IJPAM: Volume 5, No. 4 (2003)


E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it

Abstract.Let $C \subset {\bf {CP}}^3$ be a closed integral curve, which is involutive with respect to the symplectic form $dx_0\wedge dx_1+dx_2\wedge dx_3$ and $TC$ its tangent sheaf. Let $s$ be the minimal degree of a surface of ${\bf {CP}}^3$ containing $C$. Here we give a proof that $h^0(C,TC(s-2)) \ne 0$, $\mbox{deg}(TC) \ge d(2-s)$ and $\mbox{deg}(TC) = d(2-s)$ if and only if $TC \cong \mathcal {O}_C(2-s)$. Assume that $C$ is smooth. If $\omega _C \ncong \mathcal {O}_C(s-2)$ and $(s-2)h^1({\bf {CP}}^3,\mathcal {I}_C(s-1))
\ge (s-1)h^1({\bf {CP}}^3,\mathcal {I}_C(s-2))$ (resp. either $h^1({\bf {CP}}^3,\mathcal {I}_C(s-1))
= h^1({\bf {CP}}^3,\mathcal {I}_C(s-1)) = 0$, or $(s-2)h^1({\bf {CP}}^3,\mathcal {I}_C(s-1))
> s(h^1({\bf {CP}}^3,\mathcal {I}_C(s-2)))$), then $s \le 4$ (resp. $s \le 3$).

Received: March 6, 2003

AMS Subject Classification: 14H55

Key Words and Phrases: closed integral curve, symplectic form, minimal degree of a surface, involutive space curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 4