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126
Tropical curves, their Jacobians and theta functions
, 2006
"... We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems. ..."
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Cited by 92 (4 self)
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We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems.
Specialization of linear systems from curves to graphs
"... Abstract. We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry. 1. ..."
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Cited by 64 (6 self)
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Abstract. We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry. 1.
A RiemannRoch theorem in tropical geometry
, 2007
"... Recently, Baker and Norine have proven a RiemannRoch theorem for finite graphs. We extend their results to metric graphs and thus establish a RiemannRoch theorem for divisors on (abstract) tropical curves. ..."
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Cited by 59 (0 self)
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Recently, Baker and Norine have proven a RiemannRoch theorem for finite graphs. We extend their results to metric graphs and thus establish a RiemannRoch theorem for divisors on (abstract) tropical curves.
Harmonic morphisms and hyperelliptic graphs
 INTERNATIONAL JOURNAL OF URBAN AND REGIONAL RELATIONSHIPS
, 2007
"... We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graphtheoretic analogue of the classical RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1forms induced by a harmonic morphism, and present ..."
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Cited by 31 (2 self)
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We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graphtheoretic analogue of the classical RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2edgeconnected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a nonconstant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the RiemannHurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.
RANKDETERMINING SETS OF METRIC GRAPHS
, 2009
"... A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Î is an element of the free abelian group on Î. The rank of a divisor on a metric graph is a concept appearing in the RiemannRoch theorem for metric graphs (or t ..."
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Cited by 24 (1 self)
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A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Î is an element of the free abelian group on Î. The rank of a divisor on a metric graph is a concept appearing in the RiemannRoch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber [7], and Mikhalkin and Zharkov [10]. We define a rankdetermining set of a metric graph Î to be a subset A of Î such that the rank of a divisor D on Î is always equal to the rank of D restricted on A. We show constructively in this paper that there exist finite rankdetermining sets. In addition, we investigate the properties of rankdetermining sets in general and formulate a criterion for rankdetermining sets. Our analysis is a based on an algorithm to derive the v0reduced divisor from any effective divisor in the same linear system.
A tropical proof of the BrillNoether theorem
 Adv. Math
"... Abstract. We produce BrillNoether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the BrillNoether Theorem, due to Griffiths and Harris. Our proof provides an explicit criterion for a curve to be BrillNoether general over discretely valued fields of arbi ..."
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Cited by 24 (5 self)
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Abstract. We produce BrillNoether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the BrillNoether Theorem, due to Griffiths and Harris. Our proof provides an explicit criterion for a curve to be BrillNoether general over discretely valued fields of arbitrary pure or mixed characteristic. 1.
Linear series on metrized complexes of algebraic curves
, 2014
"... A metrized complex of algebraic curves over an algebraically closed field κ is, roughly speaking, a finite metric graph Γ together with a collection of marked complete nonsingular algebraic curves Cv over κ, one for each vertex v of Γ; the marked points on Cv are in bijection with the edges of Γ i ..."
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Cited by 18 (4 self)
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A metrized complex of algebraic curves over an algebraically closed field κ is, roughly speaking, a finite metric graph Γ together with a collection of marked complete nonsingular algebraic curves Cv over κ, one for each vertex v of Γ; the marked points on Cv are in bijection with the edges of Γ incident to v. We define linear equivalence of divisors and establish a RiemannRoch theorem for metrized complexes of curves which combines the classical RiemannRoch theorem over κ with its graphtheoretic and tropical analogues from [AC, BN, GK, MZ], providing a common generalization of all of these results. For a complete nonsingular curve X defined over a nonArchimedean field K, together with a strongly semistable model X for X over the valuation ring R of K, we define a corresponding metrized complex CX of curves over the residue field κ of K and a canonical specialization map τCX ∗ from divisors on X to divisors on CX which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from [B] and its weighted graph analogue from [AC], showing that the rank of a divisor cannot go down under specialization from X to CX. As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the EisenbudHarris theory [EH] of limit linear series. 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 grd in a regular family of semistable curves is a limit grd on the special fiber.
Linear systems on tropical curves
, 2009
"... Abstract. A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system D  of a divisor D on a tropical curve Γ analogously to the classical counterpart. We inv ..."
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Cited by 17 (1 self)
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Abstract. A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system D  of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of D  as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, D  defines a map from Γ to a tropical projective space, and the image can be extended to a tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system D  as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a Qtropical curve is a direct limit of critical groups of finite graphs converging to the curve. 1.