Abstract
For a (smooth irreducible) curveC of genus g and Clifford indexc>2 with a linear seriesg d r computing c (so\(r = \frac{{d - c}}{2}\)) it is well known thatc + 2 ≤d ≤2 (c + 2), and if\(d > \frac{3}{2}(c + 2)\) then 2c + 1 ≤g ≤ 2c + 4 unlessd = 2c + 4 in which caseg = 2c + 5.
Let c ≥ 0 andg be integers. If 2c + 1 ≤g ≤2c + 4 we prove that for any integerd <g such thatd ≡c mod 2 andc + 2 ≤d < 2(c + 2) there exists a curve of genus g and Clifford index c with a gd r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙr, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙr of smaller degree. In fact, if gd r computes the Clifford index c of C such thatc + 8 ≤d ≤ 2c + 3 then the birational morphism defined by this series cannot map C onto a (maybe, singular) curve contained in a surface of degree at most 2r-3 in ℙr.
Similar content being viewed by others
References
R. D. M. Accola, Plane models for Riemann surfaces admitting certain half-canonical linear series. II.Trans. Amer. Math. Soc. 263 (1981), 243–259.
E. Arbarello,M. Cornalba,P.A. Griffiths, andJ. Harris,Geometry of algebraic curves. Vol. I. Grundlehren267, Springer Verlag, 1985.
M. Coppens, The gonality of general smooth curves with a prescribed plane nodal model.Math. Ann. 289 (1991), 89–93.
M. Coppens andG. Martens, Secant spaces and Clifford’s theorem.Compos. Math. 78(1991), 193–212.
—, Divisorial complete curves.Arch. Math. 86 (2006), 409–418.
C. Ciliberto andG. Pareschi Pencils of minimal degree on curves on aK3 surface.J. reine angew. Math. 460 (1995), 15–36.
D. Eisenbud, H. Lange, G. Martens, andF.-O. Schreyer, The Clifford dimension of a projective curve.Compos. Math. 72 (1989), 173–204.
M. Green andR. Lazarsfeld, Special divisors on curves on aK3 surface.Invent. math. 89(1987), 357–370.
F. J. Gallego andB. P. Purnaprajna, Normal presentation on elliptic ruled surfaces.J. Algebra 186 (1996), 597–625.
J. Harris,Curves in projective space. (In collaboration with D. Eisenbud.) Les presses de’l Université de Montréal. Montréal, 1982.
R. Hartshorne,Algebraic geometry. Graduate Texts in Math.52, Springer Verlag, 1977.
T. Horowitz, Varieties of low Δ-genus.Duke Math. J. 50 (1983), 667–683.
A. L. Knutsen, Smooth curves on projectiveK3 surfaces.Math. Scand. 90 (2002), 215–231.
C. Keem, S. Kim, andG. Martens, On a theorem of Farkas.J. reine angew. Math. 405 (1990), 112–116.
G. Martens, On curves onK3 surfaces. In: E. Ballico, C. Ciliberto (eds.),Algebraic curves and projective geometry. Proceedings 1988. Lecture Notes in Math.1389, Springer Verlag, 1989.
---, On unexpected linear series on curves. In: M. Homma, T. Kato (eds.),Proceedings of the conference Curve 02 at Yamaguchi Univ. Publ. by Departm. of Math., Kanagawa Univ. 2002, pp. 61–67.
I. Reider, Some applications of Bogomolov’s theorem. In: F. Catanese (ed.),Problems in the theory of surfaces and their classification. Papers from the meeting at Sc. Norm. Sup. Cortona 1988, London;Acad. Press. Sympos. Math. 32 (1991), 367-410.
H. Terakawa, Higher order embeddings of algebraic surfaces of Kodaira dimension zero.Math. Z. 229 (1998), 417–433.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: O. Riemenschneider