Complexes inattendus de droites de saut (Unexpected complex of jumping lines)
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
1994
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| _version_ | 1866917852314861568 |
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| author | Valles, Jean |
| author_facet | Valles, Jean |
| contents | 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} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_alg_geom_9403013 |
| institution | arXiv |
| publishDate | 1994 |
| record_format | arxiv |
| spellingShingle | Complexes inattendus de droites de saut (Unexpected complex of jumping lines) Valles, Jean Algebraic Geometry 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} |
| title | Complexes inattendus de droites de saut (Unexpected complex of jumping lines) |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/alg-geom/9403013 |