Lifting divisors on a generic chain of loops
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In this article, it was shown that any divisor on the chain of loops that is rational over the value group lifts to a divisors of the same rank on, confirming a conjecture of Cools, Draisma, Robeva, and the third author.Abstract:
Let be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on , confirming a conjecture of Cools, Draisma, Robeva, and the third author.read more
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Book ChapterDOI
Degeneration of Linear Series from the Tropical Point of View and Applications
Matthew Baker,David Jensen +1 more
TL;DR: In this paper, the authors discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on applications to problems in Brill-Noether theory and arithmetic geometry.
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Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem
David Jensen,Sam Payne +1 more
TL;DR: In this paper, a framework to apply tropical and nonarchimedean analytic methods to multiplication maps for linear series on algebraic curves was developed, where degenerations of these multiplications maps when the special fiber is not of compact type.
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Tropical independence II: The maximal rank conjecture for quadrics
David Jensen,Sam Payne +1 more
TL;DR: In this article, the maximal rank conjecture for quadrics was shown to imply tropical independence on chains of loops, and a tropical analogue of Max Noether's theorem on quadrics containing a canonically embedded curve.
Posted Content
Brill-Noether theory for curves of a fixed gonality
David Jensen,Dhruv Ranganathan +1 more
TL;DR: In this article, Cartwright, Jensen, and Payne give a generalization of the Brill-Noether theorem for the variety of special divisors on a general curve of prescribed gonality.
Journal ArticleDOI
Tropical independence, II: The maximal rank conjecture for quadrics
David Jensen,Sam Payne +1 more
TL;DR: A tropical proof of maximal rank conjecture for quadrics is given in this paper, and a tropical analogue of Max Noether's theorem on quadrics containing a canonically embedded curve is shown.
References
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Geometry of algebraic curves
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Riemann–Roch and Abel–Jacobi theory on a finite graph
Matthew Baker,Serguei Norine +1 more
TL;DR: In this paper, a graph-theoretic analogue of the Riemann-Roch theorem is presented, and the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph is characterized.
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Limit linear series: Basic theory
David Eisenbud,Joe Harris +1 more
TL;DR: In this article, the degeneration of linear series on smooth curves is studied, as the curves degenerate to a certain type of reducible curves, curves of compact type, called limit linear series.
Posted Content
Tropical curves, their Jacobians and Theta functions
Grigory Mikhalkin,Ilia Zharkov +1 more
TL;DR: In this article, the authors studied Jacobian varieties for tropical curves, which are real tori equipped with integral affine structure and symmetric bilinear form, and defined tropical counterpart of the theta function and established tropical versions of the Abel-Jacobi and Riemann theta divisor theorems.
Journal ArticleDOI
Specialization of linear systems from curves to graphs
TL;DR: In this article, the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface is investigated, and some applications of their results to graph theory, arithmetic geometry, and tropical geometry are provided.