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

## Organizers

Matthew Baker (Georgia Institute of Technology)

Lucia Caporaso (University of Rome: Roma Tre)

Maria Angelica Cueto (The Ohio State University)

Eric Katz (University of Waterloo)

Sam Payne (Yale University)

## Objectives

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.

**Bibliography**

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- O. Amini and M. Baker,``Linear series on metrized complexes of algebraic curves," preprint, arxiv:1204.3508.
- O. Amini and L. Caporaso, ``Riemann--Roch theory for weighted graphs and tropical curves," preprint, arxiv:1112.5134, 2011.
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*Grassmannians, moduli spaces and vector bundles,*25--50, Clay Math. Proc.,**14**, Amer. Math. Soc., Providence, RI, 2011. - M. Baker, ``Specialization of linear systems from curves to graphs," Algebra Number Theory
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