Affine cones over cubic surfaces are flexible in codimension one
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Alexander Perepechko
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. 10.1215/00127094-2080132Suche 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/SM2012v203n07ABEH004248Suche 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-xSuche 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)90063-5Suche 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/622Suche 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/rnw063Suche 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. Suche in Google Scholar
[8] I. V. Dolgachev, Classical Algebraic Geometry. A Modern View, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139084437Suche 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/054/08Suche in Google Scholar
[10] Y. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North–Holland, Amsterdam, 1974. Suche 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/s00209-018-2069-2Suche 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/s40879-015-0054-4Suche 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/s10688-013-0035-7Suche in Google Scholar
[14] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.1.rc1), 2020, https://www.sagemath.org. Suche in Google Scholar
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Artikel in diesem Heft
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case
Artikel in diesem Heft
- Frontmatter
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
- Free cyclic group actions on highly-connected 2n-manifolds
- The distinction problems for Sp4 and SO3,3
- Affine cones over cubic surfaces are flexible in codimension one
- Permutations of zero-sumsets in a finite vector space
- On the finiteness of solutions for polynomial-factorial Diophantine equations
- Galois action on Fuchsian surface groups and their solenoids
- On a Lévy process pinned at random time
- Borsuk–Ulam theorem for filtered spaces
- A non-commutative differential module approach to Alexander modules
- Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
- Quantum modularity of partial theta series with periodic coefficients
- Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
- Zeros of GL2 𝐿-functions on the critical line
- Weighted boundedness of multilinear Calderón commutators
- Two characterizations of central BMO space via the commutators of Hardy operators
- Weyl 𝑛-algebras and the Swiss cheese operad
- Syzygies in equivariant cohomology in positive characteristic
- Epsilon factors of symplectic type characters in the wild case
