Linear series on metrized complexes of algebraic curves

By Omid Amini, Matthew Baker. 2012

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A metrized complex of algebraic curves over an algebraically closed field $$\kappa $$κ is, roughly speaking, a finite metric graph $$\Gamma $$Γ together with a collection of marked complete nonsingular algebraic curves $$C_v$$Cv over $$\kappa $$κ, one for each vertex $$v$$v of $$\Gamma $$Γ; the marked points on $$C_v$$Cv are in bijection with the edges of $$\Gamma $$Γ incident to $$v$$v. We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over $$\kappa $$κ with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve $$X$$X defined over a non-Archimedean field $$\mathbb {K}$$K, together with a strongly semistable model $$\mathfrak {X}$$X for $$X$$X over the valuation ring $$R$$R of $$\mathbb {K}$$K, we define a corresponding metrized complex $$\mathfrak {C}\mathfrak {X}$$CX of curves over the residue field $$\kappa $$κ of $$\mathbb {K}$$K and a canonical specialization map $$\tau ^{\mathfrak {C}\mathfrak {X}}_*$$τ∗CX from divisors on $$X$$X to divisors on $$\mathfrak {C}\mathfrak {X}$$CX which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from $$X$$X to $$\mathfrak {C}\mathfrak {X}$$CX. As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a $$\mathfrak {g}^r_d$$gdr in a regular family of semistable curves is a limit $$\mathfrak {g}^r_d$$gdr on the special fiber.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e55f0e693bea1f0b00750d9cf74761b9f57f5e6d

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Citations in the corpus

Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, methodology): “The existence of Gieseker–Petri general curves over an arbitrary algebraically closed field then follows by standard arguments from scheme theory, using the fact that the coarse moduli space of curves is defined over Spec Z, as in [Cools et al. 2012, Section 3].”
A MODULI STACK OF TROPICAL CURVES (2017, background): “thod [RSPW17b]. In [FRTU19, Theorem 4.4] it is shown that the moduli stack Mlogg,n of logarithmic curves may be viewed as a ﬁne moduli stack parameterizing of metrized curve complexes in the sense of [AB15] over the category LogSch of logarithmic schemes. Theorem 4 shows that a∗Mtropg,n functions as”
Tropical independence II: The maximal rank conjecture for quadrics (2015, background): “…building frameworks that include both classical limit linear series and also generalizations of limit linear series for curves not of compact type [Amini and Baker 2015; Osserman 2014; 2016], which are helpful for explaining connections between the tropical and limit linear series proofs of the…”