Details
| Original language | English |
|---|---|
| Pages (from-to) | 638-681 |
| Number of pages | 44 |
| Journal | Proceedings of the London Mathematical Society |
| Volume | 108 |
| Issue number | 3 |
| Publication status | Published - Mar 2014 |
| Externally published | Yes |
Abstract
We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Proceedings of the London Mathematical Society, Vol. 108, No. 3, 03.2014, p. 638-681.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Singular del Pezzo surfaces whose universal torsors are hypersurfaces
AU - Derenthal, Ulrich
N1 - Funding information: This work was supported by grant DE 1646/2-1 of the Deutsche Forschungsgemeinschaft, by grant 200021 124737/1 of the Schweizer Nationalfonds and by the Center for Advanced Studies of LMU München.
PY - 2014/3
Y1 - 2014/3
N2 - We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.
AB - We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.
UR - http://www.scopus.com/inward/record.url?scp=84896958345&partnerID=8YFLogxK
U2 - 10.1112/plms/pdt041
DO - 10.1112/plms/pdt041
M3 - Article
AN - SCOPUS:84896958345
VL - 108
SP - 638
EP - 681
JO - Proceedings of the London Mathematical Society
JF - Proceedings of the London Mathematical Society
SN - 0024-6115
IS - 3
ER -