Geodesic
Ice cream and orbifold Riemann--Roch
Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 461-486.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give an orbifold Riemann–Roch formula in closed form for the Hilbert series of a quasismooth polarized $n$-fold $(X,D)$, under the assumption that $X$ is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of $\mathrm{K3}$ surfaces and Calabi–Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications. Bibliography: 22 titles.
Keywords: orbifold, orbifold Riemann–Roch, Dedekind sum, Hilbert series, weighted projective varieties. 
@article{IM2_2013_77_3_a2,
     author = {A. Buckley and M. Reid and S. Zhou},
     title = {Ice cream and orbifold {Riemann--Roch}},
     journal = {Izvestiya. Mathematics },
     pages = {461--486},
     publisher = {mathdoc},
     volume = {77},
     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a2/}
}
TY  - JOUR
AU  - A. Buckley
AU  - M. Reid
AU  - S. Zhou
TI  - Ice cream and orbifold Riemann--Roch
JO  - Izvestiya. Mathematics 
PY  - 2013
SP  - 461
EP  - 486
VL  - 77
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a2/
LA  - en
ID  - IM2_2013_77_3_a2
ER  - 
%0 Journal Article
%A A. Buckley
%A M. Reid
%A S. Zhou
%T Ice cream and orbifold Riemann--Roch
%J Izvestiya. Mathematics 
%D 2013
%P 461-486
%V 77
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a2/
%G en
%F IM2_2013_77_3_a2
A. Buckley; M. Reid; S. Zhou. Ice cream and orbifold Riemann--Roch. Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 461-486. http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a2/

[1] A. Buckley and B. Szendrői, “Orbifold Riemann–Roch for threefolds with an application to Calabi–Yau geometry”, J. Algebraic Geom., 14:4 (2005), 601–622 | DOI | MR | Zbl

[2] S. Zhou, Orbifold Riemann–Roch and Hilbert series, University of Warwick PhD thesis, 2011

[3] M. Reid, “Young person's guide to canonical singularities”, Algebraic geometry (Brunswick, Maine 1985), Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1985, 345–414 | MR | Zbl

[4] A. R. Iano-Fletcher, “Working with weighted complete intersections”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 101–173 | MR | Zbl

[5] S. Altınok, G. Brown, and M. Reid, “Fano 3-folds, $\mathrm {K3}$ surfaces and graded rings”, Topology and geometry: commemorating SISTAG, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002, 25–53 | DOI | MR | Zbl

[6] J.-J. Chen, J. A. Chen, and M. Chen, “On quasismooth weighted complete intersections”, J. Algebraic Geom., 20:2 (2011), 239–262 | DOI | MR | Zbl

[7] M. Kawakita, “Three-fold divisorial contractions to singularities of higher indices”, Duke Math. J., 130:1 (2005), 57–126 | DOI | MR | Zbl

[8] T. Kawasaki, “The Riemann–Roch theorem for complex $V$-manifolds”, Osaka J. Math., 16:1 (1979), 151–159 | MR | Zbl

[9] B. Toen, “Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford”, K-Theory, 18:1 (1999), 33–76 | DOI | MR | Zbl

[10] F. Nironi, Riemann–Roch for weighted projective spaces, preprint, 2008

[11] D. Abramovich, T. Graber, and A. Vistoli, “Gromov–Witten theory of Deligne–Mumford stacks”, Amer. J. Math., 130:5 (2008), 1337–1398 | DOI | MR | Zbl

[12] D. Edidin, Riemann–Roch for Deligne–Mumford stacks, arXiv: 1205.4742

[13] M. F. Atiyah and G. B. Segal, “The index of elliptic operators. II”, Ann. of Math. (2), 87:3 (1968), 531–545 | DOI | MR | Zbl

[14] M. F. Atiyah and I. M. Singer, “The index of elliptic operators. III”, Ann. of Math. (2), 87:3 (1968), 546–604 | DOI | MR | Zbl

[15] A. Canonaco, The Beilinson complex and canonical rings of irregular surfaces, Mem. Amer. Math. Soc., 183, no. 862, Amer. Math. Soc., Providence, RI, 2006 | MR | Zbl

[16] M. Demazure, “Anneaux gradués normaux”, Introduction à la théorie des singularités, II, Travaux en Cours, 37, Hermann, Paris, 1988, 35–68 | MR | Zbl

[17] K. Watanabe, “Some remarks concerning Demazure's construction of normal graded rings”, Nagoya Math. J., 83 (1981), 203–211 | MR | Zbl

[18] W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math., 39, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[19] S. Mukai, “Biregular classification of Fano 3-folds and Fano manifolds of coindex 3”, Proc. Nat. Acad. Sci. U.S.A., 86:9 (1989), 3000–3002 | DOI | MR | Zbl

[20] A. Buckley, “Computing Dedekind sums using the Euclidean algorithm”, EUROSIM 2007: Proc. 6th EUROSIM Congress on Modelling and Simulation (Ljubljana, Slovenia), 2, ARGESIM, Vienna, 2007, 1–6

[21] Yu. Kawamata, “On the plurigenera of minimal algebraic 3-folds with $K\approx0$”, Math. Ann., 275:4 (1986), 539–546 | DOI | MR | Zbl

[22] Yu. Kawamata, “Boundedness of $\mathbb Q$-Fano threefolds”, Proceedings of the International Conference on Algebra (Novosibirsk 1989), Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992, 439–445 | DOI | MR | Zbl