Hostname: page-component-6898cfcb55-cbk24 Total loading time: 0 Render date: 2026-11-11T16:56:38.961Z Has data issue: false hasContentIssue false
  • English
  • Français

ORBIT PARAMETRIZATIONS FOR K3 SURFACES

Part of:Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  07 July 2016

MANJUL BHARGAVA,
WEI HO and
ABHINAV KUMAR
MANJUL BHARGAVA
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA; bhargava@math.princeton.edu
WEI HO
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; weiho@umich.edu
ABHINAV KUMAR
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USADepartment of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA; thenav@gmail.com

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

MSC classification

Primary:14J28: $K3$ surfaces and Enriques surfaces14J10: Families, moduli, classification14J50: Automorphisms of surfaces and higher-dimensional varieties

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016, e18
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

References

Société Mathématique de France, Géométrie des surfaces $K3$ : modules et périodes (Paris, 1985), Papers from the seminar held in Palaiseau, October 1981–January 1982, Astérisque No. 126 (1985).Google Scholar
Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of Algebraic Curves. Vol I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267 (Springer, New York, 1985).Google Scholar
Bădescu, L., Algebraic Surfaces (Universitext, Springer, New York, 2001), Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author.CrossRefGoogle Scholar
Beauville, A., ‘Determinantal hypersurfaces’, Michigan Math. J. 48 (2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday.CrossRefGoogle Scholar
Beauville, A., ‘Fano threefolds and K3 surfaces’, inThe Fano Conference (Univ. Torino, 2004), 175–184. Turin.Google Scholar
Bergeron, N., Li, Z., Millson, J. and Moeglin, C., ‘The Noether–Lefschetz conjecture and generalizations’, Preprint, 2014, http://arxiv.org/abs/1412.3774.Google Scholar
Bhargava, M., ‘Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations’, Ann. of Math. (2) 159(1) (2004), 217–250.CrossRefGoogle Scholar
Bhargava, M., ‘Higher composition laws II. On cubic analogues of Gauss composition’, Ann. of Math. (2) 159(2) (2004), 865–886.CrossRefGoogle Scholar
Bhargava, M., ‘Higher composition laws. III. The parametrization of quartic rings’, Ann. of Math. (2) 159(3) (2004), 1329–1360.CrossRefGoogle Scholar
Bhargava, M., ‘The density of discriminants of quartic rings and fields’, Ann. of Math. (2) 162(2) (2005), 1031–1063.Google Scholar
Bhargava, M., ‘Higher composition laws IV. The parametrization of quintic rings’, Ann. of Math. (2) 167(1) (2008), 53–94.CrossRefGoogle Scholar
Bhargava, M., ‘The density of discriminants of quintic rings and fields’, Ann. of Math. (2) 172(3) (2010), 1559–1591.CrossRefGoogle Scholar
Bhargava, M., ‘Most hyperelliptic curves over $\mathbb{Q}$ have no rational points’, Preprint, 2013,http://arxiv.org/abs/1308.0395.Google Scholar
Bhargava, M., Gross, B. and Wang, X., ‘Pencils of quadrics and the arithmetic of hyperelliptic curves’, J. Amer. Math. Soc. to appear, http://arxiv.org/abs/1310.7692.Google Scholar
Bhargava, M. and Gross, B. H., ‘The average size of the 2-selmer group of Jacobians of hyperelliptic curves having a rational weierstrass point’, inAutomorphic Representations and L-Functions, Tata Institute of Fundamental Research Studies in Mathematics, 22 (Tata Inst. Fund. Res., Mumbai, 2013), 23–91.Google Scholar
Bhargava, M. and Ho, W., ‘Coregular spaces and genus one curves’, Cambridge J. Math. 4(1) (2016), 1–119.CrossRefGoogle Scholar
Bhargava, M. and Shankar, A., ‘Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves’, Ann. of Math. (2) 181(1) (2015), 191–242.CrossRefGoogle Scholar
Bhargava, M. and Shankar, A., ‘Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0’, Ann. of Math. (2) 181(2) (2015), 587–621.Google Scholar
Bhargava, M., Shankar, A. and Tsimerman, J., ‘On the Davenport–Heilbronn theorems and second order terms’, Invent. Math. 193(2) (2013), 439–499.Google Scholar
Cantat, S., ‘Dynamique des automorphismes des surfaces K3’, Acta Math. 187(1) (2001), 1–57.CrossRefGoogle Scholar
