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Degeneration of Linear Series from the Tropical Point of View and Applications

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Nonarchimedean and Tropical Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

We discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on applications to problems in Brill–Noether theory and arithmetic geometry.

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Notes

  1. 1.

    We say C 0 is semistable if it is reduced and has only nodes as singularities. The term strongly semistable is not completely standard.

  2. 2.

    The reason for the name compact type is that the Jacobian of a nodal curve C 0 is an extension of the Jacobian of the normalization of C 0 by a torus of dimension equal to the first Betti number of the dual graph. It follows that the Jacobian of C 0 is compact if and only if its dual graph is a tree.

  3. 3.

    As a caution, some authors use the opposite sign convention.

  4. 4.

    Another natural idea would be to try to define r(D) as one less than the “dimension” of R(D) considered as a semimodule over the tropical semiring T consisting of \(\mathbb{R} \cup \{\infty \}\) together with the operations of min and plus. However, this approach also faces significant difficulties. See [73] for a detailed discussion of the tropical semimodule structure on R(D).

  5. 5.

    Sam Payne has pointed out that there is a gap in the proof of Conjectures 3.10 and 3.14 of Baker [17] given in [34]. The claim on page 82 that W d, ϕ r has nonempty fiber over b 0 does not follow from the discussion that precedes it. A priori, the fiber of \(\overline{W_{d,\phi }^{r}}\) over b 0 might be contained in the boundary \(\overline{P_{\phi }^{d}}\setminus \mathop{\mathrm{Pic}}\nolimits _{ \phi }^{d}\) of the compactified relative Picard scheme.

  6. 6.

    Although there is no known efficient algorithm; indeed, it is proved in [85] that this problem is NP-hard.

  7. 7.

    Note that there is a one-to-one correspondence between norms on K(X) and valuations on K(X), hence the terminology Val X . It is often convenient to work with (semi-)valuations rather than (semi-)norms.

  8. 8.

    See [21, Appendix B] for an introduction to the theory of \(\mathbb{R}\)-trees. For our purposes, what is most important about \(\mathbb{R}\)-trees is that there is a unique path between any two points.

  9. 9.

    A vertex-weighted finite graph (G, ω) is called stable if every vertex of weight zero has valence at least 3.

  10. 10.

    This homeomorphism is in fact an isomorphism of “generalized cone complexes with integral structure” in the sense of Abramovich et al. [1].

  11. 11.

    Note that our sign convention here for the divisor of a rational function on \(\Gamma \), which coincides with the one used in [5], is the opposite of the sign convention used in Sect. 3.2, which is also used in a number of other papers in the subject. This should not cause any confusion, but it is good for the reader to be aware of this variability when perusing the literature.

  12. 12.

    Brian Osserman [107] has recently proposed a different framework for doing this.

  13. 13.

    In fact, they consider the graph in which the lengths of the bridge edges between the loops are all zero. There is, however, a natural rank-preserving isomorphism between the Jacobian of a metric graph with a bridge and the Jacobian of the graph in which that bridge has been contracted, so their argument works equally well in this case. We consider the graph with bridges because of its use in [78, 79].

  14. 14.

    Amini has apparently made substantial progress in this direction.

  15. 15.

    This means that for every continuous function \(f: C \rightarrow \mathbb{R}\), we have ∫Cf δn = ∫ C f ω Ar.

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Acknowledgements

The authors would like to thank Omid Amini and Sam Payne for enlightening discussions, and Eric Katz and Joe Rabinoff for helpful feedback on our summary of Katz et al. [81]. They also thank Spencer Backman, Dustin Cartwright, Melody Chan, Yoav Len, and Sam Payne for their comments on an early draft of this paper.

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  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Matthew Baker

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    David Jensen

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  2. David Jensen
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Correspondence to Matthew Baker .

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  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Matthew Baker

  2. Department of Mathematics, Yale University, New Haven, Connecticut, USA

    Sam Payne

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Baker, M., Jensen, D. (2016). Degeneration of Linear Series from the Tropical Point of View and Applications. In: Baker, M., Payne, S. (eds) Nonarchimedean and Tropical Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-30945-3_11

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