Abstract
Let \(|L_g|\), be the genus g du Val linear system on a Halphen surface Y of index k. We prove that the Clifford index \({\text {Cliff}}(C)\) is constant on smooth curves \(C\in |L_g|\). Let \(\gamma (C)\) be the gonality of C. When \({\text {Cliff}}(C)<\lfloor {\frac{g-1}{2}}\rfloor \) (the relevant case), we show that \(\gamma (C)={\text {Cliff}}(C)+2=k\), and that the gonality is realized by the Weierstrass linear series \(|-{kK_Y}_{|C}|\), which is totally ramified at one point. The proof of the first statement follows closely the path indicated by Green and Lazarsfeld for a similar statement regarding K3 surfaces.
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References
Arbarello, E., Bruno, A.: Rank two vector bundles on polarized Halphen surfaces and the Gauss-Wahl map for du Val curves. J. l’École Polytech. Math. 4, 257–285 (2017)
Arbarello, E., Bruno, A., Farkas, G., Saccà, G.: Explicit Brill-Noether-Petri general curves. Commentarii Mathematici Helvetici 91(3), 477–491 (2016)
Arbarello, E., Bruno, A., Sernesi, E.: On hyperplane sections of K3 surfaces. Algebr. Geom. 4, 562–596 (2017)
Bruno, A., Lelli-Chiesa, M. :Irreducibility of Severi varieties on K3 surfaces arXiv:2112.09398 , Math. AG
Cantat, S., Dolgachev, I.: Rational surfaces with a large group of automorphisms. J. Am. Math. Soc. 25(3), 863–905 (2012)
Ciliberto, C., Pareschi, G.: Pencils of minimal degree on curves on a K3 surface. J. Reine Angew. Math. 460, 15–36 (1995)
Donagi, R., Morrison, D.: Linear systems on K3 sections. J. Diff. Geom. 29, 49–64 (1989)
Green, M., Lazarsfeld, R.: Special divisors on a K3 surface. Invent. Math. 89, 357–370 (1987)
Harbourne, R.: Complete linear systems on rational surfaces. Trans. Am. Math. Soc. 289(1), 213–226 (1985)
Knutsen, L.A.: On kth-order embeddings of K3 surfaces and Enriques surfaces. Manuscripta Math. 104, 211–237 (2001)
Martens, G.: On curves on K3 surfaces. Springer LNM 1398, 174–182 (1989)
Nagata, M.: On rational surfaces II. Memoirs of the College of Science, University of Kyoto 32, 271–293 (1960)
Saint-Donat, B.: Projective models of K3 surfaces. Am. J. Math. 96(4), 602–639 (1974)
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Arbarello, E. A remark on du Val linear systems. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01011-9
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DOI: https://doi.org/10.1007/s12215-024-01011-9