Algebra & Number Theory

Tropical independence, II: The maximal rank conjecture for quadrics

David Jensen and Sam Payne

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noether’s theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal rank conjecture for algebraic curves.

Article information

Algebra Number Theory, Volume 10, Number 8 (2016), 1601-1640.

Received: 2 October 2015
Revised: 16 April 2016
Accepted: 31 May 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H51: Special divisors (gonality, Brill-Noether theory)
Secondary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

Brill-Noether theory tropical geometry tropical independence chain of loops maximal rank conjecture Gieseker-Petri


Jensen, David; Payne, Sam. Tropical independence, II: The maximal rank conjecture for quadrics. Algebra Number Theory 10 (2016), no. 8, 1601--1640. doi:10.2140/ant.2016.10.1601.

Export citation


  • D. Abramovich, L. Caporaso, and S. Payne, “The tropicalization of the moduli space of curves”, Ann. Sci. Éc. Norm. Supér. $(4)$ 48:4 (2015), 765–809.
  • O. Amini and M. Baker, “Linear series on metrized complexes of algebraic curves”, Math. Ann. 362:1-2 (2015), 55–106.
  • E. Arbarello and C. Ciliberto, “Adjoint hypersurfaces to curves in $\mbP\sp{r}$ following Petri”, pp. 1–21 in Commutative algebra (Trento, 1981), edited by S. Greco and G. Valla, Lecture Notes in Pure and Appl. Math. 84, Dekker, New York, 1983.
  • E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, I, Grundlehren der Math. Wissenschaften 267, Springer, New York, 1985.
  • M. Baker, “Specialization of linear systems from curves to graphs”, Algebra Number Theory 2:6 (2008), 613–653.
  • M. Baker and S. Norine, “Riemann–Roch and Abel–Jacobi theory on a finite graph”, Adv. Math. 215:2 (2007), 766–788.
  • M. Baker and S. Norine, “Harmonic morphisms and hyperelliptic graphs”, Int. Math. Res. Not. 2009:15 (2009), 2914–2955.
  • E. Ballico, “Embeddings of general curves in projective spaces: the range of the quadrics”, Lith. Math. J. 52:2 (2012), 134–137.
  • E. Ballico, “Remarks on the maximal rank conjecture”, Int. J. Pure Appl. Math. 77:3 (2012), 327–336.
  • E. Ballico and P. Ellia, “Beyond the maximal rank conjecture for curves in $\mbP\sp 3$”, pp. 1–23 in Space curves (Rocca di Papa, 1985), edited by F. Ghione et al., Lecture Notes in Math. 1266, Springer, Berlin, 1987.
  • E. Ballico and P. Ellia, “The maximal rank conjecture for nonspecial curves in $\mbP\sp n$”, Math. Z. 196:3 (1987), 355–367.
  • E. Ballico and P. Ellia, “On the existence of curves with maximal rank in $\mbP\sp n$”, J. Reine Angew. Math. 397 (1989), 1–22.
  • E. Ballico and C. Fontanari, “Normally generated line bundles on general curves, II”, J. Pure Appl. Algebra 214:8 (2010), 1450–1455.
  • D. Bayer and D. Eisenbud, “Graph curves”, Adv. Math. 86:1 (1991), 1–40.
  • D. Cartwright, D. Jensen, and S. Payne, “Lifting divisors on a generic chain of loops”, Canad. Math. Bull. 58:2 (2015), 250–262.
  • G. Castelnuovo, F. Enriques, and F. Severi, “Max Noether”, Math. Ann. 93:1 (1925), 161–181.
  • A. Castorena, A. L. Martín, and M. Teixidor i Bigas, “Invariants of the Brill–Noether curve”, preprint, 2014.
  • F. Cools, J. Draisma, S. Payne, and E. Robeva, “A tropical proof of the Brill–Noether theorem”, Adv. Math. 230:2 (2012), 759–776.
