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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.
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 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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