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Stable rank-2 vector bundles on \(\mathbb{P}_2\) with \(c_1\) odd. (English) Zbl 0407.32013
This paper deals with the classification problem of stable holomorphic rank 2 vector bundles on \(\mathbb P^2\). Its starting point is that every such bundle \(F\) with \(c_1(F)=-1\) and \(c_2(F)=n\) is the cohomology of a self-dual monad: \[ \mathbb C^{n-1}\otimes\mathcal O(-1)\overset{a} \longrightarrow\mathbb C^n\otimes\Omega(1)\overset{a^T(-1)} \longrightarrow\mathbb C^{n-1}\otimes\mathcal O. \] Using this, the rationality of the moduli scheme \(M(-1,n)\) is proved. A line \(L\subset \mathbb P^2\) is called jumping line of the second kind if \(h^0(F|L^2)\neq 0\), where \(L^2\) denotes the first infinitesimal neighbourhood of \(L\). It is proved that the set \(C(F):=\{L\in {\mathbb P^3}^*;h^0(F|L^2)\neq 0\}\) forms a curve of degree \(2(n-1)\). \(C(F)\) together with a naturally defined \(\mathcal O_{C(F)}\)-sheaf \(\theta_f\) determines the bundle completely. Moreover it is proved that the set \(S(F)\subset\mathbb P^2\) of jumping lines is contained in the singularity set of \(C(F)\). For the general bundle \(F\) the equality \(S(F)=\mathrm{Sing} C(F)\) holds.

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
14E08 Rationality questions in algebraic geometry
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