Scaling of conformal blocks and generalized theta functions over \(\overline{\mathcal {M}}_{g,n}\)

Abstract

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.

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Notes

  1. 1.

    One can write down a more general sequence for the scrolls \((S(a_1,\ldots ,a_d)=\mathbb {P}(\mathcal {E}), \mathcal {O}(1))\), discussed in Sect. 6.4.

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Acknowledgments

P.B. was supported on NSF Grant DMS-0901249, and A.G. on DMS-1201268 and in part by DMS-1344994 (RTG in Algebra, Algebraic Geometry, and Number Theory, at UGA). We thank the anonymous referee for their time and thoughtful feedback.

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Correspondence to Angela Gibney.

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Belkale, P., Gibney, A. & Kazanova, A. Scaling of conformal blocks and generalized theta functions over \(\overline{\mathcal {M}}_{g,n}\) . Math. Z. 284, 961–987 (2016). https://doi.org/10.1007/s00209-016-1682-1

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Keywords

  • Conformal Blocks
  • Generalized Theta Functions
  • Chern Class
  • Ample Canonical Line Bundle
  • Projective Variety