A tropical proof of the Brill-Noether Theorem

By Filip Cools, Jan Draisma, Sam Payne, Elina Robeva. 2010

191 citations recorded by the source corpus.

We produce Brill–Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill–Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ddead6ad94eff25a80c9a328b6411e7a3b589ae3

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Brill-Noether theory for curves of a fixed gonality (2017, methodology): “ner dictated by µ. Deﬁnition 2.2. A tableau is a µ-displacement tableau if, for any two boxes containing the same symbol j, the lattice distance between them is a multiple of µj. For consistency with [15, 20, 30, 31, 32], we write our tableaux in the English style, beginning in the upper left and pr”
Tropical independence II: The maximal rank conjecture for quadrics (2015, methodology): “the classes of effective divisors D such that r(D) > deg(D) − g, are explicitly classified in [CDPR12].”
Linear series on metrized complexes of algebraic curves (2012, methodology): “In particular, the lingering lattice paths studied in [17] in connection with a tropical proof of the Brill–Noether theorem can be understood in a natural way as values of η-functions.”
Linear systems on tropical curves (2009, result): “Using this result, Cools, Draisma, Payne, and Robeva gave a new proof the Brill-Noether Theorem [8].”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “The first significant application of tropical Brill-Noether theory was the new proof of the Brill-Noether Theorem by Cools, Draisma, Payne and Robeva [CDPR12], which successfully realized the program laid out in [Bak08b].”
Tropical hyperelliptic curves (2011, methodology): “Our work ties together two strands in the recent tropical geometry literature: tropical Brill-Noether theory on the one hand [Bak08], [Cap11b], [CDPR10], [LPP11]; and tropical moduli spaces of curves on the other [BMV11], [Cap10], [Cap11a], [Cha11], [MZ07].”
Riemann–Roch theory for weighted graphs and tropical curves (2011, methodology): “ked to the algebro-geometric theory not only by the formal analogies. Indeed, a remarkable fact that connects the two theories is Baker’s Specialization Lemma, of [B]. This result has been applied in [CDPR] to obtain a new proof of the famous Brill-Noether theorem for algebraic curves, in [B] to pro”
Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, methodology): “Note that every principal divisor has degree zero, so the group Pic(0) of equivalence classes of divisors is graded by degree.”