  • D. Eisenbud and J. Harris, “A simpler proof of the Gieseker–Petri theorem on special divisors”, Invent. Math. 74:2 (1983), 269–280.
  • D. Eisenbud and J. Harris, “Limit linear series: basic theory”, Invent. Math. 85:2 (1986), 337–371.
  • D. Eisenbud and J. Harris, “Irreducibility and monodromy of some families of linear series”, Ann. Sci. École Norm. Sup. $(4)$ 20:1 (1987), 65–87.
  • D. Eisenbud and J. Harris, “The Kodaira dimension of the moduli space of curves of genus $\geq 23$”, Invent. Math. 90:2 (1987), 359–387.
  • D. Eisenbud and J. Harris, “The monodromy of Weierstrass points”, Invent. Math. 90:2 (1987), 333–341.
  • D. Eisenbud and J. Harris, “Irreducibility of some families of linear series with Brill–Noether number $-1$”, Ann. Sci. École Norm. Sup. $(4)$ 22:1 (1989), 33–53.
  • G. Farkas, “Koszul divisors on moduli spaces of curves”, Amer. J. Math. 131:3 (2009), 819–867.
  • G. Farkas and M. Popa, “Effective divisors on $\overkern41{\mathscr{M}}\sb g$, curves on $K3$ surfaces, and the slope conjecture”, J. Algebraic Geom. 14:2 (2005), 241–267.
  • M. Green and R. Lazarsfeld, “On the projective normality of complete linear series on an algebraic curve”, Invent. Math. 83:1 (1986), 73–90.
  • C. Haase, G. Musiker, and J. Yu, “Linear systems on tropical curves”, Math. Z. 270:3-4 (2012), 1111–1140.
  • J. Harris, “The genus of space curves”, Math. Ann. 249:3 (1980), 191–204.
  • J. Harris, Curves in projective space, Séminaire de Mathématiques Supérieures 85, Presses de l'Université de Montréal, 1982.
  • J. Harris, “Brill–Noether theory”, pp. 131–143 in Geometry of Riemann surfaces and their moduli spaces, edited by L. Ji et al., Surv. Differ. Geom. XIV, Int. Press, Somerville, MA, 2009.
  • D. Jensen and S. Payne, “Tropical independence, I: Shapes of divisors and a proof of the Gieseker–Petri theorem”, Algebra Number Theory 8:9 (2014), 2043–2066.
  • E. Katz, J. Rabinoff, and D. Zureick-Brown, “Uniform bounds for the number of rational points on curves of small Mordell–Weil rank”, preprint, 2015. To appear in Duke Math. J.
  • E. Larson, “The Maximal Rank Conjecture for Sections of Curves”, preprint, 2012.
  • R. Lazarsfeld, “Brill–Noether–Petri without degenerations”, J. Differential Geom. 23:3 (1986), 299–307.
  • M. Noether, “Zur Grundlegung der Theorie der algebraischen Raumcurven”, J. Reine Angew. Math. 93 (1882), 271–318.
  • B. Osserman, “Limit linear series for curves not of compact type”, preprint, 2014.
  • B. Osserman, “Dimension counts for limit linear series on curves not of compact type”, Math. Z. (online publication April 2016).
  • J. Rathmann, “The uniform position principle for curves in characteristic $p$”, Math. Ann. 276:4 (1987), 565–579.
  • F. Severi, “Sulla classificazione delle curve algebriche e sul teorema di esistenza di Riemann”, Rend. R. Acc. Naz. Lincei 24:5 (1915), 887–888.
  • M. Teixidor i Bigas, “Injectivity of the symmetric map for line bundles”, Manuscripta Math. 112:4 (2003), 511–517.
  • C. Voisin, “Sur l'application de Wahl des courbes satisfaisant la condition de Brill–Noether–Petri”, Acta Math. 168:3-4 (1992), 249–272.
  • J. Wang, “Some results on the generic vanishing of Koszul cohomology via deformation theory”, Pacific J. Math. 273:1 (2015), 47–73.