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## On full exceptional collections of line bundles on del Pezzo surfaces

arxiv.org.
math.
Cornell University
,
2015.
No. 1505.06477.

Inverse Galois problem for del Pezzo surfaces over finite fields / 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

Del Pezzo surfaces over finite fields / 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

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

On stable conjugacy of finite subgroups of the plane Cremona group, II / 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

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

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

On monodromy groups of del Pezzo surfaces / 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

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

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

On non-rational fibers of del Pezzo fibrations over curves / 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

Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families / 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

A braid group action on parking functions / Cornell University. Series "Working papers by Cornell University". 2011.

We construct an action of the braid group on n strands on the set of parking functions of n cars such that elementary braids have orbits of length 2 or 3. The construction is motivated by a theorem of Lyashko and Looijenga stating that the number of the distinguished bases for An singularity equals (n ...

Added: February 13, 2015

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

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

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

Minimal del Pezzo surfaces of degree 2 over finite fields / 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

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

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

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

Proceedings of the American Mathematical Society 2014

Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we show that the category C has finite dimensional spaces of morphisms. ...

Added: November 5, 2014

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

Cylinders in del Pezzo surfaces / 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

Michigan Mathematical Journal 2015 Vol. 64 No. 2 P. 293-318

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability. © 2015, University of Michigan. All rights reserved. ...

Added: September 8, 2015

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