Hulek, Klaus Stable rank-2 vector bundles on \(\mathbb{P}_2\) with \(c_1\) odd. (English) Zbl 0407.32013 Math. Ann. 242, 241-266 (1979). 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. Reviewer: Klaus Hulek (Hannover) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 6 ReviewsCited in 41 Documents MSC: 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 Keywords:Rationality of Moduli Scheme; Holomorphic Vector Bundle; Singularity PDFBibTeX XMLCite \textit{K. Hulek}, Math. Ann. 242, 241--266 (1979; Zbl 0407.32013) Full Text: DOI EuDML References: [1] Barth, W.: Some properties of stable rank-2 vector bundles on 266-1. Math. Ann.226, 125-150 (1977) · Zbl 0417.32013 · doi:10.1007/BF01360864 [2] Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math.42, 63-91 (1977) · Zbl 0386.14005 · doi:10.1007/BF01389784 [3] Barth, W., Hulek, K.: Monads and moduli of vector bundles. Manuscripta Math.25, 323-347 (1978) · Zbl 0395.14007 · doi:10.1007/BF01168047 [4] Hulek, K.: Forthcoming thesis. Erlangen 1979 [5] Hulsbergen, W.: Vector bundles on the projective plane. Proefschrift, Leiden (1976) [6] Kervaire, M.: Fractions rationelles invariantes [d’après H. W. Lenstra]. Sem. Bourbaki, Vol. 1973/74, pp. 170-189. Lecture Notes in Mathematics, 431. Berlin, Heidelberg, New York: Springer 1975 [7] Lenstra, H.W.: Rational functions invariant under a finite abelian group. Invent. Math.25, 299-325 (1977) · Zbl 0292.20010 · doi:10.1007/BF01389732 [8] Le Potier, J.: Fibrés stables de rang 2 sur 266-2). Math. Ann.241. 217-256 (1979) · Zbl 0405.14008 · doi:10.1007/BF01421207 [9] Maruyama, M.: Moduli of stable sheaves. II. Preprint (1977) · Zbl 0374.14002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.