29 research outputs found
The Brill-Noether rank of a tropical curve
We construct a space classifying divisor classes of a fixed degree on all
tropical curves of a fixed combinatorial type and show that the function taking
a divisor class to its rank is upper semicontinuous. We extend the definition
of the Brill-Noether rank of a metric graph to tropical curves and use the
upper semicontinuity of the rank function on divisors to show that the
Brill-Noether rank varies upper semicontinuously in families of tropical
curves. Furthermore, we present a specialization lemma relating the
Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether
locus of an algerbaic curve.Comment: 17 pages, 4 figures; v2: changed title, updated references, minor
improvements. To appear in Journal of Algebraic Combinatoric
Hyperelliptic graphs and metrized complexes
We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree 2r and rank r (for 0<r<g−1) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens' theorem for metric graphs.Publisher PDFPeer reviewe
Algebraic and combinatorial rank of divisors on finite graphs
We study the algebraic rank of a divisor on a graph, an invariant defined
using divisors on algebraic curves dual to the graph. We prove it satisfies the
Riemann-Roch formula, a specialization property, and the Clifford inequality.
We prove that it is at most equal to the (usual) combinatorial rank, and that
equality holds in many cases, though not in general.Comment: Final version to appear in Journal des Mathematiques Pures et
Appliquees. 36 page
Projective duals to algebraic and tropical hypersurfaces
We study a tropical analogue of the projective dual variety of a
hypersurface. When is a curve in or a surface in
, we provide an explicit description of in
terms of , as long as is smooth and satisfies
a mild genericity condition. As a consequence, when is a curve we describe
the transformation of Newton polygons under projective duality, and recover
classical formulas for the degree of a dual plane curve. For higher dimensional
hypersurfaces , we give a partial description of .Comment: 47 pages, 13 figures; v2 minor revisions; accepted to PLM
Kirchhoff's theorem for Prym varieties
We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is 2g−1 . Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is 2g−1 as well.Publisher PDFPeer reviewe