Cantat, S., ‘Dynamics of automorphisms of compact complex surfaces’, inFrontiers in Complex Dynamics: In Celebration of John Milnor’s 80th Birthday (eds. Sutherland, S., Bonifant, A. and Lyubich, M.) (Princeton University Press, Princeton, NJ, 2014), 463–514.CrossRefGoogle Scholar
Cassels, J. W. S., Rational Quadratic Forms, London Mathematical Society Monographs, 13 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978).Google Scholar
Cayley, A., ‘A memoir on quartic surfaces’, Proc. Lond. Math. Soc. 3 (1869/70), 19–69. [Collected Papers, VII, 133–181].Google Scholar
Cayley, A., The Collected Mathematical Papers. Volume 1, Cambridge Library Collection (Cambridge University Press, Cambridge, 2009), Reprint of the 1889 original.Google Scholar
Cossec, F. and Dolgachev, I., ‘On automorphisms of nodal Enriques surfaces’, Bull. Amer. Math. Soc. (N.S.) 12(2) (1985), 247–249.Google Scholar
Cossec, F. R., ‘Reye congruences’, Trans. Amer. Math. Soc. 280(2) (1983), 737–751.Google Scholar
Dardanelli, E. and van Geemen, B., ‘Hessians and the Moduli space of cubic surfaces’, inAlgebraic Geometry, Contemporary Mathematics, 422 (American Mathematical Society, Providence, RI, 2007), 17–36.Google Scholar
Davenport, H. and Heilbronn, H., ‘On the density of discriminants of cubic fields’, Bull. Lond. Math. Soc. 1 (1969), 345–348.CrossRefGoogle Scholar
Delone, B. N. and Faddeev, D. K., The Theory of Irrationalities of the Third Degree, Translations of Mathematical Monographs, 10 (American Mathematical Society, Providence, RI, 1964).Google Scholar
Dolgachev, I. V., ‘Mirror symmetry for lattice polarized K3 surfaces’, J. Math. Sci. 81(3) (1996), 2599–2630. Algebraic Geometry, Vol. 4.Google Scholar
Dolgachev, I. V., Classical Algebraic Geometry (Cambridge University Press, Cambridge, 2012), A modern view.CrossRefGoogle Scholar
Dolgachev, I. and Keum, J., ‘Birational automorphisms of quartic Hessian surfaces’, Trans. Amer. Math. Soc. 354(8) (2002), 3031–3057.Google Scholar
Eppstein, D., ‘The many faces of the Nauru graph’, http://11011110.livejournal.com/124705.html.Google Scholar
Festi, D., Garbagnati, A., van Geemen, B. and van Luijk, R., ‘The Cayley–Oguiso automorphism of positive entropy on a K3 surface’, J. Mod. Dyn. 7(1) (2013), 75–97.CrossRefGoogle Scholar
Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics (Birkhäuser Boston Inc., Boston, MA, 2008), Reprint of the 1994 edition.Google Scholar
Ghate, E. and Hironaka, E., ‘The arithmetic and geometry of Salem numbers’, Bull. Amer. Math. Soc. (N.S.) 38(3) (2001), 293–314.CrossRefGoogle Scholar
Greer, F., Li, Z. and Tian, Z., ‘Picard groups on moduli of K3 surfaces with Mukai models’, Int. Math. Res. Not. IMRN (16) (2015), 7238–7257.Google Scholar
Hutchinson, J. I., ‘The Hessian of the cubic surface. II’, Bull. Amer. Math. Soc. (N.S.) 6(8) (1900), 328–337.Google Scholar
Hutchinson, J. I., ‘The Hessian of the cubic surface’, Bull. Amer. Math. Soc. (N.S.) 5(6) (1899), 282–292.Google Scholar
Huybrechts, D., ‘Lectures on K3 surfaces’, http://www.math.uni-bonn.de/people/huybrech/K3.html.Google Scholar
Jessop, C. H., Quartic Surfaces with Singular Points (Cambridge University Press, Cambridge, 1916).Google Scholar
Katsylo, P. I., ‘Rationality of fields of invariants of reducible representations of the group SL2 ’, Vestnik Moskov. Univ. Ser. I. Mat. Mekh. (5) (1984), 77–79.Google Scholar
Kodaira, K., ‘On compact complex analytic surfaces I’, Ann. of Math. (2) 71 (1960), 111–152.Google Scholar
Kodaira, K., ‘On compact analytic surfaces. II, III’, Ann. of Math. (2) 77 (1963), 563–626. ibid. 78 (1963), 1–40.CrossRefGoogle Scholar
Kudla, S., ‘A note about special cycles on moduli spaces of K3 surfaces’, inArithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds, Fields Institute Communications, 67 (Springer, New York, 2013), 411–427.Google Scholar
Lombardo, G., Peters, C. and Schuett, M., ‘Abelian fourfolds of Weil type and certain K3 double planes’, Preprint, 2012, http://arxiv.org/abs/1209.5997.Google Scholar
Ma, S., ‘Rationality of fields of invariants for some representations of SL2 × SL2 ’, Compositio Math. 149(7) (2013), 1225–1234.Google Scholar
Matsumoto, K., Sasaki, T. and Yoshida, M., ‘The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type (3, 6)’, Internat. J. Math. 3(1) (1992), 1–164.Google Scholar
