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The extremal secant conjecture for curves of arbitrary gonality

Part of: Commutative algebra: Homological methods Curves

Published online by Cambridge University Press:  06 February 2017

Michael Kemeny
Michael Kemeny*
Affiliation:
Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany email [email protected]
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Abstract

We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.

Keywords

syzygiescurvesgonalitysecant conjecture

MSC classification

Primary: 14H51: Special divisors (gonality, Brill-Noether theory) 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 347 - 357
Copyright
© The Author 2017 

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References

Aprodu, M., Remarks on syzygies of d-gonal curves , Math. Res. Lett. 12 (2005), 387–400.Google Scholar
Aprodu, M. and Nagel, J., Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52 (American Mathematical Society, Providence, RI, 2010).Google Scholar
Aprodu, M. and Sernesi, E., Excess dimension for secant loci in symmetric products of curves, Preprint (2015), arXiv:1506.05281.Google Scholar
Arbarello, E. and Cornalba, M., Calculating cohomology groups of moduli spaces of curves via algebraic geometry , Publ. Math. Inst. Hautes Études Sci. 88 (1999), 97–127.Google Scholar
Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267 (Springer, 1985).CrossRefGoogle Scholar
Beauville, A., Some stable vector bundles with reducible theta divisor , Manuscripta Math. 110 (2003), 343–349.Google Scholar
Caporaso, L., Néron models and compactified Picard schemes over the moduli stack of stable curves , Amer. J. Math. 130 (2008), 1–47.CrossRefGoogle Scholar
Farkas, G., Syzygies of curves and the effective cone of M g , Duke Math. J. 135 (2006), 53–98.Google Scholar
Farkas, G., Koszul divisors on moduli spaces of curves , Amer. J. Math. 131 (2009), 819–867.Google Scholar
Farkas, G. and Kemeny, M., The generic Green–Lazarsfeld secant conjecture, Invent. Math., to appear.Google Scholar
Farkas, G., Mustaţă, M. and Popa, M., Divisors on M g, g+1 and the minimal resolution conjecture for points on canonical curves , Ann. Sci. Éc. Norm. Supér. 36 (2003), 553–581.CrossRefGoogle Scholar
Fulton, W., Harris, J. and Lazarsfeld, R., Excess linear series on an algebraic curve , Proc. Amer. Math. Soc. 92 (1984), 320–322.Google Scholar
Green, M., Koszul cohomology and the cohomology of projective varieties , J. Differential Geom. 19 (1984), 125–171.Google Scholar
Green, M. and Lazarsfeld, R., On the projective normality of complete linear series on an algebraic curve , Invent. Math. 83 (1986), 73–90.Google Scholar
Hirschowitz, A. and Ramanan, S., New evidence for Green’s conjecture on syzygies of canonical curves , Ann. Sci. Éc. Norm. Supér. 31 (1998), 145–152.Google Scholar
Voisin, C., Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface , J. Eur. Math. Soc. 4 (2002), 363–404.Google Scholar
Voisin, C., Green’s canonical syzygy conjecture for generic curves of odd genus , Compos. Math. 141 (2005), 1163–1190.Google Scholar