Knutsen, Andreas Leopold; Lopez, Angelo Felice Brill-Noether theory of curves on Enriques surfaces. II: The Clifford index. (English) Zbl 1402.14039 Manuscr. Math. 147, No. 1-2, 193-237 (2015). Summary: We complete our study of linear series on curves lying on an Enriques surface by showing that, with the exception of smooth plane quintics, there are no exceptional curves on Enriques surfaces, that is, curves for which the Clifford index is not computed by a pencil.For part I, see [Math. Z. 261, No. 3, 659–690 (2009; Zbl 1161.14022)]. Cited in 1 ReviewCited in 6 Documents MSC: 14H45 Special algebraic curves and curves of low genus 14J28 \(K3\) surfaces and Enriques surfaces 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14C20 Divisors, linear systems, invertible sheaves Citations:Zbl 1161.14022 PDFBibTeX XMLCite \textit{A. L. Knutsen} and \textit{A. F. Lopez}, Manuscr. Math. 147, No. 1--2, 193--237 (2015; Zbl 1402.14039) Full Text: DOI arXiv References: [1] Cossec F.R., Dolgachev I.V.: Enriques Surfaces I. Progress in Mathematics 76. Birkhäuser, Boston, MA (1989) · Zbl 0665.14017 [2] Coppens M., Martens G.: Secant spaces and Clifford’s theorem. Compos. Math. 78, 193-212 (1991) · Zbl 0741.14035 [3] Ciliberto C., Pareschi G.: Pencils of minimal degree on curves on a K3 surface. J. Reine Angew. Math. 460, 15-36 (1995) · Zbl 0836.14022 [4] Ein, L.: On the cohomology of projectively Cohen-Macaulay determinantal subvarieties of Pn. Geometry of complex projective varieties (Cetraro, 1990), pp. 143-152, Sem. Conf., 9, Mediterranean, Rende, 1993 · Zbl 0938.14030 [5] Eisenbud D., Koh J., Stillman M.: Determinantal equations for curves of high degree. Amer. J. Math. 110(3), 513-539 (1988) · Zbl 0681.14027 [6] Eisenbud D., Lange H., Martens G., Schreyer F.O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173-204 (1989) · Zbl 0703.14020 [7] Green M., Lazarsfeld R.: Special divisors on curves on a K3 surface. Invent. Math. 89, 357-370 (1987) · Zbl 0625.14022 [8] Giraldo L., Lopez A.F., Muñoz R.: On the projective normality of Enriques surfaces. Math. Ann. 324, 135-158 (2002) · Zbl 1081.14052 [9] Giraldo L., Lopez A.F., Muñoz R.: On the existence of Enriques-Fano threefolds of index greater than one. J. Algebraic Geom. 13, 143-166 (2004) · Zbl 1059.14051 [10] Grinenko M.M.: Normal surfaces with a numerically positive anticanonical divisor. Fundam. Prikl. Mat. 4, 757-761 (1998) · Zbl 0957.14024 [11] Hartshorne R.: Algebraic Geometry. Graduate Texts in Mathematics, 52. Springer, New York (1977) · Zbl 0367.14001 [12] Knutsen A.L., Lopez A.F.: Brill-Noether theory of curves on Enriques surfaces I: the positive cone and gonality. Math. Z. 261, 659-690 (2009) · Zbl 1161.14022 [13] Knutsen A.L., Lopez A.F.: Surjectivity of Gaussian maps for general curves on Enriques surfaces. Adv. Geom. 7, 215-247 (2007) · Zbl 1124.14035 [14] Knutsen A.L., Lopez A.F.: A sharp vanishing theorem for line bundles on K3 or Enriques surfaces. Proc. Amer. Math. Soc. 135, 3495-3498 (2007) · Zbl 1121.14033 [15] Knutsen, A.L., Lopez, A.F.: Projective normality and the generation of the ideal of an Enriques surface. Adv. Geom. (to appear) · Zbl 1323.14022 [16] Knutsen, A.L., Lopez, A.F., Muñoz, R.: On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds. J. Differ. Geom. 88(3):483-518 (2011) · Zbl 1238.14026 [17] Knutsen A.L.: On kth order embeddings of K3 surfaces and Enriques surfaces. Manuscr. Math. 104, 211-237 (2001) · Zbl 1017.14015 [18] Knutsen A.L.: Exceptional curves on Del Pezzo surfaces. Math. Nachr. 256, 58-81 (2003) · Zbl 1048.14015 [19] Knutsen A.L.: On two conjectures for curves on K3 surfaces. Int. J. Math. 20, 1547-1560 (2009) · Zbl 1197.14042 [20] Kojima H.: On normal surfaces with strictly nef anticanonical divisor. Arch. Math. 77, 517-521 (2001) · Zbl 1063.14046 [21] Laufer H.B.: On minimally elliptic singularities. Amer. J. Math. 99, 1257-1295 (1977) · Zbl 0384.32003 [22] Lazarsfeld R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299-307 (1986) · Zbl 0608.14026 [23] Lopez A.F., Pirola G.P.: On the curves through a general point of a smooth surface in \[{\mathbb{P}^3}\] P3. Math. Z. 219, 93-106 (1995) · Zbl 0842.14008 [24] Martens, G.: Über den Clifford-Index algebraischer Kurven. J. Reine Und Angew. Math. 336, 83-90 (1982) · Zbl 0484.14010 [25] Matsuki K.: Introduction to the Mori program. Universitext. Springer, New York (2002) · Zbl 0988.14007 [26] Pareschi G.: Exceptional linear systems on curves on Del Pezzo surfaces. Math. Ann. 291, 17-38 (1991) · Zbl 0723.14004 [27] Reider I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309-316 (1988) · Zbl 0663.14010 [28] Stagnaro E.: Constructing Enriques surfaces from quintics in \[{\mathbb{P}^3_K}\] PK3. Algebraic Geometry—open problems (Ravello, 1982). Lecture Notes in Math. 997. Springer, Berlin (1983) [29] Umezu Y.: Normal quintic surfaces which are birationally Enriques surfaces. Publ. Res. Inst. Math. Sci. 33, 359-384 (1997) · Zbl 0909.14023 [30] Umezu Y.: On normal projective surfaces with trivial dualizing sheaf. Tokyo J. Math. 4, 343-354 (1981) · Zbl 0496.14025 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.