The distribution of Weierstrass points on a tropical curve

Richman, David Harry

doi:10.48550/arxiv.1809.07920

Cited by 3 publications

(3 citation statements)

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“…Suppose X is a smooth algebraic curve, let Jac(X) denote the Jacobian of X, and let ι q : X → Jac(X) denote the Abel-Jacobi map with basepoint q ∈ X. The Manin-Mumford conjecture, now a theorem due to Raynaud [18], states that if X has genus g ≥ 2 then the image ι q (X) intersects only finitely many torsion points of Jac(X).…”

Section: Introductionmentioning

confidence: 99%

The tropical Manin-Mumford conjecture

Richman¹

2021

Preprint

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In analogy with the Manin-Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel-Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus g ≥ 2 which are biconnected and have edge lengths which are "sufficiently irrational" in a precise sense. Under these assumptions, the number of torsion points is bounded by 3g − 3. Next we study bounds on the number of torsion points in the image of higher-degree Abel-Jacobi embeddings, which send d-tuples of points to the Jacobian. This motivates the definition of the "independent girth" of a graph, a number which is a strict upper bound for d such that the higher-degree Manin-Mumford property holds.

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Section: Introductionmentioning

confidence: 99%

The tropical Manin-Mumford conjecture

Richman¹

2021

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“…Baker and Norine [2] proved a Riemann-Roch theorem for graphs, which was successively extended by Gathmann and Kerber [12] and Mikhalkin and Zharkov [17] to metric graphs, namely (abstract) tropical curves. The analogue notion of Weierstrass points on graphs was studied for instance in [1, Section 4] and [19].…”

Section: Introductionmentioning

confidence: 99%

Weierstrass sets on finite graphs

Borzì

2021

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We study two possible tropical analogues of Weierstrass semigroups on graphs, called rank and functional Weierstrass sets. We prove that on simple graphs, the first is contained in the second. We completely characterize the subsets of N arising as a functional Weierstrass set of some graph. Finally, we give a sufficient condition for a subset of N to be the rank Weierstrass set of some graph, allowing us to construct examples of rank Weierstrass sets that are not semigroups.

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“…Firstly, the Weierstrass points on a Riemann surface are equidistributed with respect to the Bergman measure [Nee84]. It is possible to define Weierstrass points on a Berkovich curve or on a metric graph and it turns out that they are equidistributed with respect to the Zhang measure [Ami14], [Ric18]. Secondly, recall that the Bergman measure can be obtained as a pullback of the flat metric from the Jacobian under the Abel-Jacobi map.…”

Section: Introductionmentioning

confidence: 99%