U. Derenthal, D. Loughran, “Singular del Pezzo surfaces that are equivariant compactifications”, Studies in number theory. Part 10, Zap. Nauchn. Sem. POMI, 377, POMI, St. Petersburg, 2010, 26–43; J. Math. Sci. (N. Y.), 171:6 (2010), 714

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Zapiski Nauchnykh Seminarov POMI, 2010, Volume 377, Pages 26–43 (Mi znsl3811)

This article is cited in 25 scientific papers (total in 25 papers)

Singular del Pezzo surfaces that are equivariant compactifications

U. Derenthal^a, D. Loughran^b

^a Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany
^b Department of Mathematics, University Walk, Bristol, UK

Full-text PDF (646 kB) Citations (25)

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Abstract: We determine which singular del Pezzo surfaces are equivariant compactifications of

$\mathbb G_\mathrm a^2$ , 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

$\mathbb G_\mathrm a\rtimes\mathbb G_\mathrm m$ . Bibl. 32 titles.

Key words and phrases: del Pezzo surfaces, rational points, Manin's conjecture, equivariant compactifications, Dynkin diagrams, blow-up.

Received: 09.06.2010

English version:
Journal of Mathematical Sciences (New York), 2010, Volume 171, Issue 6, Pages 714–724
DOI: https://doi.org/10.1007/s10958-010-0174-9

Bibliographic databases:

Document Type: Article

UDC: 512.752+511.515

Language: English

Citation: U. Derenthal, D. Loughran, “Singular del Pezzo surfaces that are equivariant compactifications”, Studies in number theory. Part 10, Zap. Nauchn. Sem. POMI, 377, POMI, St. Petersburg, 2010, 26–43; J. Math. Sci. (N. Y.), 171:6 (2010), 714–724

Citation in format AMSBIB

\Bibitem{DerLou10}

\by U.~Derenthal, D.~Loughran

\paper Singular del Pezzo surfaces that are equivariant compactifications

\inbook Studies in number theory. Part~10

\serial Zap. Nauchn. Sem. POMI

\yr 2010

\vol 377

\pages 26--43

\publ POMI

\publaddr St.~Petersburg

\mathnet{http://mi.mathnet.ru/znsl3811}

\transl

\jour J. Math. Sci. (N. Y.)

\yr 2010

\vol 171

\issue 6

\pages 714--724

\crossref{https://doi.org/10.1007/s10958-010-0174-9}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78650055772}

Linking options:

https://www.mathnet.ru/eng/znsl3811

https://www.mathnet.ru/eng/znsl/v377/p26

This publication is cited in the following 25 articles:

Judith Ortmann, “Integral points on a del Pezzo surface over imaginary quadratic fields”, Res. number theory, 10:4 (2024)
Anton Shafarevich, “Euler-symmetric projective toric varieties and additive actions”, Indagationes Mathematicae, 34:1 (2023), 42
I. V. Arzhantsev, Yu. I. Zaitseva, “Equivariant completions of affine spaces”, Russian Math. Surveys, 77:4 (2022), 571–650
Sergey Dzhunusov, “On uniqueness of additive actions on complete toric varieties”, Journal of Algebra, 609 (2022), 642
Pieropan M., Smeets A., Tanimoto Sh., Varilly-Alvarado A., “Campana Points of Bounded Height on Vector Group Compactifications”, Proc. London Math. Soc., 123:1 (2021), 57–101
Sergey Dzhunusov, “Additive actions on complete toric surfaces”, Int. J. Algebra Comput., 31:01 (2021), 19
Anton Shafarevich, “Additive Actions on Toric Projective Hypersurfaces”, Results Math, 76:3 (2021)
Sergey Dzhunusov, Yulia Zaitseva, “Commutative algebraic monoid structures on affine surfaces”, Forum Mathematicum, 33:1 (2021), 177
Huang Zh., Montero P., “Fano Threefolds as Equivariant Compactifications of the Vector Group”, Mich. Math. J., 69:2 (2020), 341–368
Derenthal U., Pieropan M., “The Split Torsor Method For Manin'S Conjecture”, Trans. Am. Math. Soc., 373:12 (2020), 8485–8524
Frei Ch., Loughran D., Sofos E., “Rational Points of Bounded Height on General Conic Bundle Surfaces”, Proc. London Math. Soc., 117:2 (2018), 407–440
Arzhantsev I., Romaskevich E., “Additive actions on toric varieties”, Proc. Amer. Math. Soc., 145:5 (2017), 1865–1879
Frei Ch., Pieropan M., “O-minimality on twisted universal torsors and Manin's conjecture over number fields”, Ann. Sci. Ec. Norm. Super., 49:4 (2016), 757–811
Baier S., Derenthal U., “Quadratic Congruences on Average and Rational Points on Cubic Surfaces”, Acta Arith., 171:2 (2015), 145–171
Ulrich Derenthal, Daniel Loughran, “Equivariant Compactifications of Two-Dimensional Algebraic Groups”, Proceedings of the Edinburgh Mathematical Society, 58:1 (2015), 149
Derenthal U., “Singular Del Pezzo Surfaces Whose Universal Torsors Are Hypersurfaces”, Proc. London Math. Soc., 108:3 (2014), 638–681
Derenthal U., Frei Ch., “Counting Imaginary Quadratic Points Via Universal Torsors”, Compos. Math., 150:10 (2014), 1631–1678
Le Boudec P., “Affine Congruences and Rational Points on a Certain Cubic Surface”, Algebr. Number Theory, 8:5 (2014), 1259–1296
Ivan Arzhantsev, Andrey Popovskiy, Springer Proceedings in Mathematics & Statistics, 79, Automorphisms in Birational and Affine Geometry, 2014, 17
Le Boudec P., “Manin's conjecture for two quartic del Pezzo surfaces with $3\mathbf A_1$ and $\mathbf A_1+\mathbf A_2$ singularity types”, Acta Arith., 151:2 (2012), 109–163

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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