Algebraic geometry seminar

Day & Date: Monday, December 19, 2022.

Time: 4 pm.

Venue: Online talk at meet.google.com/vzr-ctov-kbs


Title: Seshadri constants over fields of characteristic zero

Speaker: Arghya Pramanik

Abstract:  Let X be a smooth projective variety defined over a field k of
characteristic 0 and let L be a nef line bundle defined over k. In this
talk, I will show that if x ∈ X is a k-rational point then the Seshadri
constant ε(X, L, x) over \bar{k} is the same as that over k. I will also
construct families of varieties whose global Seshadri constant ε(X) is
zero. I also discuss a result on the existence of a Seshadri curve with a
natural (and necessary) hypothesis. In a recent paper, Fulger and Murayama
have defined a new version of Seshadri constants for vector bundles in a
relative setting over algebraically closed fields. We generalize the
definition for a general field and show that our results on line bundles
are also true for vector bundles. This talk is based on joint work with
Shripad M. Garge.

Title: Ovoids in PG(3,q) and Algebraic codes

Speaker; Prof. N. S. Narasimha Sastry, Formerly from ISI Bangalore and IIT Dharwad

Day, Date and Time: Tuesday, 20th December 2022 at 2 PM

Venue: Ramanujan Hall, Dept of Mathematics

Abstract:  Ovoids in PG(3,q) are the Incidence geometric analogues of spheres in Euclidean 3-space. If q is odd, 

Baralotti and Panella showed independently that elliptic quadrics are the only ovoids in PG(3,q). Further, an ovoid 

and its set of tangent lines determine each other.  However, if q is an odd power of two, then apart from elliptic 

ovoids (which exist for all q),  PG(3,q) admits one more projective class of ovoids which are not projectively equivalent 

to elliptic ovoids. They were discovered by Tits, using the graph - field outer automorphism of PSp(4,2^{2n+1!}), and its 

stabilizer in PSp(4,q)  (called the Suzuki simple group, the same as  ^2 B_2(q) in Lie notation) was discovered earlier 

by Suzuki as the final piece in the long series of works on the classification of finite Zassenhaus groups by Zassenhaus, 

Ito, Feit and Suzuki.  Further, the set of tangent lines of two ovoids can coincide  even if they are projectively nonequivalent.  

These are the only families of ovoids in PG(3,q) known and classification of ovoids in PG(3,q) is a major problem in Incidence 

Geometry.  Because of their connections to many other combinatorial structures ( like inversive planes, generalized quadrangles,  

group divisible designs, ...) and the very exceptional behavior of the Suzuki simple group and the Tits ovoid, understanding the properties of ovoids in general, and their distribution in PG(3,q) in particular, are of considerable significance.

In this talk, I will present some facts known about ovoids in general, their distribution and the role of algebraic codes in 

understanding them. An effort will be made to clarify all the basic notions involved.

Commutative algebra seminar

Date and time: Tuesday, 20 December 2022, 3pm

Venue: Ramanujan Hall

Speaker: Shravan Patankar, University of Illinois, Chicago, IL, USA

Title: Vanishing of Tors of absolute integral closure in equi-characteristic zero

Abstract: We show that the vanishing of Tors of the absolute integral closure forces regularity assuming further that the ring under consideration is IN-graded of dimension 2. This answers a question of Bhargav Bhatt, Srikanth Iyengar, and Linchuan Ma. We use almost mathematics over R+ to deduce properties of the Noetherian ring R and the theory of rational surface singularities. In particular, in spite of being a question purely in commutative algebra our proof uses algebro-geometric methods.

Mathematics Department

Virtual Commutative algebra seminar

Speaker: Shigeru Kuroda, Tokyo Metropolitan University, Hachioji, Japan
Date/Time: Friday, 23 December 2022, 5:30pm 

Gmeet link: meet.google.com/wph-nyzd-hyj

Title: Z/pZ-actions on the affine space: classification, invariant ring, and plinth ideal 

Abstract: Let k be a field of characteristic p>0. In this talk, we consider the Z/pZ-actions on the affine n-space over k, or equivalently the order p automorphisms of the polynomial ring k[X] in n variables over k. For example, every automorphism induced from a G_a-action is of order p. Hence, the famous automorphism of Nagata is of order p. Such an automorphism is important to study the automorphism group of the k-algebra k[X].

We discuss two topics: (1) classification, and (2) the relation between the polynomiality of the invariant ring and the principality of the plinth ideal. We also present some conjectures and open problems.

For more information and links to previous seminars, visit the website of VCAS: 

https://sites.google.com/view/virtual-comm-algebra-seminar