×

Smooth surfaces of degree 9 in G(1,3). (English) Zbl 0673.14026

The author gives a classification of smooth congruences of lines of degree \( 9\) in three-dimensional complex projective space. Proofs are given in terms of the associated grassmannian G(1,3), i.e. Klein’s quadric. Moreover explicit constructions for these congruences are given. The paper extends a classical result on congruences of degree \(<9\) due to G. Fano (1893).
Reviewer: H.Havlicek

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] ARBARELLO E., CORNALBA M., GRIFFITHS P., HARRIS J.: Geometry of Algebraic Curves I, Berlin-Heidelberg-New York, Springer 1984 · Zbl 0559.14017
[2] ARRONDO E., Sols I.: Classification of smooth congruences of low degree, preprint, 1987
[3] BARTH W., PETERS C., VAN De VEN A.: Compact complex surfaces, Berlin-Heidelberg-New York, Springer 1984 · Zbl 0718.14023
[4] BEAUVILLE A.: Surfaces Algebriques Complexes, Ast?risque54, 1978
[5] KLEIMAN S.: The transversality of a general translate, Comp. Math.28, 287-297 1974 · Zbl 0288.14014
[6] BORDIGA G.: Di una certa congruenza del terzo ordine e della sesta classe dello spazio ordinario, Rend. Ace. Lincei4, 8-13 1890 · JFM 22.0682.01
[7] EIN L.: Surfaces with a hyperelliptic hyperplane section, Duke math. J.50, 685-694, 1984 · Zbl 0561.14014
[8] FANO G.: Studio di alcuni sistemi di rette considerati come superficie dello spazio a cinque dimensioni, Ann. Mat. pura appl.21, 141-192, 1893 · JFM 25.1282.02
[9] GRIFFITHS P., HARRIS J.: Priciples of Algebraic Geometry, New York, Wiley and S., 1978 · Zbl 0408.14001
[10] HARTSHORNE R.: Algebraic Geometry, Berlin-Heidelberg-New York, Springer 1977 · Zbl 0367.14001
[11] LIVORNI E.L.: Classification of algebraic non ruled surfaces with sectional genus less or equal to six, Nagoya Math. J.100, 1-9, 1985 · Zbl 0594.14028
[12] LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six: Rational surfaces, Pac. J. of Math.113, 93-113, 1984 · Zbl 0573.14013
[13] LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six; ruled surfaces with dim ?K?L(X)=2, Math. Scand.58, 9-29, 1986 · Zbl 0663.14024
[14] LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six; ruled surfaces with dim ?K?L(X)=1, Can. J. Math.37, 1110-1121, 1986 · Zbl 0598.14030
[15] OKONEK C.: Flaechen vom grad 8 in P4, Math. Zeits.191, 207-223, 1986 · Zbl 0611.14032
[16] OKONEK C.: Moduli reflexiver Garben und Flaechen vom Kleinem Grad in P4, Math. Zeits.184, 549-572, 1983 · Zbl 0524.14018
[17] OKONEK C.: Ueber 2-codimensionale Untermannigfaltigkeiten vom Grad 7 in P4 und P5, Math. Zeits.187, 209-219, 1984 · Zbl 0575.14030
[18] PAPANTONOPOULOU A.: Embeddings in G(1,3), Proc. A.M.S.89,583-586, 1983
[19] RAN Z.: Surfaces of order 1 in Grassmannians, J.f.d. Reine u. Angew. Mathematik362, 1986 · Zbl 0601.14042
[20] REIDER I.: Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math.127, 309-316, 1988 · Zbl 0663.14010
[21] SAINT DONAT D.: Projective models of K3 surfaces, Am. J. Math.96, 602-639, 1974 · Zbl 0301.14011
[22] SOMMESE A.J.: Hyperplane sections of projective surfaces I - The Adjunction mapping, Duke Math. J.46, 377-401, 1979 · Zbl 0415.14019
[23] SOMMESE A.J., VAN DE VEN A.: On the Adjunction mapping, Math. Ann.278, 593-603, 1987 · Zbl 0655.14001
[24] VAN DE VEN A.: On the 2-connectedness of very ample divisors on a surface, Duke Math. J.46, 403-407, 1979 · Zbl 0458.14003
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.