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Minimal del Pezzo surfaces of degree 2 over finite fields
Cornell University
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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 surfaces of degree 2 can be achieved for given q.
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
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
Trepalin A., Moscow Mathematical Journal 2018 Vol. 18 No. 3 P. 557-597
Let 𝕜 be any field of characteristic zero, X be a del Pezzo surface of degree 2 and G be a group acting on X. In this paper we study 𝕜-rationality questions for the quotient surface X/G. If there are no smooth 𝕜-points on X/G then X/G is obviously non-𝕜-rational. Assume that the set of smooth 𝕜-points on the quotient is not empty. We find ...
Added: October 4, 2018
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., Shramov K., Park J., / Cornell University. Series math "arxiv.org". 2018.
We estimate δ-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold K\"ahler--Einstein metric. ...
Added: October 21, 2018
Galkin S., Rybakov S., Mathematical notes 2019 Vol. 106 No. 6 P. 1014-1018
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=F_{p^2}, that is optimal if p=3. ...
Added: January 29, 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
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., / 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
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
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
Zykin A. I., Ballet S., Designs, Codes and Cryptography 2019 Vol. 87 P. 517-525
We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field F_p or F_{p^2} where p denotes a prime number ≥5. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over F_{p_2} attaining the Drinfeld–Vladuts bound and on the descent of these families to ...
Added: May 12, 2020
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
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
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
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
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". 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
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
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
Galkin S., Rybakov S., / Cornell University. Series math "arxiv.org". 2019. No. 1910.14379.
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...
Added: November 6, 2019
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020