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43
RiemannRoch and AbelJacobi theory on a finite graph
 Adv. Math
"... Abstract. It is wellknown that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graphtheoretic analogue of the classic ..."
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Cited by 133 (12 self)
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Abstract. It is wellknown that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graphtheoretic analogue of the classical RiemannRoch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the AbelJacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or nonexistence of a winning strategy for a certain chipfiring game played on the vertices of a graph. 1.
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 65 (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 60 (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.
Equidistribution of small points, rational dynamics, and potential theory
 Ann. Inst. Fourier (Grenoble
, 2006
"... Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points ..."
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Cited by 46 (7 self)
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Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕcanonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the wellknown canonical measure associated to ϕ. This theorem generalizes a result of BakerHsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable ArakelovGreen’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potentialtheoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the vadic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results
Equirépartition des petits points
, 1997
"... Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont mêm ..."
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Cited by 42 (4 self)
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Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont même équidistribués
Harmonic analysis on metrized graphs
 CANAD. J. MATH
"... This paper studies the Laplacian operator on a metrized graph, and its spectral theory. ..."
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Cited by 38 (6 self)
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This paper studies the Laplacian operator on a metrized graph, and its spectral theory.
Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves
 J. Amer. Math. Soc
, 1998
"... Abstract. Let f: X → Y be a surjective and projective morphism of smooth quasiprojective varieties over an algebraically closed field of characteristic zero with dimf = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if Xy is smooth and Ey is semistable for some ..."
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Cited by 26 (3 self)
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Abstract. Let f: X → Y be a surjective and projective morphism of smooth quasiprojective varieties over an algebraically closed field of characteristic zero with dimf = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if Xy is smooth and Ey is semistable for some y ∈ Y, then f ∗ 2rc2(E) − (r − 1)c1(E) 2) is weakly positive at y. We apply this result to obtain the following description of the cone of weakly positive QCartier divisors on the moduli space of stable curves. Let Mg (resp. Mg) be the moduli space of stable (resp. smooth) curves of genus g ≥ 2. Let λ be the Hodge class and δi’s (i = 0,..., [g/2]) the boundary classes. Then, a QCartier divisor xλ+ ∑ [g/2] i=0 yiδi on Mg is weakly positive over Mg if and only if x ≥ 0, gx+(8g +4)y0 ≥ 0, and i(g − i)x + (2g + 1)yi ≥ 0 for all 1 ≤ i ≤ [g/2].
ZHANG’S CONJECTURE AND THE EFFECTIVE BOGOMOLOV CONJECTURE OVER FUNCTION FIELDS
, 2009
"... We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus ..."
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Cited by 16 (6 self)
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We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.
Bogomolov conjecture over function fields for stable curves with only irreducible fibers
"... Abstract. Let K be a function field and C a nonisotrivial curve of genus g ≥ 2 over K. In this paper, we will show that if C has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds. Contents ..."
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Cited by 15 (4 self)
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Abstract. Let K be a function field and C a nonisotrivial curve of genus g ≥ 2 over K. In this paper, we will show that if C has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds. Contents