Newton polygons and curve gonalities
Wouter Castryck
and Filip Cools
Departement Wiskunde, Afdeling Algebra, Katholieke Universiteit Leuven, Celestijnenlaan 200, 3001 Leuven (Heverlee), Belgium
DOI: 10.1007/s10801-011-0304-6
Abstract
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.
Pages: 345–366
Keywords: Newton polygons; algebraic curves; gonality; toric surfaces; degenerations
Full Text: PDF
References
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2. Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613-653 (2008)
3. Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215, 766-788 (2007)
4. Batyrev, V., Nill, B.: Multiples of lattice polytopes without interior lattice points. Mosc. Math. J. 7(2), 195-207 (2007)
5. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 235-265 (1997)
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229. Springer, New York (2005). xvi+243 pp.
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12. Gathmann, A., Kerber, M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217-230 (2008)
13. Gel'fand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser, Boston (1994)
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7. Am. Math. Mon. 116(2), 151-165 (2009)
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21. Kleiman, S.L., Laksov, D.: Another proof of the existence of special divisors. Acta Math. 132, 163- 176 (1974)
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25. Lubbes, N., Schicho, J.: Lattice polygons and families of curves on rational surfaces. J. Algebr. Comb.
2. Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613-653 (2008)
3. Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215, 766-788 (2007)
4. Batyrev, V., Nill, B.: Multiples of lattice polytopes without interior lattice points. Mosc. Math. J. 7(2), 195-207 (2007)
5. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 235-265 (1997)
6. Beelen, P.: A generalization of Baker's theorem. Finite Fields Appl. 15(5), 558-568 (2009)
7. Bruns, W., Gubeladze, J.: Polytopes, Rings, and K-theory. Springer Monographs in Mathematics Springer, Berlin (2009)
8. Castryck, W., Voight, J.: On nondegeneracy of curves. Algebra Number Theory 3(3), 255-281 (2009)
9. Cox, D.A.: Toric varieties and toric resolutions. In: Conference Proceedings of `Resolution of Singularities', Obergurgl,
1997. Progress in Mathematics, vol. 181, pp. 259-284 (2000)
10. Eisenbud, D.: The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry. Graduate Texts in Mathematics, vol.
229. Springer, New York (2005). xvi+243 pp.
11. Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173-204 (1989)
12. Gathmann, A., Kerber, M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217-230 (2008)
13. Gel'fand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser, Boston (1994)
14. Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125- 171, 279-289 (1984) J Algebr Comb (2012) 35:345-366
15. Haase, C., Schicho, J.: Lattice polygons and the number 2i +
7. Am. Math. Mon. 116(2), 151-165 (2009)
16. Hladký, J., Král', D., Norine, S.: Rank of divisors on tropical curves. Preprint (2007)
17. Lange, H., Martens, G.: On the gonality sequence of an algebraic curve. Manuscr. Math. (2011)
18. Kawaguchi, R.: The gonality conjecture for curves on certain toric varieties. Osaka J. Math. 45, 113- 126 (2008)
19. Kawaguchi, R.: The gonality conjecture for curves on toric surfaces with two P1-fibrations. Saitama Math. J. 27, 35-80 (2010)
20. Khovanski\?ı, A.G.: Newton polyhedra, and toroidal varieties. Funct. Anal. Appl. 11(4), 289-296 (1978)
21. Kleiman, S.L., Laksov, D.: Another proof of the existence of special divisors. Acta Math. 132, 163- 176 (1974)
22. Koelman, R.J.: The number of moduli of families of curves on toric surfaces. Ph.D. Thesis, Katholieke Universiteit Nijmegen (1991)
23. Koelman, R.J.: A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics. Beiträge Algebra Geom. 34(1), 57-62 (1993)
24. Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1-31 (1976)
25. Lubbes, N., Schicho, J.: Lattice polygons and families of curves on rational surfaces. J. Algebr. Comb.
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