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The gonality of smooth curves with plane models. (English) Zbl 0725.14005
Manuscr. Math. 70, No. 1, 5-25 (1990); Correction: 71, No. 3, 337-338 (1991).
Let \(\Gamma\) be an integral plane curve of degree \(d>k\geq 1\) with \(\delta\) ordinary nodes and cusps as its singularities, and let p: \(C\to \Gamma\) be its normalization. Let \({\mathbb{P}}_ k\) be the projective space parametrizing effective divisors of degree k on \({\mathbb{P}}^ 2\). The authors generalize a result of M. Namba [“Families of meromorphic functions on compact Riemann surfaces”, Lect. Notes Math. 767 (1979; Zbl 0417.32008)]) concerning linear systems on a smooth curve to the case with ordinary nodes and cusps as follows.
Let \(g^ 1_ n\) be a fixed point free linear system on C with \(n+\delta <k(d-k)\) for some integer \(k>0\). Then there exists a pencil \({\mathbb{P}}\subset {\mathbb{P}}_{k-1}\) including \(g^ 1_ n\) on C. From this main lemma the authors deduce several theorems, and give examples to show the sharpness of the theorems. The results are used by the first author in Math. Ann. 289, No.1, 89-93 (1991; Zbl 0697.14019).
The proof of the main lemma is corrected in Manuscr. Math. 71, No.3, 337- 338 (1991)].
Reviewer: R.Horiuchi (Kyoto)

14C20 Divisors, linear systems, invertible sheaves
14H20 Singularities of curves, local rings
Full Text: DOI EuDML
[1] Ballico, E. and Ciliberto C. (Eds.): Open Problemsin Algebraic Curves and Projective Geometry, Lecture Notes in Math.1389, Springer-Verlag, Berlin, (1989) 276–285
[2] Coppens, M.: A study of the schemesW e 1 of smooth plane curves, hoc. 1o Belgium-Spanish week on Algebra and Geometry 1988, R. U. C. A., 29–63
[3] Coppens, M.: Free linear systems on integral Gorenstein curves,to appear in J. of Algebra · Zbl 0770.14002
[4] Farkas H. M. and Kra, I.:Riemann Surfaces, Graduate texts in Math.,71, Springer-Verlag, 1980 · Zbl 0475.30001
[5] Fulton, W.:Algebraic Curves, W. A. Benjamin, Reading, Massachusetts, 1969 · Zbl 0181.23901
[6] Griffiths P. and Harris, J.:Principle of Algebraic Geometry, John Wiley & Sons, New York, 1978 · Zbl 0408.14001
[7] Hartshorne, R.:Algebraic Geometry, Springer-Verlag, Berlin, 1977 · Zbl 0367.14001
[8] Homma, M.: Funny plane curves in characteristicp>0,Comm. in algebra,15 (1987) 1469–1501 · Zbl 0623.14014
[9] Martens, H. H.: Varieties of special divisors on a curve II,J. reine angew. Math.,233 (1968) 89–100 · Zbl 0221.14004
[10] Namba, M.:Families of meromorphic functions on compact Riemann surfaces, Lecture Notes in Math.767, Springer-Verlag, Berlin, 1979 · Zbl 0417.32008
[11] Walker, R. J.:Algebraic Curves, Princeton Univ. Press, Princeton, New Jersey, 1950 · Zbl 0039.37701
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