The Manin–Peyre conjecture for smooth spherical Fano threefolds

• Valentin Blomer ^[2] ; Jörg Brüdern ^[3] ; Ulrich Derenthal ^[1] ; Giuliano Gagliardi ^[1]
  1. [1] University of Hannover

     University of Hannover

     Region Hannover, Alemania

     
  2. [2] Mathematisches Institut, Universität Bonn, Germany
  3. [3] Mathematisches Institut, Universität Göttingen, Germany

  Mostrar afiliaciones +
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-61
• Idioma: inglés
• DOI: 10.1007/s00029-024-00952-4
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings. Cambridge Studies in Advanced Mathematics 144. Cambridge University Press,...
  • Batyrev, V.V., Tschinkel, Y.: Tamagawa numbers of polarized algebraic varieties. Astérisque 251, 299–340 (1998).
  • Blomer, V., Brüdern, J., Derenthal, U., Gagliardi, G.: The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one....
  • Blomer, V., Brüdern, J., Salberger, P.: On a certain senary cubic form. Proc. Lond. Math. Soc. 108, 911–964 (2014).
  • Bonolis, D., Browning, T.D., Huang, Z.: Density of Rational Points on Some Quadric Bundle Threefolds. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02854-4.
  • Bravi, P., Pezzini, G.: Primitive wonderful varieties. Math. Z. 282, 1067–1096 (2016).
  • Browning, T.D., Heath-Brown, D.R.: Density of rational points on a quadric bundle in 𝑃 3 × 𝑃 3 P 3 ×P 3 . Duke Math....
  • Colliot-Thélène, J.-L., Sansuc, J.-J.: Torseurs sous des groupes de type multiplicatif; applications à l’étude des points rationnels de certaines...
  • Colliot-Thélène, J.-L., Sansuc, J.-J.: La descente sur les variétés rationnelles II. Duke Math. J. 54(2), 375–492 (1987).
  • Cox, D.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995).
  • Cox, D., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics 124. American Mathematical Society, Providence (2011).
  • Duke, W., Friedlander, J., Iwaniec, H.: Bounds for automorphic L-functions. Invent. Math. 112, 1–8 (1993).
  • Heath-Brown, D.R.: A new form of the circle method, and its application to quadratic forms. J. Reine Angew. Math. 481, 149–206 (1996).
  • Hofscheier, J.: Spherical Fano varieties. Ph.D. thesis, Universität Tübingen (2015).
  • Huang, Z., Montero, P.: Fano threefolds as equivariant compactifications of the vector group. Michigan Math. J. 69(2), 341–368 (2020).
  • Kollár, J., Kovács, S.: Singularities of the Minimal Model Program. Cambridge University Press, Cambridge (2013).
  • Lehmann, B., Sengupta, A., Tanimoto, S.: Geometric consistency of Manin’s conjecture. Compos. Math. 158(6), 1375–1427 (2022).
  • Luna, D.: Variétés sphériques de type A. Publ. Math. Inst. Hautes Études Sci. 161–226 (2001).
  • Luna, D., Vust, Th.: Plongements d’espaces homogènes. Comment. Math. Helv. 58, 186–245 (1983).
  • Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79, 101–218 (1995).
  • Peyre, E.: Terme principal de la fonction zêta des hauteurs et torseurs universels. Astérisque 251, 259–298 (1998).
  • Peyre, E.: Torseurs universels et méthode du cercle. Progr. Math. 199, Birkhäuser Verlag, Basel, pp. 221–274 (2001).
  • Peyre, E.: Counting points on varieties using universal torsors. Progr. Math. 226, Birkhäuser Boston Inc, Boston, MA, 61–81, p. 2002. Arithmetic...
  • Salberger, P.: Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251 (1998), 91–258. Nombre...