Complexes inattendus de droites de saut (Unexpected complex of jumping lines)

Details Complexes inattendus de droites de saut (Unexpected complex of jumping lines) Complexes inattendus de droites de saut (Unexpected complex of jumping lines)

1994-03-17

arxiv.org/abs/alg-geom/9403013v1

Mathematics

Algebraic Geometry

7 pages, Latex

Scientific paper

We prove here the following results: \begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, if $C$ is a reduced irreducible curve of ${\bf P}_3^{\vee}$ such that $E_H$ is unstable for all $H\in C$ then $C$ is a line. \end{th} We define now the set $W(E)$ as the set of planes $H$ such that the restricted bundle $E_H$ is unstable (that means non semi-stable). \begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, with first chern class $c_1=c_1(E)$, $L$ a line and an integer $n\ge 0$. The following conditions are equivalent: \begin{description} \item[(i)] $L^{\vee}\subset W(E)$ and $H^0(E_H(-n+[-c_1/2]))\neq 0$ for a general point $H\in L^{\vee}$. \item[(ii)] There exist $m>0$ and a section $t\in H^0(E(m+[-c_1/2]))$ such that the zero variety of $t$ contains the infinitesimal neighbourhood of order $(m+n-1)$ of $L$. \end{description}

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