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On smooth surfaces in Gr$$(1,\mathbb{P}^ 3)$$ with a fundamental curve. (English) Zbl 0803.14019
Let $$Y$$ be a smooth congruence of lines in complex projective 3-space. Assume that $$Y$$ has a curve $$C$$ of singular points. The authors give a complete classification of all such congruences. These are four classes which are – roughly speaking as follows:
(I) $$C$$ is a line;
(II) $$Y$$ is the set of bisecants of $$C$$, with $$C$$ a twisted cubic or an elliptic quartic;
(III) $$C$$ is a scroll with $$C$$ being a conic or a smooth plane cubic;
(IV) $$Y$$ is a conic bundle over a smooth plane cubic.
In this classification there is one type of congruences that is missing in A. Verra’s paper [ Manuscr. Math. 62, No. 4, 417-435 (1988; Zbl 0673.14026)].
Reviewer: H.Havlicek (Wien)

##### MSC:
 14J25 Special surfaces 14M15 Grassmannians, Schubert varieties, flag manifolds
##### Keywords:
congruence of lines; fundamental curve
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##### References:
 [1] Arbarello, E., Cornalba, M., Griffiths, P., and Harris, J. Geometry of Algebraic Curves, Springer-Verlag, 1985 · Zbl 0559.14017 [2] Arrondo, E., and Sols, I., On Congruences of Lines in the Projective Space, 1990, to appear Mém. Soc. Math. France · Zbl 0804.14016 [3] Arrondo, E.; Sols, I., Classification of Smooth Congruences of Low Degree, J. Reine Angew. Math, 393, 199-219, (1989) · Zbl 0649.14027 [4] Aure, A. and Ranestad, K., The smooth surfaces of degree 9 inP4, preprint (1990) [5] Barth, W., Peters, C. and Van de Ven, A., Compact Complex Surfaces, Ergebnisse de Mathematik und ihrer Grenzgebiete, series 3, vol. 4, Springer (1984) [6] Cossec, F, Dolgachev, I. and Verra, A, unpublished manuscript [7] Ein, L., Nondegenerate surfaces of degree $$n + 3$$ inP3c, J. Reine Angew. Math., 351, 1-11, (1984) · Zbl 0529.14020 [8] Fano, G., Nuove Ricerche sulle Congruenze di Rette del 3o Ordine Prive di Linea Singolare, Mem. Della Reale Acc. delle Sci. di Torino,51, 1-80 (1902) · JFM 33.0685.02 [9] Fulton, W., Intersection Theory, Ergebnisse de Mathematik und ihrer Grenzgebiete, series 3, vol. 2, Springer (1984) · Zbl 0541.14005 [10] Goldstein, N., Scroll Surfaces in Gr(1,P3), Conference on Algebraic Varieties of Small Dimension (Turin, 1985), Rend. Sem. Mat. Univers. Politecn. Special Issue, 69-75 (1987) [11] Gross, M., Surfaces in the Four-Dimensional Grassmannian, Ph.D. Thesis, U.C. Berkeley, 1990 [12] Gross, M., Surfaces of Bidegree $$(3,n)$$ in Gr (1,P3), 1990, to appear in Math. Z. [13] Gross, M., The Distribution of Bidegrees of Smooth Surfaces in Gr(1,P3), Math. Ann., 292, 127-147, (1992) · Zbl 0741.14017 [14] Hartshorne, R., Algebraic Geometry, Springer-Verlag (1977) [15] Hernández, R.; Sols, I., Line Congruences of Low Degree, 141-154, (1987), Paris · Zbl 0632.14038 [16] Jessop, C.M., A Treatise on the Line Complex, Cambridge University Press, (1903), reprinted Chelsea Publishing Company, 1969 · JFM 34.0702.06 [17] Kleiman, S., Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles, Publ. Math. IHES, 36, 281-298, (1969) · Zbl 0208.48501 [18] Ranestad, K., On Smooth Plane Curve Fibrations inP4, preprint (1990) [19] Ranestad, K., Surfaces of Degree 10 inP4, Ph.D. Thesis, Oslo (1990) [20] Verra, A., Smooth Surfaces of Degree 9 in $$G$$(1, 3), Manuscripta Mathematica, 62, 417-435, (1988) · Zbl 0673.14026
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