On the jumping conics of a semistable rank two vector bundle on P2

Stromme (see [S], (1.7)) introduced the notion of jumping conic of a normalized semistable rank two vector bundle E on ℙ² and he remarked that the locus of jumping conics of E has codimension ≤3+2c₁(E), with equality for general E. Here we introduce a concept of jumping conic of a semistable rank two vector bundle on ℙ² (see (I.1)) by generalizing the notion of jumping line of the second kind introduced by Hulek in [H]. Our definition agrees with Stromme's for c₁=−1, but not for c₁=0. In contrast with the case of jumping lines, where we have a different behaviour in the case c₁ even or c₁ odd, the set of jumping conics according to our definition is always a divisor (possibly empty) in the ℙ⁵ of all conics of ℙ² (see th. (I.8)), whose degree depends on c₂(E).