We determine which singular del Pezzo surfaces are equivariant compactifications of \( \mathbb{G}_{\text{a}}^2 \), to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \( {\mathbb{G}_{\text{a}}} \) ⋊ \( {\mathbb{G}_{\text{m}}} \). Bibliography: 32 titles.
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V. Alexeev and V. V. Nikulin, Del Pezzo and K3 Surfaces, MSJ Memoirs 15, Mathematical Society of Japan, Tokyo (2006).
R. de la Bretèche and T. D. Browning, “On Manin's conjecture for singular del Pezzo surfaces of degree 4. I,” Michigan Math. J., 55, No. 1, 51–80 (2007).
R. de la Bretèche and T. D. Browning, “Manin's conjeture for quartic del Pezzo surfaces with a conic fbration,” with an appendix by U. Derenthal, arXiv:0808.1616(2008).
R. de la Bretèche, T. D. Browning, and U. Derenthal, “On Manin's conjeturec for a certain singular cubic surface.” Ann. Sci. École Norm. Sup. (4), 40, No. 1, 1–50 (2007).
R. de la Bretèche, T. D. Browning, and E. Peyre, “On Manin's conjecture for a family of Châtelet surfaces,” arXiv:1002.0255 (2010).
T. D. Browning and U. Derenthal, “Manin's conjecture for a cubic surface with D 5 singularity,” Internat. Math. Res. Notices, 14, 2620–2647 (2009).
T. D. Browning and U. Derenthal, “Manin's conjecture for a quartic del Pezzo surface with A 4 singularity,” Ann. Inst. Fourier (Grenoble), 59, No. 3, 1231–1265 (2009).
R. de la Bretèche and E. Fouvry, “L'éclaté du plan projectif en quatre points dont deux conjugués,”J. reine angew. Math., 576, 63–122 (2004).
P. Le Boudec, “Manin's conjecture for two quartic del Pezzo surfaces with 3A 1 and A 1 + A 2 singularity types,” arXiv:1006.0691 (2010).
R. de la Bretèche, “Sur le nombre de points de hauteur bornée dúne certaine surface cubique singulière,” Astérisque, 251, 51–77 (1998).
R. de la Bretèche, “”Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5,” Duke Math. J., 113, No. 2, 421–464 (2002).
V. V. Batyrev and Yu. Tschinkel, “Manin’s conjecture for toric varieties,” J. Algebraic Geom.,7, No. 1, 15–53 (1998).
J. W. Bruce and C. T. C. Wall, “On the classification of cubic surfaces,” J. London Math. Soc. (2), 19, No. 2, 245–256 (1979).
A. Chambert-Loir and Yu. Tschinkel, “On the distribution of points of bounded height on equivariant compatifications of vector groups,” Invent. Math., 148, No. 2, 421–452 (2002).
D. F.Coray and M. A. Tsfasman, “Arithmetic on singular Del Pezzo surfaces,” Proc. London Math. Soc. (3), 57, No. 1, 25–87 (1988).
U. Derenthal, forthoming.
U. Derenthal, “Singular del Pezzo surfaces whose universal torsors are hypersurfaces,” arXiv:math.AG/0604194 (2006).
U. Derenthal, “Manin's conjecture for a quintic del Pezzo surface with A 2 singularity,” arXiv:0710.1583 (2007).
U. Derenthal, “Counting integral points on universal torsors,” Internat. Math. Res. Notices, 14, 2648–2699 (2009).
U. Derenthal, M. Joyce, and Z. Teitler, “The nef cone volume of generalized del Pezzo surfaces,” Algebra Number Theory, 2, No. 2, 157–182 (2008).
M. Demazure and H. C. Pinkham (eds.), Séminaire sur les Singularités des Surfaces, Lect. Notes Math., 777, Springer, Berlin (1980).
U. Derenthal and Yu. Tshinkel, “Universal torsors over del Pezzo surfaces and rational points,” in: Equidistribution in Number Theory, an Introduction, Springer, Dordreht (2007), pp. 169–196.
J. Franke, Yu. I. Manin, and Yu. Tschinkel, “Rational points of bounded height on Fano varieties,” Invent. Math.,95, No. 2, 421–435 (1989).
É. Fouvry, “Sur la hauteur des points d'une certaine surface cubique singulière,” Astérisque, 251, 31–49 (1998).
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York (1977).
D. R. Heath-Brown and B. Z. Moroz, The density of rational points on the cubic surface X 3o = X 1 X 2 X 3,” Math. Proc. Cambridge Philos. Soc., 125, No. 3, 385–395 (1999).
B. Hassett and Yu. Tschinkel, “Geometry of equivariant compactifications of G n a ,” Internat. Math. Res. Notices, 22, 1211–1230 (1999).
B. Hassett and Yu. Tschinkel, “Universal torsors and Cox rings,” in: Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002), Progr. Math., 226, Birkhäuser Boston, Boston (2004), pp. 149–173.
D. T. Loughran, “Manin's conjecture for a singular sextic del Pezzo surface,” J. Théor. Nombres Bordeaux, to appear.
Y. Sakamaki, “Automorphism groups on normal singular cubic surfaces with no parameters,” Trans. Amer. Math. Soc., 362, No. 5, 2641–2666 (2010).
P. Salberger, “Tamagawa measures on universal torsors and points of bounded height on Fano varieties,” Astérisque, 251, 91–258 (1998).
Q. Ye, “On Gorenstein log del Pezzo surfaces,” Japan. J. Math. (N.S.), 28, No. 1, 87–136 (2002).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 377, 2010, pp. 26–43.
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Derenthal, U., Loughran, D. Singular del Pezzo surfaces that are equivariant compactifications. J Math Sci 171, 714–724 (2010). https://doi.org/10.1007/s10958-010-0174-9
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DOI: https://doi.org/10.1007/s10958-010-0174-9