## Algebra & Number Theory

### Tropical independence, II: The maximal rank conjecture for quadrics

#### Abstract

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

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

Dates
Revised: 16 April 2016
Accepted: 31 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842582

Digital Object Identifier
doi:10.2140/ant.2016.10.1601

Mathematical Reviews number (MathSciNet)
MR3556794

Zentralblatt MATH identifier
1379.14020

#### Citation

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. https://projecteuclid.org/euclid.ant/1510842582

#### References

• 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.