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Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families
Cornell University
,
2017.
We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.
Publication based on the results of:
Cheltsov I., Park J., Won J., / Cornell University. Series math "arxiv.org". 2015.
On del Pezzo surfaces, we study effective ample R-divisors such that the complements of their supports are isomorphic to A1-bundles over smooth affine curves. ...
Added: November 18, 2015
Cheltsov I., Park J., Won J., International Mathematics Research Notices 2017 No. 4 P. 1179-1230
On del Pezzo surfaces, we study effective ample ℝ -divisors such that the complements of their supports are isomorphic to 𝔸1 -bundles over smooth affine curves. All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic 0 throughout this article. ...
Added: July 31, 2017
Cheltsov I., Park J., Won J., Compositio Mathematica 2016 Vol. 152 P. 1198-1224
For each del Pezzo surface S with du Val singularities, we determine whether it admits a (−K S )-polar cylinder or not. If it allows one, then we present an effective Q-divisor D that is Q-linearly equivalent to −K S and such that the open set S\Supp(D) is a cylinder. As a corollary, we classify ...
Added: August 31, 2016
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., Transactions of the American Mathematical Society 2018 Vol. 370 No. 9 P. 6097-6124
In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field 𝕜 of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface X contains a point defined over the ground field and the degree of X is at least five then the ...
Added: June 14, 2017
Trepalin A., Central European Journal of Mathematics 2014
Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...
Added: October 14, 2013
Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2011.
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 11, 2013
Prokhorov Y., Kyoto Journal of Mathematics 2019 Vol. 59 No. 4 P. 1041-1073
We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings. ...
Added: August 1, 2017
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
Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313-326
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...
Added: January 27, 2014
Andrey S. Trepalin, Central European Journal of Mathematics 2014 Vol. 12 No. 2 P. 229-239
Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...
Added: December 3, 2013
Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597-613
We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...
Added: August 30, 2019
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
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
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
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
Rybakov S., Trepalin A., Математический сборник 2017 Т. 208 № 9 С. 148-170
Пусть X -- минимальная поверхность над полем F_q. Образ Г группы Галуа Gal ( \bar{F_q}, F_q ) в группе автоморфизмов Aut ( Pic X ) является циклической подгруппой группы Вейля W ( E_6 ). В этой подгруппе 25 классов сопряженности циклических подгрупп, и пять из них соответствуют минимальным кубическим поверхностям. Возникает естественный вопрос: какие классы сопряженности ...
Added: October 23, 2017
Trepalin A., / Cornell University. Series arXiv "math". 2018.
Let X be a del Pezzo surface of degree 2 or greater 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 preserving the anticanonical class and the intersection form. The conjugacy class of Γ in the subgroup of Aut(Pic(\bar{X})) preserving the anticanonical class and the intersection form is a natural invariant of X. We say that the ...
Added: December 2, 2018
Alexey Elagin, Lunts V., / Cornell University. Series math "arxiv.org". 2015. No. 1505.06477.
Added: October 15, 2015
Cheltsov I., Prokhorov Y., Algebraic Geometry 2021 Vol. 8 No. 3 P. 319-357
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...
Added: September 7, 2021
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
Trepalin A., Loughran D., / Cornell University. Series arXiv "math". 2019.
We completely solve the inverse Galois problem for del Pezzo surfaces of degree 2 and 3 over all finite fields. ...
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
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