Details
| Original language | English |
|---|---|
| Article number | 69 |
| Journal | Selecta Mathematica, New Series |
| Volume | 30 |
| Issue number | 4 |
| Early online date | 12 Jul 2024 |
| Publication status | Published - Sept 2024 |
Abstract
The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.
Keywords
- 11G35, 14G05, Cox rings, Fano threefolds, harmonic analysis, Manin–Peyre conjecture, Primary 11D45, Rational points, Secondary 14M27, Spherical varieties
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Physics and Astronomy(all)
- General Physics and Astronomy
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In: Selecta Mathematica, New Series, Vol. 30, No. 4, 69, 09.2024.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Manin–Peyre conjecture for smooth spherical Fano threefolds
AU - Blomer, Valentin
AU - Brüdern, Jörg
AU - Derenthal, Ulrich
AU - Gagliardi, Giuliano
N1 - Publisher Copyright: © The Author(s) 2024.
PY - 2024/9
Y1 - 2024/9
N2 - The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.
AB - The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.
KW - 11G35
KW - 14G05
KW - Cox rings
KW - Fano threefolds
KW - harmonic analysis
KW - Manin–Peyre conjecture
KW - Primary 11D45
KW - Rational points
KW - Secondary 14M27
KW - Spherical varieties
UR - http://www.scopus.com/inward/record.url?scp=85198443167&partnerID=8YFLogxK
U2 - 10.1007/s00029-024-00952-4
DO - 10.1007/s00029-024-00952-4
M3 - Article
AN - SCOPUS:85198443167
VL - 30
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
SN - 1022-1824
IS - 4
M1 - 69
ER -