zbMATH — the first resource for mathematics

The tangent bundle of a ruled surface. (English) Zbl 0541.14035
Let Z be a ruled surface. The authors settle, in the case of decomposable Z, the questions of existence and classification of sublinebundles of the tangent bundle \(T_ Z\). They also show, in the indecomposable case, that \(T_ Z\) splits in general in the sense of moduli (if the ground field has characteristic not 2). The ”integrable” sublinebundles of \(T_ Z\) give rise to ”quotient” surfaces Y of Z, which exhibit various interesting ”pathological” features. One obtains, for instance, surfaces Y of general type with global vector fields. These quotient surface examples were worked out in collaboration with H. Kurke; a more detailed analysis of them is forthcoming.

14J25 Special surfaces
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J17 Singularities of surfaces or higher-dimensional varieties
Full Text: DOI EuDML
[1] Atiyah, M.F.: Complex fibre bundles and ruled surfaces. Proc. Lond. Math. Soc. III. Ser.5, 407-434 (1955). · Zbl 0174.52804 · doi:10.1112/plms/s3-5.4.407
[2] Ganong, R.: Plane Frobenius sandwiches. Proc. Am. Math. Soc.48, 474-478 (1982) · Zbl 0504.13012 · doi:10.1090/S0002-9939-1982-0643732-1
[3] Ganong, R., Russell, P.: Derivations with only divisorial singularities on rational and ruled surfaces. J. Pure App. Alg.26, 165-182 (1982) · Zbl 0501.14009 · doi:10.1016/0022-4049(82)90022-6
[4] Ghione, F.: Quelques résultats de C. Segre sur les surfaces réglées. Math. Ann.255, 77-95 (1981) · Zbl 0446.14017 · doi:10.1007/BF01450557
[5] Ghione, F.: Quotient schemes over a smooth curve. Napoli Publ. Ist. Mat.33, (1981/82) · Zbl 0616.14021
[6] Hartshorne, R.: Algebraic geometry Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[7] Kurke, H., Russell, P.: Examples of false-ruled surfaces. To appear
[8] Maruyama, M.: On classification of ruled surfaces. Tokyo: Kinokuniya 1970 · Zbl 0214.20103
[9] Matsumura, H.: Geometric structure of the cohomology rings in abstract algebraic geometry. Mem. Coll. Sci. Univ. Kyoto (A)32, 33-84 (1959) · Zbl 0119.36903
[10] Nagata, M.: On self-intersection number of a section on a ruled surface. Nagoya Math. J.37, 191-196 (1970) · Zbl 0193.21603
[11] Rudakov, A.N., ?afarevi?, I.R.: Inseparable morphisms of algebraic surfaces. Math. USSR Izv.10, 1205-1237 (1976) · Zbl 0379.14006 · doi:10.1070/IM1976v010n06ABEH001833
[12] Russell, P.: Forms of the affine line and its additive group. Pac. J. Math.32, 527-539 (1970) · Zbl 0199.24502
[13] Segre, C.: Recherches générales sur les courbes et les surfaces réglées algébriques. Math. Ann.34, 1-25 (1889) · JFM 21.0664.01 · doi:10.1007/BF01446790
[14] Seshadri, C.S.: L’opération de Cartier. Applications, exposé 6, dans: Séminaire Chevalley (1958/59)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.