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Del Pezzo surfaces with infinite automorphism groups
Algebraic Geometry. 2021. Vol. 8. No. 3. P. 319-357.
Cheltsov I., Prokhorov Y.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
Publication based on the results of:
Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45-52
We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...
Added: September 26, 2019
Popov V. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.
We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...
Added: August 8, 2018
Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185-213
This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...
Added: April 28, 2014
Kishimoto T., Yuri Prokhorov, Zaidenberg M., Algebraic Geometry 2014 Vol. 1 No. 1 P. 46-56
In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface ...
Added: October 10, 2013
Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.
Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...
Added: September 12, 2022
Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2013.
We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability. ...
Added: October 10, 2013
Trepalin A., / Cornell University. Series arXiv "math". 2017.
Let X be a minimal del Pezzo surface of degree 2 over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q/𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup of the Weyl group W(E_7). There are 60 conjugacy classes of cyclic subgroups in W(E_7) and 18 of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo ...
Added: December 2, 2018
Cheltsov I., Kuznetsov A., Shramov K., Algebra & Number Theory 2020 Vol. 14 No. 1 P. 213-274
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all 𝔖6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, ...
Added: May 10, 2020
Prokhorov Y., Annales de l'Institut Fourier 2015 No. 65 P. 1-16
We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated. ...
Added: October 17, 2014
Avilov A., Sbornik Mathematics 2016 Vol. 307 No. 3 P. 315-330
We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid ...
Added: July 6, 2016
Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.
We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...
Added: June 8, 2019
Loginov K., / Cornell University. Series arXiv "math". 2018.
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations ...
Added: December 1, 2018
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.
This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...
Added: July 21, 2013
Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549-568
We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...
Added: March 17, 2014
Cheltsov Ivan, Wilson A., Journal of Geometric Analysis 2013 Vol. 23 No. 3 P. 1257-1289
We classify smooth del Pezzo surfaces whose α-invariant of Tian is bigger than 1. ...
Added: November 14, 2013
Cheltsov I., Известия РАН. Серия математическая 2014 Т. 78 № 2 С. 167-224
We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities. ...
Added: December 6, 2013
Prokhorov Y., Cheltsov I., / Cornell University. Series arXiv "math". 2020.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...
Added: August 19, 2020
Serge Lvovski, / Cornell University. Series arXiv "math". 2017.
We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of ...
Added: June 14, 2017
Loginov K., Moscow Mathematical Journal 2018 Vol. 18 No. 4 P. 721-737
We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations π: X→C of degree 1 and relative Picard number 1. We also embed the standard model into the relative weighted projective space ℙ_C(1,1,2,3). Our construction works in the G-equivariant category where G is a finite group. ...
Added: October 11, 2019
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.
We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...
Added: January 3, 2014
Shramov K., Przyjalkowski V., Proceedings of the Steklov Institute of Mathematics 2019 Vol. 307 P. 198-209
We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. ...
Added: August 12, 2020
Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723-3731
We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...
Added: August 18, 2020
Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088
The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...
Added: September 28, 2014
Avilov A., Математические заметки 2020 Т. 107 № 1 С. 3-10
The forms of the Segre cubic over non-algebraically closed fields, their automorphisms groups, and equivariant birational rigidity are studied. In particular, it is shown that all forms of the Segre cubic over any field have a point and are cubic hypersurfaces. ...
Added: May 11, 2020