Skip to main content Accessibility help

Equivariant Compactifications of Two-Dimensional Algebraic Groups

  • Ulrich Derenthal (a1) and Daniel Loughran (a1)


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 .



Hide All
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), 429471.
2.Baier, S. and Derenthal, U., Quadratic congruences on average and rational points on cubic surfaces, preprint (arXiv:1205.0373, 2012).
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), 2743.
4.Batyrev, V. V. and Tschinkel, Yu., Manin's conjecture for toric varieties, J. Alg. Geom. 7(1) (1998), 1553.
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. 299340 (Société Mathématique de France, Paris, 1998).
6.Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer, 1991).
7.Browning, T. D., Quantitative arithmetic of projective varieties, Progress in Mathematics, Volume 277 (Birkhäuser, 2009).
8.Bruce, J. W. and Wall, C. T. C., On the classification of cubic surfaces, J. Lond. Math. Soc. 19(2) (1979), 245256.
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), 421452.
10.Coray, D. F. and Tsfasman, M. A., Arithmetic on singular del Pezzo surfaces, Proc. Lond. Math. Soc. 57(1) (1988), 2587.
11.Demazure, M. and Pinkham, H. C. (Eds), Séminaire sur les singularités des surfaces, Lecture Notes in Mathematics, Volume 777 (Springer, 1980).
12.Derenthal, U., Geometry of universal torsors, Doctoral Dissertation, Universitat Gottingen (2006).
13.Derenthal, U., Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. 108(3) (2014), 638681.
14.Derenthal, U. and Loughran, D., Singular del Pezzo surfaces that are equivariant compactifications, J. Math. Sci. 171(6) (2010), 714724.
15.Dolgachev, I., Lectures on invariant theory, London Mathematical Society Lecture Note Series, Volume 296 (Cambridge University Press, 2003).
16.Grothendieck, A., Éléments de géométrie algébrique, IV, Étude locale des schémas et des morphismes de schémas, Publ. Math. IHES 20(1) (1964), 5259.
17.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).
18.Hassett, B. and Tschinkel, Yu., Geometry of equivariant compactifications of , Int. Math. Res. Not. 22 (1999), 12111230.
19.Derenthal, U. and Loughran, D., Singular del Pezzo surfaces that are equivariant compactifications, J. Math. Sci. 171(6) (2010), 714724
20.Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, 3rd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Volume 34 (Springer, 1994).
21.Sakamaki, Y., Automorphism groups on normal singular cubic surfaces with no parameters, Trans. Am. Math. Soc. 362(5) (2010), 26412666.
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. 119157 (American Mathematical Society, Providence, RI, 2012).
23.Ye, Q., On Gorenstein log del Pezzo surfaces. Jpn J. Math. 28(1) (2002), 87136.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed