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Algebraic Geometry

Moduli spaces, Toric varieties,birational geometry, Bridgeland stability, Principal bundles, Vector bundles.

IRIS Webinar

In algebraic geometry the spaces parametrizing geometric objects, such as varieties with given invariants, or sheaves on varieties, are themselves endowed with a natural algebraic structure which reflects the properties, both abstract and projective, of the parametrized objects. Moduli theory is the study of these structures in the various situations of interest, and it is the main topic of this project. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety, called a moduli space, in its own right.

  • Thus in algebraic geometry, the metaphor of thinking about a community of organisms as itself being an organism is not a just a metaphor but a rigorous and quite useful fact.

  • Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety.

  • Next to it, we study algebraic varieties up to birational equivalence, which is an equivalence relation of fundamental importance for algebraic varieties of dimension at least 2 and their moduli theory.

Algebraic Geometry

Arijit Dey

Principal Investigator

People

Arijit Dey

Area of Interest

Arijit Dey

Principal Investigator

Department of Mathematics

Website
arijit@iitm.ac.in
Algebraic Geometry

Project

Toric geometry provides a bridge between algebraic geometry and combinatorics of fans and polytopes. For each polarized toric variety (X,L) one can associate a polytope P. One effectively uses this correspondence to study birational geometry for toric varieties. To this end, one studies such as Minimal Model Program, Mori fiber spaces, and chamber structures on the cone of effective divisors of a toric variety.

A toric variety is a normal algebraic variety over a field k that contains an algebraic torus T as an open dense subset so that the natural action of T on itself extends to an action T £ X !X. Every n-dimensional toric variety gives rise to a certain collection of cone §X of cones in Rn called a fan. Many algebrogeometric properties of the variety X can be translated to combinatorial properties of §X that are, in general, simpler to be studied. Thanks to this combinatorial structure, toric varieties provide a “testing ground” for general conjectures. For instance, one of the greatest achievements in the problem of classification of complex projective manifolds is the Minimal Model Program (MMP). Roughly speaking, the goal of the MMP is perform successive birational transforma-tions in a given complex algebraic variety to obtain a birational model which is “as simple as possible”. In 1988, Mori established the MMP for 3-folds. With this result Mori earned the Fields medal in 1990. In the toric context, the MMP was established by Reid in 1983 for varieties of arbitrary dimension using the combinatorics of the associated fans.

Moduli spaces are one of the fundamental objects in algebraic geometry and they arise in connection with classification problems. Roughly speaking a moduli space for a collection of objects A and an equivalence relation » is a space M (suitably defined) such that every point corresponds to one and only one object of A/ » . In this project we are mostly interested in moduli space M(r,c1,c2) (M(r,c1,c2)) of rank r (semi)stable vector bundles (torsion-free sheaves) with fixed Chern classes c1 and c2 on a smooth polarized projective surface (X,H).

The original construction of the moduli space is due to Mumford, Seshadri (for curves), Gieseker (for surface), Maruyama (for arbitrary dimension) which corepresents the moduli functor. The closed points of M(r,c1,c2) (M(r,c1,c2)) corresponds to isomorphism classes (S-equivalence) of (semi)stable vector bundles (torsion-free sheaves). Once the existence of moduli space is established, the question arises as what can be said about its local and global algebraic and topological nature of these moduli spaces? For example one can ask following questions: when it is non-empty? Is it connected, irreducible, rational, smooth? What is the birational type of these spaces in particular Kodaira dimension? Can one compute numerical invariants like Betti, Hodge numbers?

Expected deliverables of the research

  • Study the moduli space of T -equivariant vector bundles/principal bundles on a toric variety.
  • Try to prove the rationality conjecture, i.e. M(r,c1,c2) is rationalwhenever it is nonempty and underlying surface is rational.
  • Find topological as well as algebraic invariants forM(r,c1,c2), when underlying surface is P2 and r >= 3 and more generally for any surface other than K3.
  • Study MMP for the moduli space of vector bundles.

Current status

Currently working on a problem related to finding algebraic invariants for certain moduli spaces of vector bundles. A post-doc has joined under this project who is also working on the problem. Making groundwork for solving other problems and discussing with collaborators.

Collaborations

International Collaborators

  • Marco Antei, Steven Ryan.

Technical/ Scientific Progress

New work done in the project

  • Brauer group of parabolic moduli.

Output

  • Preprint available on request

Mobility

Visits planned for PI, co-PIs, international collaborators and students (both inbound and outbound)

Jorge Arturo Esquivel Araya a Ph.D student of my collaborator Marco Antei from University of CostaRica is visiting currently, he will be in IIT Madras till May middle. Together with Marco ANtei, we are working on a joint project related to Fundamental group scheme. In coming March one of my collaborator Ozhan Genc has agreed to visit IIT Madras for one month on a collaborative research program. Other than that I am working with IPDF Sujoy Chakroborty on some problems related to parabolic moduli over algebraic curves.

Updates

Relevant Updates

Along with IPDF Sujoy Chakraborty we are organising a lecture series on some topics in Algebraic geometry. Details can be found in link