×

The moduli spaces of rank 2 stable vector bundles over Veronesean surfaces. (English) Zbl 0802.14018

Let \(\pi : S \to \mathbb{P}^ 2\) be the blow-up of a reduced subscheme, and let \(H : = d \pi^*{\mathcal O}_{\mathbb{P}^ 2} (1) - \sum_ i E_ i\), where the \(E_ i\) are the exceptional divisors of \(\pi\). Assume \(H\) is ample. For \(c_ i \in H^{2i} (S; \mathbb{Z})\) let \({\mathcal M} (c_ 1,c_ 2)\) be the moduli space of rank-two \(H\)-slope-stable vector-bundles with the given Chern classes. The author gives a necessary and sufficient condition for \({\mathcal M} (c_ 1, c_ 2)\) to be non-empty. Certain extremal cases are described explicitly.

MSC:

14J10 Families, moduli, classification: algebraic theory
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] B. F.A. Bogomolov,Holomorphic tensors and vector bundles on projective manifolds, Math. USSR Izv.13 (1979), 499-555 · Zbl 0439.14002
[2] Ba. E. Ballico,On the moduli of vector bundles on rational surfaces, Arch. Math.49 (1987), 267-272 · Zbl 0611.14008
[3] Br. R. Brusee,Stable bundles on blown up surfaces, Math. Z.205 (1990), 551-565 · Zbl 0695.14005
[4] C. F. Catanese,Footnotes to a Theorem of I. Reider, L.N.M.1417 (1990), 67-74 · Zbl 0705.14015
[5] D-G-M. E.D. Davis, A.V. Geramita and P. Maroscia,Perfect homogeneous ideals: Dubreil’s theorems revisited, Bull. Sci. Math.108 (1984), 143-185 · Zbl 0559.14034
[6] D-G. E.D. Davis and A.V. Geramita,Birational morphism toP2:An ideal-theoretic perspective, Math. Ann.279 (1988), 435-448 · Zbl 0657.14003
[7] G. D. Gieseker,On the moduli of vector bundles on an algebraic surface, Ann. Math.106 (1977), 45-60 · Zbl 0381.14003
[8] G-M. A.V. Geramita and P. Maroscia,The ideal of forms vanishing at a finite set of points inPn, J. of Alg.90 (1984), 528-555 · Zbl 0547.14001
[9] G-Gi. A.V. Geramita and A. Gimigliano,Generators for the defining ideal of certain rational surfaces, Duke Math. J.62 (1999), 61-83 · Zbl 0731.14031
[10] G-Gi-H. A.V. Geramita, A. Gimigliano, and B. Harbourne,Intrinsic and Extrinsic Properties of Special blowings up ofP2, Preprint No. 1991-04, Queen’s University
[11] Gi. A. Gimigliano,On Veronesean suefaces, Proc. Konin. Ned. Akads. van Wetenschappen, Ser. A92 (1989), 71-85 · Zbl 0701.14014
[12] H. R. Hartshorne,Algebraic Geometry, GTM52 · Zbl 0367.14001
[13] M. M. Maruyama,Stable vector bundles on an algebraic surface, Nagoya Math. J.58 (1975), 25-68 · Zbl 0337.14026
[14] Q. Z. Qin,Simple sheaves versus stable sheaves on algebraic surfaces, Math. Z.209 (1992), 559-579 · Zbl 0735.14014
[15] Mi. rosa M. Miró-Roig,Sharp bounds for the p-th module of syzygies of the ideal of certain rational surfaces, Preprint, Abril 1992
[16] S. R.L.E. Schwarzenberger,Vector bundles on an algebraic surface, Proc. London Math. Soc.11 (1961), 601-622 · Zbl 0212.26003
[17] O-S-S. C. Okonek, M. Scheneider and H. Spindler,Vector Bundles and Complex Projective Spaces, Progress in Math.3
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.