Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if there are no smooth $\Bbbk$-points on $X / G$ then it is not $\Bbbk$-rational. Therefore under assumption that the set of smooth $\Bbbk$-points on $X / G$ is not empty we show that there are few possibilities for non-$\Bbbk$-rational quotients.
The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author's previous papers. In this paper we study the quotients of del Pezzo surfaces of degree $1$. We show that they can be non-$\Bbbk$-rational only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the trivial group and the group of order $2$ we show that both $X$ and $X / G$ are not $\Bbbk$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$ we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non-$\Bbbk$-rational del Pezzo surfaces of degree $1$ such that the $G$-invariant Picard number of $X$ is $1$.
As a result of complete classification of non-$\Bbbk$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $\Bbbk$-rational surfaces, and obtain some corollaries concerning fields of invariants of $\Bbbk(x , y)$.
In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field $\Bbbk$ 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 quotient is always $\Bbbk$-rational. If the degree of $X$ is equal to four then the quotient can be non-$\Bbbk$-rational only if the order of the group is $1$, $2$ or $4$. For these groups we construct examples of non-$\Bbbk$-rational quotients.
Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.
Let $\Bbbk$ 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 $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously non-$\Bbbk$-rational. Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We find a list of groups, such that the quotient surface can be non-$\Bbbk$-rational. For these groups we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non…
We consider geometrically cellular varieties $X$ over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group $H^3_{nr}(X,\mathbb{Q}/\mathbb{Z}(2))$ by its constant part. For $X$ a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient if finite. For $X$ a del Pezzo surface of degree $\geq 5$, we show that this quotient is zero, unless $X$ is a del Pezzo surface of degree 8 of a special type.
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 Kähler--Einstein metric.
We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.
We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.
We consider the problem of interpolating projective varieties through points and linear spaces. We show that del Pezzo surfaces satisfy weak interpolation.
We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \neq 2$. In characteristic $0$, every RDP del Pezzo surface $X$ is equivariant, that is, ${\rm Aut}_X = {\rm Aut}_{\widetilde{X}}$, where $\widetilde{X}$ is the minimal resolution of $X$, hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if $p \neq 2,3,5,7$, then it is still true that every RDP del Pezzo surface is equivariant…
In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.
In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.
In this paper we prove that del Pezzo fibrations admit $\mathbb{G}_{a}^{3}$-structures if and only if they are $\mathbb{P}^2$-bundles over $\mathbb{P}^{1}$.
In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.
We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.
In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of Kähler-Einstein Fano varieties constructed in [OSS16].
Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general de…
We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling, while for some weak del Pezzo surfaces of degree $2$ the above is not true. We classify smooth projective surfaces possessing a cyclic strong exceptional collection of line bundles of maximal length: we prove that they are weak del Pezzo surfaces and find all types of weak del Pezzo surfaces admitting such a collection. We find simple criteria of exceptional…
