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Andrey Trepalin

Publications and source records attributed to Andrey Trepalin.

14 recordsLinked to original sources
arXiv2026

On the coregularity of del Pezzo surfaces with du Val singularities

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with some special properties. We also prove results about coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic $0$. Our results confirm the expectation that "most" del Pezzo surfaces have coregularity $0$, while del Pezzo surfaces with positive coregularity enjoy some special properties.

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arXiv2025

Birational geometry of del Pezzo surfaces of degree 4

It is known that any Mori fiber space birational to a minimal smooth del Pezzo surface $S$ of degree $4$ is either a del Pezzo surface of degree $4$ itself, or a smooth cubic surface with a structure of a relatively minimal conic bundle. We show that any del Pezzo surface of degree $4$ birational to $S$ is actually isomorphic to $S$. Also, we sketch an equivariant version of this fact. On the way, we review the biregular classification of del Pezzo surfaces of degree $4$ obtained by A. N. Skorobogatov.

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arXiv2025

G-coregularity of del Pezzo surfaces

We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute $G$-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with $G$-coregularity 0. We describe the relations between the notions of $G$-coregularity, $G$-log-canonical thresholds, $G$-rigidity, and exceptional quotient singularities.

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arXiv2022

Birational classification of pointless del Pezzo surfaces of degree 8

Let k be a perfect field. Recently J.-L. Colliot-Th\'el\`ene showed that two pointless quadric surfaces over k are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces of degree $8$ with the Picard number $1$, and describe minimal surfaces birationally equivalent to a given pointless del Pezzo surface of degree $8$.

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arXiv2021

Quotients of Severi-Brauer surfaces

We show that a quotient of a non-trivial Severi-Brauer surface $S$ over arbitrary field $\Bbbk$ of characteristic $0$ by a finite group $G \subset \operatorname{Aut}(S)$ is $\Bbbk$-rational, if and only if $|G|$ is divisible by $3$. Otherwise, the quotient is birationally equivalent to $S$.

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arXiv2019

Quotients of del Pezzo surfaces

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 $\Bbbk(x , y)$.

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arXiv2018

Del Pezzo surfaces over finite fields

Let $X$ be a del Pezzo surface of degree $2$ or greater over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of $\Gamma$ in the subgroup of $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ preserving the anticanonical class and the intersection form is a natural invariant of $X$. We say that the conjugacy class of $\Gamma$ in $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ is the \textit{type} of a del Pezzo surface. In this paper we study which types of del Pezzo surfaces of degree $2$ or greater can be realized for given $q$. We collect known results about this problem and fill the gaps.

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arXiv2017

Quotients of del Pezzo surfaces of degree 2

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-$\Bbbk$-rational del Pezzo surfaces of degree $2$ such that the $G$-invariant Picard number of $X$ is $1$. For all other groups we show that the quotient $X / G$ is always $\Bbbk$-rational.

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arXiv2016

Minimal del Pezzo surfaces of degree $2$ over finite fields

Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ 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 $q$.

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arXiv2016

Minimal cubic surfaces over finite fields

Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.

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arXiv2015

Quotients of cubic surfaces

Let $\Bbbk$ be any field of characteristic zero, $X$ be a cubic surface in $\mathbb{P}^3_{\Bbbk}$ and $G$ be a group acting on $X$. We show that if $X(\Bbbk) \ne \varnothing$ and $G$ is not trivial and not a group of order $3$ acting in a special way then the quotient surface $X / G$ is rational over $\Bbbk$. For the group $G$ of order $3$ we construct examples of both rational and nonrational quotients of both rational and nonrational $G$-minimal cubic surfaces over $\Bbbk$.

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arXiv2015

Quotients of conic bundles

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over projective line by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group of order $2^k$, dihedral group of order $2^k$, alternating group of degree $4$, symmetric group of degree $4$ or alternating group of degree $5$ effectively acting on the base of conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.

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arXiv2013

Quotients of del Pezzo surfaces of high degree

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.

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