6,920 research outputs found
About the stability of the tangent bundle restricted to a curve
Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and
let L be a line bundle on C generated by its global sections. The morphism i:C
-->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent
bundle T of the projective space P. Sharpening a theorem by Paranjape, we show
that if deg L>2g-c(C)-1 then i*T is semi-stable, specifying when it is also
stable. We then prove the existence on many curves of a line bundle L of degree
2g-c(C)-1 such that i*T is not semi-stable. Finally, we completely characterize
the (semi-)stability of i*T when C is hyperelliptic.Comment: 5 page
Computing in Jacobians of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over
finite fields, and with their Jacobians, using the algorithmic representation
of projective curves developed by Khuri-Makdisi. We show that many desirable
operations can be done efficiently in this setting: decomposing divisors into
prime divisors; computing pull-backs and push-forwards of divisors under finite
morphisms, and hence Picard and Albanese maps on Jacobians; generating
uniformly random divisors and points on Jacobians; computing Frobenius maps and
Kummer maps; and finding a basis for the -torsion of the Picard group, where
is a prime number different from the characteristic of the base field.Comment: 42 page
Projective Normality Of Algebraic Curves And Its Application To Surfaces
Let be a very ample line bundle on a smooth curve of genus with
. Then is normally generated if . Let be a triple
covering of genus curve with and a
divisor on with . Then
becomes a very ample line bundle which is normally generated. As an
application, we characterize some smooth projective surfaces.Comment: 7 pages, 1figur
Curves of genus g on an abelian variety of dimension g
In this paper we prove a general theorem concerning the number of translation
classes of curves of genus belonging to a fixed cohomology class in a
polarized abelian variety of dimension . For we recover results of
G\"ottsche and Bryan-Leung. For we deduce explicit numbers for these
classes.Comment: 12 page
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