6,920 research outputs found

    About the stability of the tangent bundle restricted to a curve

    Get PDF
    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

    Full text link
    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 ll-torsion of the Picard group, where ll 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

    Get PDF
    Let LL be a very ample line bundle on a smooth curve CC of genus gg with 3g+32<degL2g5\frac{3g+3}{2}<\deg L\le 2g-5. Then LL is normally generated if degL>max{2g+24h1(C,L),2gg162h1(C,L)}\deg L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}. Let CC be a triple covering of genus pp curve CC' with CϕCC\stackrel{\phi}\to C' and DD a divisor on CC' with 4p<degD<g162p4p<\deg D< \frac{g-1}{6}-2p. Then KC(ϕD)K_C(-\phi^*D) 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

    Get PDF
    In this paper we prove a general theorem concerning the number of translation classes of curves of genus gg belonging to a fixed cohomology class in a polarized abelian variety of dimension gg. For g=2g = 2 we recover results of G\"ottsche and Bryan-Leung. For g=3g = 3 we deduce explicit numbers for these classes.Comment: 12 page
    corecore