A tropical proof of the Brill–Noether Theorem

Abstract

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.

Chip-firing groups of iterated cones

“…The technology transfer flows both ways; chip-firing (and variants and tools from tropical geometry) have emerged as a central tool in recent results across several subfields of algebraic/arithmetic geometry and number theory, including the maximal rank conjecture for quadrics [JP16], the Gieseker-Petri theorem [JP14], the Brill-Noether theorem [CDPR12], and the uniform boundedness conjecture [KRZB16]; see [BJ15] for an extensive survey.…”

The locus of Brill–Noether general graphs is not dense

“…The Baker-Norine theory of divisors on metric graphs gives us an analogous notion of Brill-Noether general graphs [Bak08,BN07]. As in the classical case, the locus of Brill-Noether general graphs in the moduli space of tropical curves M trop g is open [LPP12,Len14] and non-empty [CDPR12], but this does not imply that it is dense. Specifically, M The top-dimensional strata of M trop g correspond to trivalent graphs, and it is a straightforward exercise to construct a trivalent graph G with the property that every metric graph Γ ∈ M trop G is Brill-Noether special.…”

Degeneration of Linear Series from the Tropical Point of View and Applications

“…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]. In [CDPR12], the authors consider the family of graphs pictured in Figure 13, colloquially known as the chain of loops.…”

“…In [CDPR12], the authors consider the family of graphs pictured in Figure 13, colloquially known as the chain of loops.…”

“…Using Theorem 4.6 as the only input from algebraic geometry, the authors of [CDPR12] employ an intricate combinatorial argument to prove the following:…”