Maulik, D. and Pandharipande, R., ‘Gromov–Witten theory and Noether–Lefschetz theory’, inA Celebration of Algebraic Geometry, Clay Mathematics Proceedings, 18 (American Mathematical Society, Providence, RI, 2013), 469–507.Google Scholar
Mayer, A. L., ‘Families of K - 3 surfaces’, Nagoya Math. J. 48 (1972), 1–17.Google Scholar
McMullen, C. T., ‘Dynamics on K3 surfaces: Salem numbers and Siegel disks’, J. Reine Angew. Math. 545 (2002), 201–233.Google Scholar
McMullen, C. T., ‘K3 surfaces, entropy and glue’, J. Reine Angew. Math. 658 (2011), 1–25.CrossRefGoogle Scholar
Mukai, S., ‘Curves, K3 surfaces and Fano 3-folds of genus ⩽10’, inAlgebraic Geometry and Commutative Algebra, I (Kinokuniya, Tokyo, 1988), 357–377.Google Scholar
Mukai, S., ‘Polarized K3 surfaces of genus 18 and 20’, inComplex Projective Geometry (Trieste, 1989/Bergen, 1989), London Mathematical Society Lecture Note Series, 179 (Cambridge University Press, Cambridge, 1992), 264–276.Google Scholar
Mukai, S., ‘Polarized K3 surfaces of genus thirteen’, inModuli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, 45 (Mathematical Society of Japan, Tokyo, 2006), 315–326.Google Scholar
Ng, K. O., ‘The classification of (3, 3, 4) trilinear forms’, J. Korean Math. Soc. 39(6) (2002), 821–879.Google Scholar
Nikulin, V. V., ‘Kummer surfaces’, Izv. Akad. Nauk SSSR Ser. Mat. 39(2) (1975), 278–293, 471. English translation: Math. USSR. Izv. 9(2) (1975), 261–275.Google Scholar
Nikulin, V. V., ‘Finite groups of automorphisms of Kählerian K3 surfaces’, Tr. Mosk. Mat. Obs. 38 (1979), 75–137. English translation: Trans. Moscow Math. Soc. 38 (1980), 71–135.Google Scholar
Nikulin, V. V., ‘Integer symmetric bilinear forms and some of their geometric applications’, Izv. Akad. Nauk SSSR Ser. Mat. 43(1) (1979), 111–177, 238.Google Scholar
Oguiso, Keiji, ‘The third smallest Salem number in automorphisms of K3 surfaces’, inAlgebraic Geometry in East Asia—Seoul 2008, Advanced Studies in Pure Mathematics, 60 (Mathematical Society of Japan, Tokyo, 2010), 331–360.Google Scholar
Oguiso, Keiji, ‘Free automorphisms of positive entropy on smooth Kähler surfaces’, Preprint, 2012, http://arxiv.org/abs/arXiv:1202.2637v3.Google Scholar
Poonen, B. and Stoll, M., ‘Most odd degree hyperelliptic curves have only one rational point’, Ann. of Math. (2) 180(3) (2014), 1137–1166.Google Scholar
Reschke, P., ‘Salem numbers and automorphisms of complex surfaces’, Math. Res. Lett. 19(2) (2012), 475–482.Google Scholar
Rivin, I., ‘Algorithm to compute the integral orthogonal group, MathOverflow’, http://mathoverflow.net/q/143543 (version: 2013-09-29).Google Scholar
Room, T. G., ‘Self-transformations of determinantal quartic surfaces. I’, Proc. Lond. Math. Soc. (2) 51 (1950), 348–361.Google Scholar
Saint-Donat, B., ‘Projective models of K - 3 surfaces’, Amer. J. Math. 96 (1974), 602–639.Google Scholar
Shankar, A. and Wang, X., ‘Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves’, Preprint, 2013, http://arxiv.org/abs/1307.3531.Google Scholar
Snyder, V. and Sharpe, F. R., ‘Certain quartic surfaces belonging to infinite discontinuous Cremonian groups’, Trans. Amer. Math. Soc. 16(1) (1915), 62–70.Google Scholar
Taniguchi, T. and Thorne, F., ‘Secondary terms in counting functions for cubic fields’, Duke Math. J. 162(13) (2013), 2451–2508.Google Scholar
Thorne, J., ‘The arithmetic of simple singularities’, PhD Thesis, Harvard University, 2012.Google Scholar
Tjurin, A. N., ‘The intersection of quadrics’, Uspekhi Mat. Nauk 30(6(186)) (1975), 51–99.Google Scholar
Wall, C. T. C., ‘Pencils of binary quartics’, Rend. Semin. Mat. Univ. Padova 99 (1998), 197–217.Google Scholar
Wilson, K., ‘Three perspectives on $n$ points in $\mathbb{P}^{n-2}$ ’, PhD Thesis, Princeton University, 2012.Google Scholar
Wood, M. M., ‘Rings and ideals parameterized by binary n-ic forms’, J. Lond. Math. Soc. (2) 83(1) (2011), 208–231.Google Scholar
Wood, M. M., ‘Parametrization of ideal classes in rings associated to binary forms’, J. Reine Angew. Math. 689 (2014), 169–199.Google Scholar
Wright, D. J. and Yukie, A., ‘Prehomogeneous vector spaces and field extensions’, Invent. Math. 110(2) (1992), 283–314.Google Scholar