Let $S$ be a non-singular cubic surface in the projective 3-space $P^{3}$ defined over an algebraically closed field of arbitrary characteristic. In [H2], we have constructed universal families of moduli of semi-stable 2-bundles with $\\Delta=C_{1}^{2}-4C_{2}=-4$ on $S$. In this article, we will treat the case of $\\Delta=-5$ . For a torsion free sheaf, the semi-stability may vary by twisting line bundles. In fact, for a modulo 2 class in Pic $S$, we had to consider 243 species of moduli of semi-stable 2-bundles with $\\Delta=-4$ . On the other hand, a semi-stable sheaf of rank 2 with $\\Delta=-5$ is always a $\\mu$-stable bundle (Proposition 2.2). We can normalize the first Chem class to reduce two cases: (i) $C_{1}=e,$ $C_{2}=1$ and (ii) $C_{1}=h,$ $C_{2}=2$ where $e$ is an exceptional class and $h$ is the hyperplane class. The moduli of the first case is obtained by contracting the line, corresponding to the first Chem class $C_{1}$ , on the cubic surface $S$ (Theorem 3.4). The moduli of the second case is the cubic surface $S$ itself (Theorem 6.3). In each case, we will construct a universal family. It is worth while to mention that the universal family of the first case which we will construct is obtained by modification of the family considered in the case of $\\Delta=-4$ . For the second case, to construct a universal family, we will use the method introduced in [H2, \\S 3 Limits of extensions]. We shall use notation and terminology in [Hl, H2] freely.

/pdf/moduli-of-stable-2-bundles-with-delta-5-on-a-non-singular-3u1ilfm4bs.pdf

Stable vector bundles of rank $2$ on a $3$-dimensional rational scroll

Moduli of Stable 2-Bundles with $\Delta=-5$ on a Non-Singular Cubic Surface

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1230395343

Vector Bundles on Complex Projective Spaces

L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/elements-de-geometrie-algebrique-iii-etude-cohomologique-des-2qqwx6ujza.pdf

Stable vector bundles of rank $2$ on a $3$-dimensional rational scroll

Citations

Moduli of Stable 2-Bundles with $\Delta=-5$ on a Non-Singular Cubic Surface

References

Vector Bundles on Complex Projective Spaces

Éléments de géométrie algébrique : III. Étude cohomologique des faisceaux cohérents, Seconde partie

Stable vector bundles of rank 2 on P 3

Related Papers (5)

On stable and uniform rank-2 vector bundles on P2 in characteristic p

Some vanishing for the cohomology of stable rank two vector bundles on P3.

Vector Bundles on the Moduli Stack of Elliptic Curves

Some properties of stable rank-2 vector bundles on ℙ n

Positive vector bundles on complex surfaces