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## A note on degenerations of del Pezzo surfaces

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.

Publication based on the results of:

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

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

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093–1113

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...

Added: October 10, 2013

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

Glutsyuk A., / Cornell University. Series math "arxiv.org". 2014. No. 1309.1843.

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete ...

Added: September 29, 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

Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215–229

We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...

Added: January 24, 2014

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, 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

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, Documenta Mathematica 2010 Vol. 15 P. 843–872

We study Q-Fano threefolds of large Fano index. In
particular, we prove that the maximum possible Fano index is attained
only by the weighted projective space P(3,4,5,7). ...

Added: December 6, 2013

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

Galkin S., Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.07001.

Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3). ...

Added: October 19, 2018

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

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

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Transformation Groups 2013 Vol. 18 No. 4 P. 1137–1153

We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety. ...

Added: October 10, 2013

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

Prokhorov Y., Shramov K., Mathematical Research Letters 2018 Vol. 25 No. 3 P. 957–972

We classify threefolds with non-Jordan birational automorphism groups. ...

Added: October 4, 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

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

Prokhorov Y., / Cornell University. Series arXiv "math". 2017.

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings. ...

Added: August 28, 2017

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

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

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