Specialization of Linear Series for Algebraic and Tropical Curves (14w5133)

Arriving in Banff, Alberta Sunday, March 30 and departing Friday April 4, 2014


(Georgia Institute of Technology)

(University of Rome: Roma Tre)

(The Ohio State University)

(University of Waterloo)

(Yale University)


This workshop will bring together leading experts and young researchers from tropical geometry and the classical theory of linear series on algebraic curves. These are two largely separate mathematical research communities that rarely meet together at conferences, but recent breakthroughs have created important links connecting fundamental techniques, theorems, and open problems on both sides. As explained in the overview, new specialization lemmas in tropical geometry simultaneously generalize both the classical theory of limit linear series, due to Eisenbud and Harris, as well as the original tropical specialization lemma, due to Baker. Both approaches lead to proofs of the Brill--Noether Theorem, and the extent to which combining the two approaches may lead to significant new results is not yet known, but experts on both sides are very interested in exploring the possibilities. In particular, we hope to investigate whether, and how, a combination of tropical and classical linear series techniques might be used to resolve such outstanding open problems as the Maximal Rank Conjectures, the Kodaira dimension of $M_{23}$, and Caporaso's Conjecture for an algebro-geometric interpretation of the Baker-Norine rank of a divisor on a tropical curve.

Objective 1. Create an opportunity for tropical geometers and researchers in the classical theory of linear series to learn the state of the art and fundamental techniques on both sides, and foster mutual understanding across these two disciplines.

Objective 2. Explore possibilities for combining classical and tropical techniques to resolve outstanding problems on both sides, such as those mentioned above.

Objective 3. Help advanced graduate students and recent PhDs working in these two areas to connect with peers and experts across these two subjects, and encourage an understanding of the rigorous connections between tropical geometry and classical linear series.

Introducing young researchers in tropical geometry to experts from the algebraic theory of linear series is perhaps particularly important, to help those working primarily with combinatorial statements to focus their efforts in directions most closely connected to questions of classical interest. This meeting should also accelerate the increasing appreciation among algebraic geometers for the potential power of tropical techniques. We expect that more and more will encourage their students and postdocs to learn these new methods alongside more classical approaches.

We have contacted several of the leading experts, both from tropical geometry and from the classical theory of linear series, and are pleased to report widespread support for the proposed meeting. We are confident that many others, including advanced graduate students and recent PhDs, will respond positively to the opportunity to participate in this workshop.

We conclude with a selected bibliography of research articles most relevant to the proposed workshop.


  1. O. Amini, ``Reduced divisors and embeddings of tropical curves," to appear in Trans. Amer. Math. Soc., preprint, arxiv:1007.5364

  2. O. Amini and M. Baker,``Linear series on metrized complexes of algebraic curves," preprint, arxiv:1204.3508.

  3. O. Amini and L. Caporaso, ``Riemann--Roch theory for weighted graphs and tropical curves," preprint, arxiv:1112.5134, 2011.

  4. M. Aprodu and G. Farkas, ``Koszul cohomology and applications to moduli," In Grassmannians, moduli spaces and vector bundles, 25--50, Clay Math. Proc., 14, Amer. Math. Soc., Providence, RI, 2011.

  5. M. Baker, ``Specialization of linear systems from curves to graphs," Algebra Number Theory 2 (2008), no.~6, 613--653.

  6. M. Baker and X. Faber, ``Metric properties of the tropical Abel--Jacobi map," J. Algebraic Combin. 33 (2011), no.~3, 349--381.

  7. M. Baker, S. Norine, ``Riemann--Roch and Abel--Jacobi theory on a finite graph," Adv. Math. 215 (2007), no.~2, 766--788.

  8. M. Baker, S. Payne and J. Rabinoff, ``Nonarchimedean geometry, tropicalization, and metrics on curves," preprint, arXiv:1104.0320.

  9. V. Baratham, D. Jensen, C. Mata, D. Nguyen and S. Parekh, ``Towards a tropical Proof of the Gieseker-Petri Theorem," preprint, arxiv: 1205.3987.

  10. S. Brannetti, M. Melo, and F. Viviani. ``On the tropical Torelli map," Adv. Math. 226 (2011), no.~3, 2546--2586.

  11. L. Caporaso, ``Algebraic and combinatorial Brill--Noether theory," In Compact moduli spaces and vector bundles, Contemp. Math. 564, p. 69--85, Amer. Math. Soc. 2012.

  12. L. Caporaso. ``Algebraic and Tropical Curves: Comparing their Moduli Spaces." In Handbook of Moduli, edited by Farkas and Morrison; (2011).

  13. L. Caporaso, ``Linear series on semistable curves," Int. Math. Res. Not. 13 (2011), 2921--2969.

  14. L. Caporaso and F. Viviani. ``Torelli theorem for graphs and tropical curves," Duke Math. J. 153 (2010), no.~1, 129--71.

