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Algebra & N Umber Theory Algebra & Number Theory Algebra & Number Theory Volume 3 No. 3 2009 Volume 3 2009 2009 No. 3 Vol. 3, No. 3 Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, USA Yujiro Kawamata University of Tokyo, Japan Michel van den Bergh Hasselt University, Belgium Janos´ Kollar´ Princeton University, USA Marie-France Vigneras´ Universite´ Paris VII, France Hendrik W. Lenstra Universiteit Leiden, The Netherlands Kei-Ichi Watanabe Nihon University, Japan Yuri Manin Northwestern University, USA Andrei Zelevinsky Northeastern University, USA Barry Mazur Harvard University, USA Efim Zelmanov University of California, San Diego, USA PRODUCTION [email protected] Paulo Ney de Souza, Production Manager Silvio Levy, Senior Production Editor See inside back cover or www.jant.org for submission instructions. Regular subscription rate for 2009: $200.00 a year ($140.00 electronic only). Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA. Algebra & Number Theory, ISSN 1937-0652, at Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. PUBLISHED BY mathematical sciences publishers http://www.mathscipub.org A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2009 by Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 3:3(2009) On nondegeneracy of curves Wouter Castryck and John Voight We study the conditions under which an algebraic curve can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field nd is nondegenerate in the above sense. More generally, let ᏹg be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we nd nd show that dim ᏹg D min.2g C1; 3g −3/, except for g D 7 where dim ᏹ7 D 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4. Let k be a perfect field with algebraic closure k. Let f 2 kTx±1; y±1U be an P i j irreducible Laurent polynomial, and write f D .i; j/2Z2 ci j x y . We denote by 2 supp. f / D f.i; j/ 2 Z V ci j 6D 0g the support of f , and we associate to f its New- ton polytope 1 D 1. f /, the convex hull of supp. f / in R2. We assume throughout P i j that 1 is 2-dimensional. For a face τ ⊂ 1, let f jτ D .i; j/2τ ci j x y . We say that f is nondegenerate if, for every face τ ⊂ 1 (of any dimension), the system of equations @ f j @ f j f j D x τ D y τ D 0 (1) τ @x @y has no solutions in k∗2. From the perspective of toric varieties, the condition of nondegeneracy can be rephrased as follows. The Laurent polynomial f defines a curve U. f / in the torus 2 ±1 ±1 2 Tk D Spec kTx ; y U, and Tk embeds canonically in the projective toric surface X.1/k associated to 1 over k. Let V . f / be the Zariski closure of the curve U. f / inside X .1/k. Then f is nondegenerate if and only if for every face τ ⊂ 1, the intersection V . f / \ Tτ is smooth of codimension 1 in Tτ , where Tτ is the toric component of X.1/k associated to τ. (See Proposition 1.2 for alternative characterizations.) Nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry [Batyrev and Cox 1994]: A wealth of geometric information about V . f / is contained in the combinatorics of the Newton polytope 1. f /. The notion was first used by Kouchnirenko [1976], MSC2000: primary 14M25; secondary 14H10. Keywords: nondegenerate curve, toric surface, Newton polytope, moduli space. 255 256 Wouter Castryck and John Voight who studied nondegenerate polynomials in the context of singularity theory. Non- denegerate polynomials also emerge naturally in the theory of sparse resultants [Gel’fand et al. 1994] and admit a linear effective Nullstellensatz [Castryck et al. 2006, Section 2.3]. They make an appearance in the study of real algebraic curves in maximal position [Mikhalkin 2000] and in the problem of enumerating curves through a set of prescribed points [Mikhalkin 2003]. In the case where k is a finite field, they arise in the construction of curves with many points [Beelen and Pellikaan 2000; Kresch et al. 2002], in the p-adic cohomology theory of Adolphson and Sperber [1989], and in explicit methods for computing zeta functions of vari- eties over k [Castryck et al. 2006]. Despite their utility and seeming ubiquity, the intrinsic property of nondegeneracy has not seen detailed study, with the exception of the otherwise unpublished PhD thesis of Koelman [1991]; see Section 12 below. We are therefore led to the central problem of this article: Which curves are nondegenerate? To the extent that toric varieties are generalizations of projective space, this question essentially asks us to generalize the characterization of nonsin- gular plane curves amongst all curves. An immediate provocation for this question was to understand the locus of curves to which the point counting algorithm of Castryck, Denef, and Vercauteren [Castryck et al. 2006] actually applies. Our results are collected in two parts. In the first part, comprising Sections 3–7, we investigate the nondegeneracy of some interesting classes of curves (hyperelliptic, Cab, and low genus curves). Our conclusions can be summarized as follows. Theorem. Let V be a curve of genus g over a perfect field k. Suppose that one of these conditions holds: (i) g D 0; (ii) g D 1 and V .k/ 6D ?; (iii) g D 2; 3, and either 17 ≤ #k < 1, or #k D 1 and V .k/ 6D ?; (iv) g D 4 and k D k. Then V is nondegenerate. Remark. The condition #k ≥ 17 in (iii) ensures that k is large enough to allow nontangency to the toric boundary of X .1/k, but is most likely superfluous; see Remark 7.2. In the second part, consisting of Sections 8–12, we restrict to algebraically nd closed fields k D k and consider the locus ᏹg of nondegenerate curves inside the coarse moduli space of all curves of genus g ≥ 2. We prove the following: nd Theorem. We have dim ᏹg D min.2g C 1; 3g − 3/, except for g D 7 where nd D dim ᏹ7 16. In particular, a generic curve of genus g is nondegenerate if and only if g ≤ 4. On nondegeneracy of curves 257 Our methods combine ideas of Bruns and Gubeladze [2002] and Haase and Schicho [2009] and are purely combinatorial — only the universal property of the coarse moduli space is used. Conventions and notations. Throughout, 1 ⊂ R2 will denote a polytope with dim 1 D 2. The coordinate functions on the ambient space R2 will be denoted by X and Y .A facet or edge of a polytope is a face of dimension 1. A lattice polytope is a polytope with vertices in Z2. Two lattice polytopes 1 and 10 are equivalent if there is an affine map ' V R2 ! R2; v 7! Av C b 0 2 such that '.1/ D 1 with A 2 GL2.Z/ and b 2 Z . Two Laurent polynomials f and f 0 are equivalent if f 0 can be obtained from f by applying such a map to the exponent vectors. Note that equivalence preserves nondegeneracy. For a polytope 1 ⊂ R2, we let int.1/ denote the interior of 1. We denote the standard 2-simplex in R2 by 6 D conv.f.0; 0/; .1; 0/; .0; 1/g/. 1. Nondegenerate Laurent polynomials In this section, we review the geometry of nondegenerate Laurent polynomials. We retain the notation used in the introduction. In particular, k is a perfect field, P i j ±1 ±1 f D ci j x y 2 kTx ; y U is an irreducible Laurent polynomial, and 1 is its Newton polytope. Our main implicit reference on toric varieties is [Fulton 1993]. Let kT1U denote the graded semigroup algebra over k generated in degree d by the monomials that are supported in d1, that is, 1 M i j d 2 kT1UD hx y t j .i; j/ 2 .d1 \ Z /ik: dD0 Then X D X .1/k D Proj kT1U is the projective toric surface associated to 1 over k.
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