Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T20:24:05.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant Compactifications of Two-Dimensional Algebraic Groups

Part of: Surfaces and higher-dimensional varieties Algebraic groups Diophantine equations

Published online by Cambridge University Press:  27 October 2014

Ulrich Derenthal  and
Daniel Loughran
Ulrich Derenthal
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Weifengerten 1, 30167 Hannover, Germany, ([email protected]; [email protected])
Daniel Loughran
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Weifengerten 1, 30167 Hannover, Germany, ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

Keywords

del Pezzo surfacesalgebraic groupsequivariant compactificationsManin's conjectures

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients)
Secondary: 14J26: Rational and ruled surfaces 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 149 - 168
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arzhantsev, I., Hausen, J., Herppich, E. and Liendo, A., The automorphism group of a variety with torus action of complexity one, Moscow Math J. 14(3) (2014), 429–471.Google Scholar
2.Baier, S. and Derenthal, U., Quadratic congruences on average and rational points on cubic surfaces, preprint (arXiv:1205.0373, 2012).Google Scholar
3.Batyrev, V. V. and Manin, Yu. I., Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Annalen 286(1) (1990), 27–43.CrossRefGoogle Scholar
4.Batyrev, V. V. and Tschinkel, Yu., Manin's conjecture for toric varieties, J. Alg. Geom. 7(1) (1998), 15–53.Google Scholar
5.Batyrev, V. V. and Tschinkel, Yu., Tamagawa numbers of polarized algebraic varieties, in Nombre et répartition de points de hauteur bornée (Paris, 1996), Asterisqué, pp. 299–340 (Société Mathématique de France, Paris, 1998).Google Scholar
6.Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer, 1991).CrossRefGoogle Scholar
7.Browning, T. D., Quantitative arithmetic of projective varieties, Progress in Mathematics, Volume 277 (Birkhäuser, 2009).Google Scholar
8.Bruce, J. W. and Wall, C. T. C., On the classification of cubic surfaces, J. Lond. Math. Soc. 19(2) (1979), 245–256.CrossRefGoogle Scholar
9.Chambert-Loir, A. and Tschinkel, Yu., On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math. 148(2) (2002), 421–452.CrossRefGoogle Scholar
10.Coray, D. F. and Tsfasman, M. A., Arithmetic on singular del Pezzo surfaces, Proc. Lond. Math. Soc. 57(1) (1988), 25–87.Google Scholar
11.Demazure, M. and Pinkham, H. C. (Eds), Séminaire sur les singularités des surfaces, Lecture Notes in Mathematics, Volume 777 (Springer, 1980).CrossRefGoogle Scholar
12.Derenthal, U., Geometry of universal torsors, Doctoral Dissertation, Universitat Gottingen (2006).Google Scholar
13.Derenthal, U., Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. 108(3) (2014), 638–681.Google Scholar
14.Derenthal, U. and Loughran, D., Singular del Pezzo surfaces that are equivariant compactifications, J. Math. Sci. 171(6) (2010), 714–724.Google Scholar
15.Dolgachev, I., Lectures on invariant theory, London Mathematical Society Lecture Note Series, Volume 296 (Cambridge University Press, 2003).CrossRefGoogle Scholar
16.Grothendieck, A., Éléments de géométrie algébrique, IV, Étude locale des schémas et des morphismes de schémas, Publ. Math. IHES 20(1) (1964), 5–259.Google Scholar
17.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).CrossRefGoogle Scholar
18.Hassett, B. and Tschinkel, Yu., Geometry of equivariant compactifications of , Int. Math. Res. Not. 22 (1999), 1211–1230.Google Scholar
19.Derenthal, U. and Loughran, D., Singular del Pezzo surfaces that are equivariant compactifications, J. Math. Sci. 171(6) (2010), 714–724Google Scholar
20.Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, 3rd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Volume 34 (Springer, 1994).Google Scholar
21.Sakamaki, Y., Automorphism groups on normal singular cubic surfaces with no parameters, Trans. Am. Math. Soc. 362(5) (2010), 2641–2666.Google Scholar
22.Tanimoto, S. and Tschinkel, Yu., Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups, in Zeta functions in algebra and geometry, Contemporary Mathematics, Volume 566, pp. 119–157 (American Mathematical Society, Providence, RI, 2012).CrossRefGoogle Scholar
23.Ye, Q., On Gorenstein log del Pezzo surfaces. Jpn J. Math. 28(1) (2002), 87–136.Google Scholar