The Martens–Mumford theorem and the Green–Lazarsfeld secant conjecture
Author:
Daniele Agostini
Journal:
J. Algebraic Geom. 33 (2024), 629-654
DOI:
https://doi.org/10.1090/jag/819
Published electronically:
January 12, 2024
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References |
Additional Information
Abstract: The Green–Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill–Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.
References
- Daniele Agostini, Asymptotic syzygies and higher order embeddings, Int. Math. Res. Not. IMRN 4 (2022), 2934–2967. MR 4381936, DOI 10.1093/imrn/rnaa208
- Marian Aprodu, On the vanishing of higher syzygies of curves, Math. Z. 241 (2002), no. 1, 1–15. MR 1930982, DOI 10.1007/s002090100403
- Marian Aprodu, Remarks on syzygies of $d$-gonal curves, Math. Res. Lett. 12 (2005), no. 2-3, 387–400. MR 2150892, DOI 10.4310/MRL.2005.v12.n3.a9
- Marian Aprodu and Jan Nagel, Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI, 2010. MR 2573635, DOI 10.1090/ulect/052
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Lawrence Ein and Robert Lazarsfeld, The gonality conjecture on syzygies of algebraic curves of large degree, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 301–313. MR 3415069, DOI 10.1007/s10240-015-0072-2
- Lawrence Ein, Wenbo Niu, and Jinhyung Park, Singularities and syzygies of secant varieties of nonsingular projective curves, Invent. Math. 222 (2020), no. 2, 615–665. MR 4160876, DOI 10.1007/s00222-020-00976-5
- Gavril Farkas and Michael Kemeny, The generic Green-Lazarsfeld secant conjecture, Invent. Math. 203 (2016), no. 1, 265–301. MR 3437872, DOI 10.1007/s00222-015-0595-7
- Gavril Farkas and Michael Kemeny, Linear syzygies on curves with prescribed gonality, Adv. Math. 356 (2019), no. 1, 265–301.
- Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
- Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
- M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), no. 3, 301–314. MR 959214
- Michael Kemeny, The extremal secant conjecture for curves of arbitrary gonality, Compos. Math. 153 (2017), no. 2, 347–357. MR 3705227, DOI 10.1112/S0010437X16008198
- Michael Kemeny, Betti numbers of curves and multiple-point loci, J. Pure Appl. Algebra 226 (2022), no. 11, Paper No. 107090, 40. MR 4403637, DOI 10.1016/j.jpaa.2022.107090
- Michael Kemeny, Projecting syzygies of curves, Algebr. Geom. 7 (2020), no. 5, 561–580. MR 4156418, DOI 10.14231/ag-2020-020
- Andreas Krug, Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles, Math. Ann. 371 (2018), no. 1-2, 461–486. MR 3788855, DOI 10.1007/s00208-018-1660-5
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
References
- Daniele Agostini, Asymptotic syzygies and higher order embeddings, Int. Math. Res. Not. IMRN 4 (2022), 2934–2967. MR 4381936, DOI 10.1093/imrn/rnaa208
- Marian Aprodu, On the vanishing of higher syzygies of curves, Math. Z. 241 (2002), no. 1, 1–15. MR 1930982, DOI 10.1007/s002090100403
- Marian Aprodu, Remarks on syzygies of $d$-gonal curves, Math. Res. Lett. 12 (2005), no. 2-3, 387–400. MR 2150892, DOI 10.4310/MRL.2005.v12.n3.a9
- Marian Aprodu and Jan Nagel, Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI, 2010. MR 2573635, DOI 10.1090/ulect/052
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Lawrence Ein and Robert Lazarsfeld, The gonality conjecture on syzygies of algebraic curves of large degree, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 301–313. MR 3415069, DOI 10.1007/s10240-015-0072-2
- Lawrence Ein, Wenbo Niu, and Jinhyung Park, Singularities and syzygies of secant varieties of nonsingular projective curves, Invent. Math. 222 (2020), no. 2, 615–665. MR 4160876, DOI 10.1007/s00222-020-00976-5
- Gavril Farkas and Michael Kemeny, The generic Green–Lazarsfeld secant conjecture, Invent. Math. 203 (2016), no. 1, 265–301. MR 3437872, DOI 10.1007/s00222-015-0595-7
- Gavril Farkas and Michael Kemeny, Linear syzygies on curves with prescribed gonality, Adv. Math. 356 (2019), no. 1, 265–301.
- Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
- Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
- M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), no. 3, 301–314. MR 959214
- Michael Kemeny, The extremal secant conjecture for curves of arbitrary gonality, Compos. Math. 153 (2017), no. 2, 347–357. MR 3705227, DOI 10.1112/S0010437X16008198
- Michael Kemeny, Betti numbers of curves and multiple-point loci, J. Pure Appl. Algebra 226 (2022), no. 11, Paper No. 107090, 40. MR 4403637, DOI 10.1016/j.jpaa.2022.107090
- Michael Kemeny, Projecting syzygies of curves, Algebr. Geom. 7 (2020), no. 5, 561–580. MR 4156418, DOI 10.14231/ag-2020-020
- Andreas Krug, Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles, Math. Ann. 371 (2018), no. 1-2, 461–486. MR 3788855, DOI 10.1007/s00208-018-1660-5
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
Additional Information
Daniele Agostini
Affiliation:
Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10 (C-Bau), 72076 Tübingen, Germany
MR Author ID:
1135681
Email:
daniele.agostini@uni-tuebingen.de
Received by editor(s):
October 22, 2021
Received by editor(s) in revised form:
February 14, 2023
Published electronically:
January 12, 2024
Additional Notes:
This research was supported by the Deutscher Akademischer Austauschdienst and the Berlin Mathematical School.
Article copyright:
© Copyright 2024
University Press, Inc.
