Spin canonical rings of log stacky curves
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2339-2383.

Consider modular forms arising from a finite-area quotient of the upper-half plane by a Fuchsian group. By the classical results of Kodaira–Spencer, this ring of modular forms may be viewed as the log spin canonical ring of a stacky curve. In this paper, we tightly bound the degrees of minimal generators and relations of log spin canonical rings. As a consequence, we obtain a tight bound on the degrees of minimal generators and relations for rings of modular forms of arbitrary integral weight.

Considérons les formes modulaires d’un quotient d’aire finie du demi-plan de Poincaré par un groupe fuchsien. D’après un résultat classique de Kodaira–Spencer, cet anneau de formes modulaires peut être considéré comme l’anneau log-canonique à spin d’une courbe champêtre. Dans cet article, nous obtenons une borne optimale pour les degrés des générateurs minimaux et des relations minimales d’un tel anneau, et donc des anneaux de formes modulaires de poids entier arbitraire.

Published online:
DOI: 10.5802/aif.3065
Classification: 14Q05, 11F11
Keywords: Modular forms, canonical rings, theta characteristic, Petri’s theorem, stacks, Groebner basis
Mot clés : Formes modulaires, anneaux canoniques, thêta-caractéristiques, théorème de Pétri, champs algébriques, base de Gröbner
Landesman, Aaron 1; Ruhm, Peter 2; Zhang, Robin 2

1 Department of Mathematics Harvard University One Oxford Street Cambridge MA 02138 (USA)
2 Department of Mathematics Stanford University Building 380 Stanford CA 94305 (USA)
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     title = {Spin canonical rings of log stacky curves},
     journal = {Annales de l'Institut Fourier},
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Landesman, Aaron; Ruhm, Peter; Zhang, Robin. Spin canonical rings of log stacky curves. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2339-2383. doi : 10.5802/aif.3065. https://aif.centre-mersenne.org/articles/10.5802/aif.3065/

[1] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Volume 15 (2002) no. 1, pp. 27-75 | DOI

[2] Adler, Allan; Ramanan, S. Moduli of abelian varieties, Lecture Notes in Mathematics, 1644, Springer-Verlag, Berlin, 1996, vi+196 pages

[3] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A.; Harris, Joseph Geometry of algebraic curves. Volume I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer, Heidelberg, 1985, xvi+386 pages

[4] Behrend, Kai; Noohi, Behrang Uniformization of Deligne-Mumford curves, J. Reine Angew. Math., Volume 599 (2006), pp. 111-153

[5] Buckley, A.; Reid, M.; Zhou, S. Ice cream and orbifold Riemann-Roch, Izv. Ross. Akad. Nauk Ser. Mat., Volume 77 (2013) no. 3, pp. 29-54 | DOI

[6] Cox, David; Little, John; O’Shea, Donald Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer, New York, 2007, xvi+551 pages

[7] Eisenbud, David Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages

[8] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977, xvi+496 pages (Graduate Texts in Mathematics, No. 52)

[9] Milnor, John On the 3-dimensional Brieskorn manifolds M(p,q,r), Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N. J., 1975, p. 175-225. Ann. of Math. Studies, No. 84

[10] Neves, Jorge Halfcanonical rings on algebraic curves and applications to surfaces of general type, University of Warwick, UK (2003), 155 pages (Ph. D. Thesis)

[11] O’Dorney, Evan Canonical rings of divisors on 1 , Annals of Combinatorics, Volume 19 (2015) no. 4, pp. 765-784 | DOI

[12] Popescu-Pampu, Patrick The geometry of continued fractions and the topology of surface singularities, Singularities in geometry and topology 2004 (Adv. Stud. Pure Math.), Volume 46, Math. Soc. Japan, Tokyo, 2007, pp. 119-195

[13] Reid, Miles Infinitesimal view of extending a hyperplane section—deformation theory and computer algebra, Algebraic geometry (L’Aquila, 1988) (Lecture Notes in Math.), Volume 1417, Springer, Berlin, 1990, pp. 214-286 | DOI

[14] Saint-Donat, B. On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann., Volume 206 (1973), pp. 157-175 | DOI

[15] Voight, John; Zureick-Brown, David The canonical ring of a stacky curve (2015) (http://arxiv.org/abs/1501.04657)

[16] Watanabe, Keiichi Some remarks concerning Demazure’s construction of normal graded rings, Nagoya Math. J., Volume 83 (1981), pp. 203-211 http://projecteuclid.org/euclid.nmj/1118786485 | DOI

[17] Zhou, Shengtian Orbifold Riemann-Roch and Hilbert Series, The University of Warwick, UK (2011) http://wrap.warwick.ac.uk/49768/1/WRAP_THESIS_Zhou_2011.pdf (Ph. D. Thesis)

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