On the jumping conics of a semistable rank two vector bundle on \(\mathbb P^2\) (Q2640662)

The study of restrictions of vector bundles to lines, double lines, etc. have played an important role in the classification of vector bundles E of rank 2 on \({\mathbb{P}}^ 2\) [e.g. \textit{W. Barth}, Math. Ann. 226, 125-150 (1977; Zbl 0332.32021) and \textit{K. Hulek}, Math. Ann. 242, 241-266 (1977; Zbl 0407.32013)]. In this paper restriction to smooth conics C are considered. The author calls C a jumping conic of E (with \(c_ 1(E)=0\) or \(-1)\) if \(E|_ C\) is not a trivial bundle of \(c_ 1(E)=0\) and \(h^ 0(E|_ C)\neq 0\) for \(c_ 1(E)=-1\). He shows that the set of jumping conics is a divisor (if nonempty) in \({\mathbb{P}}^ 5\) of degree \(c_ 2(E)\) or \(c_ 2(E)-1\) according to \(c_ 1(E)\) is even or odd. In case \(c_ 1(E)=-1\) the proof is based on a construction of a complete simultaneous deformation of a couple of lines Y to a smooth conic C of \({\mathcal O}_ Y(0,k)\oplus {\mathcal O}_ Y(0,-k)\) to the trivial rank two bundle on C. As a corollary the author gets that if E has the same splitting type over all smooth conics, then E is uniform. \textit{S. A. Strømme} [Math. Z. 187, 405-423 (1984; Zbl 0533.14006)] defines a jumping conic by \(h^ 1(E|_ C)\neq 0\) which agrees with the above definition only for \(c_ 1(E)=-1\). He uses jumping conics to get free ample generators of Pic(M), M being the moduli space of semistable sheaves E on \({\mathbb{P}}^ 2\) of rank 2, \(c_ 1(E)=-1\) (respectively \(c_ 1(E)=0)\), \(c_ 2(E)\geq 2\) (respectively \(c_ 2(E)\geq 3\) and odd).