ADE surfaces and their moduli

Alexeev, Valery; Thompson, Alan

doi:10.1090/jag/762

Most recent issue | All issues | Recently published articles

ADE surfaces and their moduli

Authors: Valery Alexeev and Alan Thompson
Journal: J. Algebraic Geom.
DOI: https://doi.org/10.1090/jag/762
Published electronically: November 19, 2020
Full-text PDF

Abstract | References | Additional Information

Abstract: We define a class of surfaces corresponding to the root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular families over these moduli spaces, which extend to families of stable pairs over the compactifications. One simple application is a geometric compactification of the moduli of rational elliptic surfaces that is a finite quotient of a projective toric variety.

[AET19] Valery Alexeev, Philip Engel, and Alan Thompson, Stable pair compactification of moduli of K3 surfaces of degree 2, arXiv:1903.09742, 2019.
[Ale02] Valery Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708. MR 1923963, https://doi.org/10.2307/3062130
[Ale06] Valery Alexeev, Higher-dimensional analogues of stable curves, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 515–536. MR 2275608
[AN88] V. A. Alekseev and V. V. Nikulin, Classification of del Pezzo surfaces with log-terminal singularities of index ≤2, involutions on 𝐾3 surfaces, and reflection groups in Lobachevskiĭ spaces, Lectures in mathematics and its applications, Vol. 2, No. 2 (Russian), Ross. Akad. Nauk, Inst. Mat. im. Steklova (MIAN), Moscow, 1988, pp. 51–150 (Russian, with Russian summary). MR 1787240
[AN89] V. A. Alekseev and V. V. Nikulin, Classification of del Pezzo surfaces with log-terminal singularities of index \le2 and involutions on 𝐾3 surfaces, Dokl. Akad. Nauk SSSR 306 (1989), no. 3, 525–528 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 3, 507–511. MR 1009466
[AN06] Valery Alexeev and Viacheslav V. Nikulin, Del Pezzo and 𝐾3 surfaces, MSJ Memoirs, vol. 15, Mathematical Society of Japan, Tokyo, 2006. MR 2227002
[Arn72] V. I. Arnol′d, Normal forms of functions near degenerate critical points, the Weyl groups 𝐴_{𝑘},𝐷_{𝑘},𝐸_{𝑘} and Lagrangian singularities, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 3–25 (Russian). MR 0356124
[BB00] Lionel Bayle and Arnaud Beauville, Birational involutions of 𝑃², Asian J. Math. 4 (2000), no. 1, 11–17. Kodaira’s issue. MR 1802909, https://doi.org/10.4310/AJM.2000.v4.n1.a2
[BB11] Victor Batyrev and Mark Blume, On generalisations of Losev-Manin moduli spaces for classical root systems, Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1053–1084. MR 2918154, https://doi.org/10.4310/PAMQ.2011.v7.n4.a2
[Bou05] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
[Dol12] Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027
[DPT80] Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.), Séminaire sur les Singularités des Surfaces, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980 (French). Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977. MR 579026
[Dyn52] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
[EOR07] Pavel Etingof, Alexei Oblomkov, and Eric Rains, Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, Adv. Math. 212 (2007), no. 2, 749–796. MR 2329319, https://doi.org/10.1016/j.aim.2006.11.008
[Fuj12] Osamu Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371. MR 2928144, https://doi.org/10.2977/PRIMS/71
[GHK15] Mark Gross, Paul Hacking, and Sean Keel, Moduli of surfaces with an anti-canonical cycle, Compos. Math. 151 (2015), no. 2, 265–291. MR 3314827, https://doi.org/10.1112/S0010437X14007611
[Hac04a] Paul Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), no. 2, 213–257. MR 2078368, https://doi.org/10.1215/S0012-7094-04-12421-2
[Hac04b] Paul Hacking, A compactification of the space of plane curves, math/0104193, 2004.
[HP10] Paul Hacking and Yuri Prokhorov, Smoothable del Pezzo surfaces with quotient singularities, Compos. Math. 146 (2010), no. 1, 169–192. MR 2581246, https://doi.org/10.1112/S0010437X09004370
[Kol13] János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950
[Kol15] János Kollár, Book on moduli of surfaces -- ongoing project, 2015, available at https://web.math.princeton.edu/~kollar.
[KSB88] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, https://doi.org/10.1007/BF01389370
[LM00] A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of flat connections, Michigan Math. J. 48 (2000), 443–472. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786500, https://doi.org/10.1307/mmj/1030132728
[MP86] Rick Miranda and Ulf Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), no. 4, 537–558. MR 867347, https://doi.org/10.1007/BF01160474
[Nak07] Noboru Nakayama, Classification of log del Pezzo surfaces of index two, J. Math. Sci. Univ. Tokyo 14 (2007), no. 3, 293–498. MR 2372472
[Per90] Ulf Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990), no. 1, 1–47. MR 1069483, https://doi.org/10.1007/BF02571223
[Sage17] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 7.5.1), 2017, http://www.sagemath.org
[Sho92] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, https://doi.org/10.1070/IM1993v040n01ABEH001862
[Ste98] John R. Stembridge, The partial order of dominant weights, Adv. Math. 136 (1998), no. 2, 340–364. MR 1626860, https://doi.org/10.1006/aima.1998.1736
[Tju70] G. N. Tjurina, Resolution of singularities of flat deformations of double rational points, Funkcional. Anal. i Priložen. 4 (1970), no. 1, 77–83 (Russian). MR 0267129

Additional Information

Valery Alexeev
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: valery@uga.edu

Alan Thompson
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
Email: a.m.thompson@lboro.ac.uk

DOI: https://doi.org/10.1090/jag/762
Received by editor(s): January 7, 2019
Received by editor(s) in revised form: October 6, 2019, and December 20, 2019
Published electronically: November 19, 2020
Additional Notes: The first author was partially supported by the NSF under DMS-1603604 and DMS-1902157.