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Seminar by Suratno Basu

Thursday, October 24, 2019 - 11:35 to 12:35
SMS Seminar Room
Suratno Basu
IMSc Chennai
Degeneration of intermediate Jacobians and the Torelli type theorems
Let $X$ be a smooth, irreducible, projective curve of genus $g\geq 2$ defined over the field of complex numbers. The Jacobian of $X$ is defined as the ``moduli space'' of rank $1$, degree $0$ line bundles. This variety has close relationship with $X$. Namely, Torelli showed that the Jacobian $J(X)$ together with its canonical polarization recovers $X$ upto isomorphism. The study of analogous question for the ``moduli space'' of higher rank bundles was first started by Mumford and Newstead. To be more precise, let $M_{X}(2,L)$ be the moduli space of rank $2$ stable bundles with determinant $L$. We assume that the degree of $L$ is odd. Then they showed that the isomorphism class of $M_{X}(2,L)$ determines the isomorphism class of $X$. This theorem was subsequently generalized for any rank by Narashiman and Ramanan. In recent times we have initiated a study to understand analogous questions for certain ``moduli spaces of sheaves'' over singular nodal curves. In the case when the curve has two smooth projective components meeting at a simple node we show that from certain moduli space of rank $2$, stable torsion free sheaves we can recover the curve. More recently, in collaboration with Ananyo Dan and Inder Kaur we show that from the certain moduli space of rank $2$ sheaves over irreducible nodal curves (with single node) we can recover the curve. Finally, in a joint work with Sourav Das, we have shown from a certain moduli space (constructed by D Gieseker) of sheaves of any rank and ``degree'' we can recover the nodal curve. In this talk we shall demonstrate some of these results. We will begin by recalling some basic background notions needed to explain these results. The talk will be as self contained as possible.

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