446 research outputs found
Scrolls and hyperbolicity
Using degeneration to scrolls, we give an easy proof of non-existence of
curves of low genera on general surfaces in P3 of degree d >=5. We show, along
the same lines, boundedness of families of curves of small enough genera on
general surfaces in P3. We also show that there exist Kobayashi hyperbolic
surfaces in P3 of degree d = 7 (a result so far unknown), and give a new
construction of such surfaces of degree d = 6. Finally we provide some new
lower bounds for geometric genera of surfaces lying on general hypersurfaces of
degree 3d > 15 in P4.Comment: 17
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
Geometry of vector bundle extensions and applications to a generalised theta divisor
Let E and F be vector bundles over a complex projective smooth curve X, and
suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a
subbundle of F, and D an effective divisor on X. We give a criterion for the
subsheaf G(-D) \subset F to lift to W, in terms of the geometry of a scroll in
the extension space \PP H^1 (X, Hom(F, E)). We use this criterion to describe
the tangent cone to the generalised theta divisor on the moduli space of
semistable bundles of rank r and slope g-1 over X, at a stable point. This
gives a generalisation of a case of the Riemann-Kempf singularity theorem for
line bundles over X. In the same vein, we generalise the geometric Riemann-Roch
theorem to vector bundles of slope g-1 and arbitrary rank.Comment: Main theorem slightly weakened; statement and proof significantly
more compac
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
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
Congruences of lines in , quadratic normality, and completely exceptional Monge-Amp\`ere equations
The existence is proved of two new families of locally Cohen-Macaulay sextic
threefolds in , which are not quadratically normal. These
threefolds arise naturally in the realm of first order congruences of lines as
focal loci and in the study of the completely exceptional Monge-Amp\`ere
equations. One of these families comes from a smooth congruence of multidegree
which is a smooth Fano fourfold of index two and genus 9.Comment: 16 page
Galois descent for the gonality of curves
We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls
- …