187 research outputs found
Newton polygons and curve gonalities
We give a combinatorial upper bound for the gonality of a curve that is
defined by a bivariate Laurent polynomial with given Newton polygon. We
conjecture that this bound is generically attained, and provide proofs in a
considerable number of special cases. One proof technique uses recent work of
M. Baker on linear systems on graphs, by means of which we reduce our
conjecture to a purely combinatorial statement.Comment: 29 pages, 18 figures; erratum at the end of the articl
Linear pencils encoded in the Newton polygon
Let be an algebraic curve defined by a sufficiently generic bivariate
Laurent polynomial with given Newton polygon . It is classical that the
geometric genus of equals the number of lattice points in the interior of
. In this paper we give similar combinatorial interpretations for the
gonality, the Clifford index and the Clifford dimension, by removing a
technical assumption from a recent result of Kawaguchi. More generally, the
method shows that apart from certain well-understood exceptions, every
base-point free pencil whose degree equals or slightly exceeds the gonality is
'combinatorial', in the sense that it corresponds to projecting along a
lattice direction. We then give an interpretation for the scrollar invariants
associated to a combinatorial pencil, and show how one can tell whether the
pencil is complete or not. Among the applications, we find that every smooth
projective curve admits at most one Weierstrass semi-group of embedding
dimension , and that if a non-hyperelliptic smooth projective curve of
genus can be embedded in the th Hirzebruch surface
, then is actually an invariant of .Comment: This covers and extends sections 1 to 3.4 of our previously posted
article "On the intrinsicness of the Newton polygon" (arXiv:1304.4997), which
will eventually become obsolete. arXiv admin note: text overlap with
arXiv:1304.499
The lattice size of a lattice polygon
We give upper bounds on the minimal degree of a model in and
the minimal bidegree of a model in of the
curve defined by a given Laurent polynomial, in terms of the combinatorics of
the Newton polygon of the latter. We prove in various cases that this bound is
sharp as soon as the polynomial is sufficiently generic with respect to its
Newton polygon
On the relation between weighted trees and tropical Grassmannians
In this article, we will prove that the set of 4-dissimilarity vectors of
n-trees is contained in the tropical Grassmannian G_{4,n}. We will also propose
three equivalent conjectures related to the set of m-dissimilarity vectors of
n-trees for the case m > 4. Using a computer algebra system, we can prove these
conjectures for m = 5.Comment: 11 pages, to appear in Journal of Symbolic Computatio
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