104 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
Nondegenerate curves of low genus over small finite fields
In a previous paper, we proved that over a finite field of sufficiently
large cardinality, all curves of genus at most 3 over k can be modeled by a
bivariate Laurent polynomial that is nondegenerate with respect to its Newton
polytope. In this paper, we prove that there are exactly two curves of genus at
most 3 over a finite field that are not nondegenerate, one over F_2 and one
over F_3. Both of these curves have remarkable extremal properties concerning
the number of rational points over various extension fields.Comment: 8 pages; uses pstrick
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
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
On nondegeneracy of curves
A curve is called nondegenerate if it can be modeled by a Laurent polynomial
that is nondegenerate with respect to its Newton polytope. We show that up to
genus 4, every curve is nondegenerate. We also prove that the locus of
nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is
min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional
Point counting on curves using a gonality preserving lift
We study the problem of lifting curves from finite fields to number fields in
a genus and gonality preserving way. More precisely, we sketch how this can be
done efficiently for curves of gonality at most four, with an in-depth
treatment of curves of genus at most five over finite fields of odd
characteristic, including an implementation in Magma. We then use such a lift
as input to an algorithm due to the second author for computing zeta functions
of curves over finite fields using -adic cohomology
The holomorphy conjecture for nondegenerate surface singularities
The holomorphy conjecture states roughly that Igusa's zeta function
associated to a hypersurface and a character is holomorphic on
whenever the order of the character does not divide the order of any eigenvalue
of the local monodromy of the hypersurface. In this article we prove the
holomorphy conjecture for surface singularities which are nondegenerate over
with respect to their Newton polyhedron. In order to provide
relevant eigenvalues of monodromy, we first show a relation between the
normalized volume (which appears in the formula of Varchenko for the zeta
function of monodromy) of faces in a simplex in arbitrary dimension. We then
study some specific character sums that show up when dealing with false poles.
In contrast with the context of the trivial character, we here need to show
fakeness of certain poles in addition to the candidate poles contributed by
-facets.Comment: 21 pages, 3 figure
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