[Paper Review] Linear series on metrized complexes of algebraic curves

Omid Amini, Matthew Baker|arXiv (Cornell University)|Apr 16, 2012

Algebraic Geometry and Number Theory36 references4 citations

TL;DR

This paper introduces metrized complexes of algebraic curves as a unifying framework that combines classical algebraic geometry, graph theory, and tropical geometry. It establishes a Riemann-Roch theorem for these complexes and proves a specialization inequality, showing that divisor rank cannot decrease when specializing from a curve over a non-Archimedean field to its associated metrized complex, thereby generalizing both Eisenbud-Harris limit linear series and Baker's specialization lemma.

ABSTRACT

A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its graph-theoretic and tropical analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also establish generalizations of the second author's specialization lemma and its weighted graph analogue due to Caporaso and the first author, showing that the rank of a divisor cannot go down under specialization from curves to metrized complexes. As an application of these considerations, we formulate a generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type.

Motivation & Objective

• To unify classical Riemann-Roch theorems over algebraically closed fields with tropical and graph-theoretic analogues.
• To define linear equivalence and rank of divisors on metrized complexes of algebraic curves.
• To establish a specialization map from divisors on a curve over a non-Archimedean field to its associated metrized complex.
• To prove that the rank of a divisor cannot decrease under specialization, generalizing Baker’s specialization lemma.
• To link divisor specialization to Eisenbud-Harris theory of limit linear series, extending it beyond curves of compact type.

Proposed method

• Constructs metrized complexes as finite metric graphs with marked algebraic curves at each vertex, where markings correspond to incident edges.
• Defines divisors, linear equivalence, and rank on metrized complexes using reductions and cut conditions.
• Introduces a canonical specialization map from divisors on a curve over a non-Archimedean field to divisors on the metrized complex of its special fiber.
• Applies the theory of rank-determining sets and reduced divisors on metric graphs to prove the specialization inequality.
• Uses the reduction map and continuity arguments to show that if a divisor has rank ≥1 at all points of the complex, then it is effective in a generalized sense.
• Establishes a correspondence between effective divisors on the metrized complex and limit linear series on semistable curves.

Experimental results

Research questions

• RQ1 Can a common framework unify classical Riemann-Roch, graph-theoretic Riemann-Roch, and tropical Riemann-Roch theorems?
• RQ2 Does the rank of a divisor on a curve over a non-Archimedean field remain bounded below under specialization to its metrized complex?
• RQ3 How does the specialization map interact with linear equivalence and divisor rank?
• RQ4 Can Eisenbud-Harris’s theory of limit linear series be generalized to curves that are not of compact type?
• RQ5 What is the precise relationship between specialization of divisors to metrized complexes and the existence of limit linear series?

Key findings

• A Riemann-Roch theorem is established for metrized complexes of curves, combining classical, tropical, and graph-theoretic versions.
• The specialization map from a curve to its metrized complex preserves both degree and linear equivalence of divisors.
• The rank of a divisor cannot decrease under specialization, generalizing Baker’s specialization lemma and its weighted graph analogue.
• A concrete link is established between divisor specialization to metrized complexes and Eisenbud-Harris limit linear series.
• The theory generalizes limit linear series to curves that are not necessarily of compact type, proving that any degeneration of a g^r_d in a regular family of semistable curves is a limit g^r_d on the special fiber.
• For any rank-determining set in the complex, the rank over the full complex equals the rank over the subgroup of divisors supported on that set, under appropriate conditions.