Abstract
We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill–Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill–Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over \({\mathbb {Q}}\) which are Brill–Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.
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References
Arbarello, E.: Weierstrass points and moduli of curves. Compos. Math. 29(3), 325–342 (1973)
Arbarello, E., Bruno, A., Farkas, G., Saccà, G.: Explicit Brill–Noether–Petri general curves. Comment. Math. Helv. 91(3), 477–491 (2016)
Arbarello, E., Bruno, A., Sernesi, E.: On hyperplane sections of K3 surfaces. To Appear in Algebraic Geometry, arXiv:1507.05002
Cukierman, F.: Families of Weierstrass points. Duke Math. J. 58(2), 317–346 (1989)
Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math. 85(2), 337–371 (1986)
Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus \(\ge 23\). Invent. Math. 90(2), 359–387 (1987)
Eisenbud, D., Harris, J.: Irreducibility of some families of linear series with Brill–Noether number \(-1\). Ann. Sci. École Norm. Sup. (4) 22(1), 33–53 (1989)
Epema, D.H.J.: Surfaces with Canonical Hyperplane Sections, CWI Tract, vol. 1. Stichting Mathematisch Centrum, Centrumvoor Wiskunde en Informatica, Amsterdam (1984)
Farkas, G.: Syzygies of curves and the effective cone of \(\overline{\cal{M}}_g\). Duke Math. J. 135(1), 53–98 (2006)
Farkas, G., Kemeny, M.: The Prym-Green conjecture for curves of odd genus. Preprint
Farkas, G., Kemeny, M.: The Prym-Green conjecture for torsion line bundles of high order. Appear Duke Math. J., arXiv:1509.07162
Farkas, G., Popa, M.: Effective divisors on \(\overline{\cal{M}}_g\), curves on \(K3\) surfaces, and the slope conjecture. J. Algebraic Geom. 14(2), 241–267 (2005)
Farkas, G., Tarasca, N.: Pointed castelnuovo numbers. Math. Res. Lett. 23(2), 389–404 (2016)
Fuentes-Garcia, L., Pedreira, M.: The projective theory of ruled surfaces. Note di Matematica 24, 25–63 (2005)
Griffiths, P., Harris, J.: On the variety of special linear systems on a general algebraic curve. Duke Math. J. 47(1), 233–272 (1980)
Harris, J., Morrison, I.: Slopes of effective divisors on the moduli space of stable curves. Invent. Math. 99(2), 321–355 (1990)
Lazarsfeld, R.: Brill–Noether–Petri without degenerations. J. Differ. Geom. 23(3), 299–307 (1986)
Osserman, B.: A simple characteristic-free proof of the Brill-Noether theorem. Bull. Braz. Math. Soc. (N.S.) 45(4), 807–818 (2014)
Polishchuk, A.: Contracting the Weierstrass locus to a point. arXiv:1611.04243
Treibich, A.: Revêtements tangentiels et condition de Brill-Noether. C. R. Acad. Sci. Paris Sér. I Math. 316(8), 815–817 (1993)
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Farkas, G., Tarasca, N. Du Val curves and the pointed Brill–Noether Theorem. Sel. Math. New Ser. 23, 2243–2259 (2017). https://doi.org/10.1007/s00029-017-0329-3
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DOI: https://doi.org/10.1007/s00029-017-0329-3