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Modifications of Hodge bundles and enumerative geometry: the stable hyperelliptic locus

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Abstract

We study the stable hyperelliptic locus, i.e. the closure, in the Deligne–Mumford moduli of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated with a family of stable curves, we construct a bundle map (‘degree-2 Brill–Noether’) from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton–MacPherson residual intersection theory, the class of the structure sheaf and various other sheaves supported on the stable hyperelliptic locus can be computed by the Porteous formula and similar tools.

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Notes

  1. The presence of extraneous, often excessive, boundary components is a difficulty in Gromov–Witten theory as well.

  2. Without loss of significant generality.

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Acknowledgments

I thank the referee for his very careful reading of the paper and highly detailed comments and corrections, including his insistence that we reproduce the Faber–Pandharipande formula in genus 4; these have resulted in a much improved paper. I thank Ann Kostant for her patient and determined assistance.

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  1. Mathematics Department, University of California, Big Springs Rd. Surge Facility, Riverside, CA, 92521, USA

    Ziv Ran

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  1. Ziv Ran
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Ran, Z. Modifications of Hodge bundles and enumerative geometry: the stable hyperelliptic locus. Sel. Math. New Ser. 21, 1203–1269 (2015). https://doi.org/10.1007/s00029-015-0184-z

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