3,098 research outputs found
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
Quantum indices and refined enumeration of real plane curves
We associate a half-integer number, called {\em the quantum index}, to
algebraic curves in the real plane satisfying to certain conditions. The area
encompassed by the logarithmic image of such curves is equal to times
the quantum index of the curve and thus has a discrete spectrum of values. We
use the quantum index to refine real enumerative geometry in a way consistent
with the Block-G\"ottsche invariants from tropical enumerative geometry.Comment: Version 4: exposition improvement, particularly in the proof of
Theorem 5 (following referee suggestions
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
A Viro Theorem without convexity hypothesis for trigonal curves
A cumbersome hypothesis for Viro patchworking of real algebraic curves is the
convexity of the given subdivision. It is an open question in general to know
whether the convexity is necessary. In the case of trigonal curves we interpret
Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a
coherent way we prove that no convexity hypothesis is required to patchwork
such curves.Comment: 26 pages, 18 figure
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