  15. W. Castryck and F. Cools, ``Newton polygons and curve gonalities," J. Algebr. Comb. 35 (2012), no.~3, 345--366.

  16. M. Chan, ``Combinatorics of the tropical Torelli map," to appear in Algebra Number Theory, preprint arXiv:1012.4539v2.

  17. F. Cools, J. Draisma, S. Payne, and E. Robeva, ``A tropical proof of the Brill--Noether Theorem," Adv. Math. 230 (2012), 759--776.

  18. M. A. Cueto and S. Lin, ``Tropical secant graphs of monomial curves," In 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 669--680, Discrete Math. Theor. Comput. Sci. Proc., Nancy, 2010.

  19. D. Eisenbud and J. Harris, ``A simpler proof of the Gieseker--Petri theorem on special divisors," Invent. Math. 74 (1983), no.~2, 269--280.

  20. D. Eisenbud and J. Harris, ``Limit linear series: Basic theory," Invent. Math. 85 (1986), 337--371.

  21. D. Eisenbud and J. Harris, ``The monodromy of Weierstrass points," Invent. Math. 90 (1987), no.~2, 333--341.

  22. E. Esteves, ``Linear systems and ramification points on reducible nodal curves," Math. Contemp. 14 (1998), 21--35.

  23. L. Facchini, ``On tropical Clifford's theorem." Ric. Mat. 59 (2010), no. 2, 343--349.

  24. G. Farkas, ``The geometry of the moduli space of curves of genus 23," Math. Ann. 318 (2000), no. 1, 43--65.

  25. G. Farkas and M. Popa, ``Effective divisors on $M_g$, curves on $K3$ surfaces, and the slope conjecture," J. Algebraic Geom. 14 (2005), no. 2, 241--267.

  26. A. Gathmann and M. Kerber, ``A Riemann--Roch theorem in tropical geometry," Math. Z. 259 (2008), no.~1, 217--230.

  27. A. Gathmann, M. Kerber, and H. Markwig, ``Tropical fans and the moduli spaces of tropical curves," Compos. Math. 145 (2009), no. 1, 173--195.

  28. A. Gathmann and H. Markwig, ``The numbers of tropical plane curves through points in general position," J. Reine Angew. Math. 602 (2007), 155--177.

  29. D. Gieseker, ``Stable curves and special divisors: Petri's conjecture," Invent. Math. 66 (1982), no.~2, 251--275.

  30. P. Griffiths and J. Harris, ``On the variety of special linear systems on a general algebraic curve," Duke Math. J. 47 (1980), no.~1, 233--272.

  31. C. Haase, G. Musiker and J. Yu, ``Linear systems on tropical curves," Math. Z. 270 (2012), nos. 3--4, 1111--1140.

  32. E. Katz and D. Zureick-Brown, ``The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions," preprint, arXiv:1204.3335.

  33. R. Kawaguchi, ``The gonality conjecture for curves on certain toric surfaces," Osaka J. Math. 45 (2008), no.~1,113--126.

  34. R. Kawaguchi, ``The gonality conjecture for curves on toric surfaces with two $mathbf{P}^1$-fibrations," Saitama Math. J. 27 (2010), 35--80.

  35. S. Kleiman and D. Laksov, ``On the existence of special divisors," Amer. J. Math. 94 (1972), 431--436.

  36. R. Lazarsfeld, ``Brill--Noether--Petri without degenerations," J. Differential Geom. 23 (1986), no.~3, 299--307.

  37. C. M. Lim, S. Payne and N. Potashnik, ``A note on Brill--Noether theory and rank-determining sets for metric graphs," to appear in Int. Math. Res. Not., preprint, arXiv:1106.5519.

  38. Y. Luo, ``Rank-determining sets of metric graphs," J. Combin. Theory Ser. A 118 (2011), no.~6, 1775--1793.

  39. D. Maclagan, ``Introduction to tropical algebraic geometry," to appear in Contemp. Math., preprint, arXiv:1207.1925.

  40. M. Manjunath and B. Sturmfels, ``Monomials, binomials, and Riemann-Roch," to appear in J. Algebr. Comb, preprint, arXiv:1201.4357.

  41. G. Mikhalkin and I. Zharkov, ``Tropical curves, their Jacobians and theta functions," In Curves and abelian varieties, Contemp. Math., 465, 203--230, Amer. Math. Soc., Providence, RI, 2008.

  42. B. Osserman, ``A limit linear series moduli scheme," Ann. Inst. Fourier 56 (2006), no.~4, 1165--1205.

  43. T. Nishinou and B. Siebert, ``Toric degenerations of toric varieties and tropical curves," Duke Math. J. 135 (2006), no. 1, 1--51.

  44. R. Vakil, ``The enumerative geometry of rational and elliptic curves in projective space," J. Reine Angew. Math. 529 (2000), 101--153.

  45. R. Vakil, ``Enumerative geometry of curves via degeneration methods," Thesis (Ph.D.), Harvard University, 1997.