IPDF Seminar

Tuesday, 25 April 2023, 11.30 pm

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Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Arusha C

Affiliation:  TIFR, Mumbai

Title: Vector Bundles and Parabolic Bundles on Nodal Curves

Abstract:  A great deal is known about vector bundles and parabolic bundles in the case of smooth curves but a similar study on singular curves has been relatively slow.
Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and much more difficult as there are torsion free
sheaves on a nodal curve that are not locally free unlike the smooth case. Ramanan proved in the 70s that a universal family (also called a Poincar ́e bundle) exists for the moduli problem vector bundles on smooth curves if and only if the rank and degree are coprime. One of the key elements in the proof is the computation of the Picard group of the moduli space. First, we prove the non-existence of a Poincar ́e bundle for the moduli problem of vector bundles on nodal curves when the degree and rank are not coprime closely following that of Ramanan. When the degree is sufficiently high, the pushforward of a Poincar ́e bundle to the moduli space is a vector bundle, called the Picard bundle. Although the existence of Poincar ́e bundles (hence Picard bundles) depend on the rank and degree being relatively prime, there always exists a Poincar ́e family of projective bundles called the projective Poincar ́e bundle. Similarly, there is a projective Picard bundle. Next, we discuss the stability of these bundles. Finally, we move from vector bundles to parabolic bundles. Mehta-Seshadri theorem gives a one to one correspondence between irreducible unitary representations of the fundamental group of a punctured compact Riemann surface and stable parabolic bundles on the compact Riemann surface with a parabolic structure at the punctures. We prove that such a correspondence does not hold for nodal curves.

Algebraic Groups seminar

Tuesday, 25 April 2023, 4.30 pm

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Venue: Room 105

Host: Shreepad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Study of diagonalizable groups