Details
| Original language | English |
|---|---|
| Pages (from-to) | 157-182 |
| Number of pages | 26 |
| Journal | Algebra and Number Theory |
| Volume | 2 |
| Issue number | 2 |
| Publication status | Published - 1 Jan 2008 |
| Externally published | Yes |
Abstract
We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.
Keywords
- Del Pezzo surface, Manin’s conjecture, Nef cone, Root system
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Algebra and Number Theory, Vol. 2, No. 2, 01.01.2008, p. 157-182.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The nef cone volume of generalized del pezzo surfaces
AU - Derenthal, Ulrich
AU - Joyce, Michael
AU - Teitler, Zachariah
N1 - Funding information: MSC2000: primary 14J26; secondary 14C20, 14G05. Keywords: Del Pezzo surface, Manin’s conjecture, nef cone, root system. The first author was partially supported by a Feodor Lynen Research Fellowship of the Alexander von Humboldt Foundation.
PY - 2008/1/1
Y1 - 2008/1/1
N2 - We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.
AB - We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over an arbitrary perfect field, a similar result holds, except that the associated root system is no longer necessarily simply-laced.
KW - Del Pezzo surface
KW - Manin’s conjecture
KW - Nef cone
KW - Root system
UR - http://www.scopus.com/inward/record.url?scp=68149129809&partnerID=8YFLogxK
U2 - 10.2140/ant.2008.2.157
DO - 10.2140/ant.2008.2.157
M3 - Article
AN - SCOPUS:68149129809
VL - 2
SP - 157
EP - 182
JO - Algebra and Number Theory
JF - Algebra and Number Theory
SN - 1937-0652
IS - 2
ER -