Abstract
Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.
Funding source: Russian Science Foundation
Award Identifier / Grant number: RSF-19-11-00172
Funding statement: The research was supported by the grant RSF-19-11-00172.
Acknowledgements
The author is grateful to M. G. Zaidenberg for everlasting motivation, numerous discussions and remarks, to I. Cheltsov and J. Park for useful discussions on the subject, and to I. Arzhantsev for valuable remarks and suggestions. The author thanks the referee for useful comments.
References
[1] I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch and M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767–823. 10.1215/00127094-2080132Search in Google Scholar
[2] I. V. Arzhantsev, M. G. Zaidenberg and K. G. Kuyumzhiyan, Flag varieties, toric varieties, and suspensions: Three examples of infinite transitivity, Sb. Math. 203 (2012), no. 7, 923–949. 10.1070/SM2012v203n07ABEH004248Search in Google Scholar
[3] A. Buckley and T. Košir, Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115–140. 10.1007/s10711-007-9144-xSearch in Google Scholar
[4] P. J. Cameron, 6-transitive graphs, J. Combin. Theory Ser. B 28 (1980), no. 2, 168–179. 10.1016/0095-8956(80)90063-5Search in Google Scholar
[5] I. Cheltsov, J. Park and J. Won, Affine cones over smooth cubic surfaces, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 7, 1537–1564. 10.4171/JEMS/622Search in Google Scholar
[6] I. Cheltsov, J. Park and J. Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not. IMRN 2017 (2017), no. 4, 1179–1230. 10.1093/imrn/rnw063Search in Google Scholar
[7] I. V. Dolgachev, Luigi Cremona and cubic surfaces, Luigi Cremona (1830–1903), Incontr. Studio 36, Istituto Lombardo di Scienze e Lettere, Milan (2005), 55–70. Search in Google Scholar
[8] I. V. Dolgachev, Classical Algebraic Geometry. A Modern View, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139084437Search in Google Scholar
[9] T. Kishimoto, Y. Prokhorov and M. Zaidenberg, Group actions on affine cones, Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, American Mathematical Society, Providence (2011), 123–163. 10.1090/crmp/054/08Search in Google Scholar
[10] Y. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North–Holland, Amsterdam, 1974. Search in Google Scholar
[11] M. Michałek, A. Perepechko and H. Süß, Flexible affine cones and flexible coverings, Math. Z. 290 (2018), no. 3–4, 1457–1478. 10.1007/s00209-018-2069-2Search in Google Scholar
[12] J. Park and J. Won, Flexible affine cones over del Pezzo surfaces of degree 4, Eur. J. Math. 2 (2016), no. 1, 304–318. 10.1007/s40879-015-0054-4Search in Google Scholar
[13] A. Y. Perepechko, Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5 (in Russian), Funktsional. Anal. i Prilozhen. 47 (2013), no. 4, 45–52; English translation, Funct. Anal. Appl. 47 (2013), no. 4, 284–289. 10.1007/s10688-013-0035-7Search in Google Scholar
[14] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.1.rc1), 2020, https://www.sagemath.org. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
