD Student | IIT Kanpur | 2011 | |

5 | 29534 | Mr. Anoop Singh | Male | Harish-Chandra Research Institute | PhD | University of Allahabad | 2013 | |

6 | 29613 | Mr. Anindya Ganguly | Male | IIT Kharagpur | MS(by Research) | IIT Kharagpur | 2016 | |

7 | 29627 | Mr. Saikat Maity | Male | University of Calcutta | B.Sc complete year student | |||

8 | 29691 | Mr. Ritankar Nath | Male | Tata Institute of Fundamental Research, Mumbai | Integrated PhD | Tata Institute of Fundamental Research, Mumbai | ||

9 | 30174 | Dr. Shameek Paul | Male | RKMVERI | Assistant Professor | University of Mumbai | 2005 | 19 Jan 2012 |

10 | 30221 | Ms. Ananya Chaturvedi | Female | Tata Institute of Fundamental Research | Postdoctoral Fellow | University of Houston | 2012 | 15 Dec 2016 |

11 | 30282 | Mr. Debabrota Mondal | Male | Indian institute of technology Bhubaneswar | MSc Student | IIT Bhubaneswar | App / Awt Res. | |

12 | 30359 | Dr. Parangama Sarkar | Female | Chennai Mathematical Institute | Postdoc | IIT Madras | 2011 | 3 Jun 2016 |

13 | 30482 | Mr. Amit Kumar Singh | Male | IIT Madras | PhD | University of Allahabad | 2011 | |

14 | 30483 | Ms Arusha C | Female | IISER Bhopal | PhD Student | M.S University of Baroda | 2015 | |

15 | 30507 | Mr. Sujoy Chakraborty | Male | TIFR Mumbai | Research Scholar | CMI, chennai | 2015 | |

16 | 30537 | Mr. Areeb Shah | Male | Chennai Mathematical Institute | B.Sc. Student |

**How to reach**