Secant planes of a general curve via degenerations

Abstract

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite.

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Notes

  1. 1.

    For example, if \(X_0\) is compact type, a line bundle on \(X_0\) with concentrated mutidegree restricts to a degree zero line bundle over all but one component.

  2. 2.

    See the discussion at the beginning of this section.

  3. 3.

    More concretely, let \(\ell \) denote the unique path in the directed dual graph \(\Gamma \) from \(v_0\) to v and let \(X_{\ell }\) denote the sub-curve of \(X_0\) corresponding to \(\ell \); then \(P^*_v\) is the intersection of \(C_v\) with the closure of its complement inside \(X_{\ell }\).

  4. 4.

    Note the change of variables relative to the preceding sections: \(s=r_1\) and \(s-d+r=r_2\). Recall that \(d=d_1-d_2\).

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Acknowledgements

We thank Melody Chan, Javier Gargiulo, Alberto López, Brian Osserman and Nathan Pflueger for useful conversations; the Brazilian CNPq, whose postdoctoral scheme allowed the first and third authors to meet; and the anonynomous referee, for his/her careful reading and corrections.

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Correspondence to Ethan Cotterill.

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Naizhen Zhang was supported by the Methusalem Project Pure Mathematics at KU Leuven during the preparation of this paper. Xiang He is supported by the ERC Consolidator Grant 770922 - BirNonArchGeom.

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Cotterill, E., He, X. & Zhang, N. Secant planes of a general curve via degenerations. Geom Dedicata (2020). https://doi.org/10.1007/s10711-020-00545-3

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Keywords

  • Moduli of curves
  • Linear series
  • Degenerations

Mathematics Subject Classification

  • 14D20
  • 14H10
  • 14H51
  • 14N10
  • 05A19