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1 results

  • 9 - Picard Group of Moduli Space of G-Bundles

  • Shrawan Kumar, University of North Carolina, Chapel Hill
  • Book:
    Conformal Blocks, Generalized Theta Functions and the Verlinde Formula
    Published online:
    19 November 2021
    Print publication:
    25 November 2021, pp 379-408

  • Summary

    The main aim of this chapter is to determine the Picard group of the moduli space M(G) of semistable G-bundles over a smooth projective curve ? explicitly and show that it is generated by the theta bundles. In fact, it is shown that the theta bundle corresponding to the fundamental representation with the minimal Dynkin index freely generates the Picard group. In particular, it is isomorphic with the group of integers. We further prove that M(G) is Gorenstein and we identify its dualizing line bundle. Moreover, we prove the vanishing of the higher cohomology of the theta bundles over M(G). The moduli space M(G) is identified as a weighted projective space for ? an elliptic curve. This identification allows us to directly prove the above results in the case of genus-1 curves.