5,270 research outputs found
Equivalence of families of singular schemes on threefolds and on ruled fourfolds
The main purpose of this paper is twofold. We first want to analyze in
details the meaningful geometric aspect of the method introduced in the
previous paper [12], concerning regularity of families of irreducible, nodal
"curves" on a smooth, projective threefold . This analysis highlights
several fascinating connections with families of other singular geometric
"objects" related to and to other varieties.
Then, we generalize this method to study similar problems for families of
singular divisors on ruled fourfolds suitably related to .Comment: 22 pages, Latex 2e, submitted preprin
P^r-scrolls arising from Brill-Noether theory and K3-surfaces
In this paper we study examples of P^r-scrolls defined over primitively
polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of
the general curve in the primitive linear system on S and from classical
Lazarsfeld's results in.
We show that such scrolls form an open dense subset of a component H of their
Hilbert scheme; moreover, we study some properties of H (e.g. smoothness,
dimensional computation, etc.) just in terms of the moduli space of such K3's
and of the moduli space of semistable torsion-free sheaves of a given
Mukai-vector on S.
One of the motivation of this analysis is to try to introducing the use of
projective geometry and degeneration techniques in order to studying possible
limits of semistable vector-bundles of any rank on a general K3 as well as
Brill-Noether theory of vector-bundles on suitable degenerations of projective
curves.
We conclude the paper by discussing some applications to the Hilbert schemes
of geometrically ruled surfaces whose base curve has general moduli.Comment: published in Manuscripta Mathematic
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
On the linear bounds on genera of pointless hyperelliptic curves
An irreducible smooth projective curve over is called
\textit{pointless} if it has no -rational points. In this paper
we study the lower existence bound on the genus of such a curve over a fixed
finite field . Using some explicit constructions of
hyperelliptic curves, we establish two new bounds that depend linearly on the
number . In the case of odd characteristic this improves upon a result of R.
Becker and D. Glass. We also provide a similar new bound when is even
Uniqueness of low genus optimal curves over F_2
A projective, smooth, absolutely irreducible algebraic curve X of genus g
defined over a finite field F_q is called optimal if for every other such genus
g curve Y over F_q one has . In this paper we show that
for there is a unique optimal genus g curve over F_2. For g=6 there
are precisely two and for g=7 there are at least two.Comment: 21 page
On the convex hull of a space curve
The boundary of the convex hull of a compact algebraic curve in real 3-space
defines a real algebraic surface. For general curves, that boundary surface is
reducible, consisting of tritangent planes and a scroll of stationary
bisecants. We express the degree of this surface in terms of the degree, genus
and singularities of the curve. We present algorithms for computing their
defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change
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