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## Del Pezzo zoo

Experimental Mathematics. 2013. Vol. 22. No. 3. P. 313-326.

Cheltsov Ivan, Shramov Constantin

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.

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

Ю. Г. Прохоров, Известия РАН. Серия математическая 2013 Т. 77 № 3 С. 199-222

We study elements $\tau$ of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements. ...

Added: July 1, 2013

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

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

Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 389-418

We classify Fano threefolds with only terminal singularities whose canonical class is
Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor
class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of
examples of Fano 3-folds with “many” symmetries. ...

Added: October 7, 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

Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 419-434

We classify Fano threefolds with only Gorenstein terminal singularities and Picard
number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil
divisor class group is of rank 1 with respect to an action of some group G. ...

Added: October 7, 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

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

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

Przyjalkowski V., Shramov K., Communications in Number Theory and Physics 2020 Vol. 14 No. 3 P. 511-553

We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound ...

Added: October 13, 2020

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., 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

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

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

Cheltsov Ivan, Park J., Won J., Mathematische Zeitschrift 2014 No. 276 P. 51-79

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general. ...

Added: November 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

Cheltsov I., Zhang K., European Journal of Mathematics 2019 Vol. 5 P. 729-762

We prove that 𝛿δ-invariants of smooth cubic surfaces are at least 6/5. ...

Added: May 10, 2020

Prokhorov Y., Sbornik Mathematics 2013 Vol. 204 No. 3 P. 347-382

We classify $\mathbb Q$-Fano threefolds of Fano index > 2 and sufficiently big degree. ...

Added: October 7, 2013

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

Coates T., Galkin S., Kasprzyk A. et al., / Cornell University. Series math "arxiv.org". 2014. No. 1406.4891.

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: June 20, 2014

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014