Searcharxiv⌕ Search

arXiv subjects

Alexey Elagin

Publications and source records attributed to Alexey Elagin.

17 recordsLinked to original sources
arXiv2026

A categorical Torelli theorem for quartic del Pezzo surfaces

We solve categorical Torelli problem for quartic del Pezzo surfaces. That is, we prove that a del Pezzo surface of degree $4$ can be canonically reconstructed from its Kuznetsov component, which is the orthogonal subcategory to the structure sheaf in the derived category of the surface. Our methods work in equivariant setting and over arbitrary perfect fields. Using recent theory of atomic semi-orthogonal decompositions arXiv:2512.05064, we conclude that two minimal quartic del Pezzo surfaces are birational if and only if they are isomorphic. We also verify that the Kuznetsov component of a minimal quartic del Pezzo surface is semi-orthogonally indecomposable, confirming a conjecture by Auel and Bernardara.

math.AG↗
arXiv2025

Atomic decompositions for derived categories of G-surfaces

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

math.AG↗
arXiv2025

Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops

We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $\Sigma$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min \Omega)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max \Omega)t$, for $t\leq 0$, where $\Omega = \{\frac{\omega_1}{m_1} \ldots, \frac{\omega_b}{m_b},0\}$, and $\omega_i$ is the winding number of the $i$th boundary component $\partial_i\Sigma$ of the surface with $b$ boundary components and $m_i$ stops on $\partial_i \Sigma$. It then follows that the upper and lower Serre dimensions are given by $1-\min \Omega$ and $1-\max \Omega$, respectively. Furthermore, in the case of a finite dimensional gentle algebra $A$, we show that a Gromov-Yomdin-like equality holds by relating the categorical entropy of the Serre functor of the perfect derived category of $A$ to the logarithm of the spectral radius of the Coxeter transformation.

math.RT↗
arXiv2024

Thick subcategories on weighted projective curves and nilpotent representations of quivers

We continue the study of thick triangulated subcategories, started by Valery Lunts and the author in arXiv:2007.02134, and consider thick subcategories in the derived category of coherent sheaves on a weighted projective curve and the corresponding abelian thick subcategories. Our main result is that any thick subcategory on a weighted projective curve either is equivalent to the derived category of nilpotent representations of some quiver (we call such categories quiver-like) or is the orthogonal to an exceptional collection of torsion sheaves (we call such subcategories big). We examine the structure of thick subcategories: in particular, for weighted projective lines we prove that any admissible subcategory is generated by an exceptional collection and any exceptional collection is a part of a full one. We show that the derived categories of weighted projective curves satisfy Jordan-Holder property and do not contain phantoms. Finally, we extend and simplify results from arXiv:2007.02134, providing sufficient criteria for a triangulated or abelian category to be quiver-like.

math.RT↗
arXiv2020

Thick subcategories on curves

We classify thick subcategories $\mathcal T \subset D^b(\mathrm{coh}\,C)$ for smooth projective curves $C$ over an algebraically closed field.

math.AG↗
arXiv2020

On cyclic strong exceptional collections of line bundles on surfaces

We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of L.Hille and M.Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling.

math.AG↗
arXiv2020

Three notions of dimension for triangulated categories

In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.

math.AG↗
arXiv2020

Calculating dimension of triangulated categories: path algebras, their tensor powers and orbifold projective lines

This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.

math.AG↗
arXiv2020

Derived categories of coherent sheaves on some zero-dimensional schemes

Let $X_N$ be the second infinitesimal neighborhood of a closed point in $N$-dimensional affine space. In this note we study $D^b(coh\, X_N)$, the bounded derived category of coherent sheaves on $X_N$. We show that for $N\geq 2$ the lattice of triangulated subcategories in $D^b(coh\, X_N)$ has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. We also establish a relation between triangulated subcategories in $D^b(coh\, X_N)$ and universal localizations of a free graded associative algebra in $N$ variables. Our homological methods produce some applications to the structure of such universal localizations.

math.AG↗
arXiv2020

Regular subcategories in bounded derived categories of affine schemes

Let $R$ be a commutative Noetherian ring such that $X=Spec R$ is connected. We prove that the category $D^b(coh X)$ contains no proper full triangulated subcategories which are regular. We also bound from below the dimension of a regular category $T$, if there exists a triangulated functor $T \to D^b(coh X)$ with certain properties. Applications are given to cohomological annihilator of $R$ and to point-like objects in $T$.

math.AG↗
arXiv2019

Smoothness of Derived Categories of Algebras

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.

math.AG↗
arXiv2017

On exceptional collections of line bundles on weak del Pezzo surfaces

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 exceptionality/strong exceptionality for collections of line bundles on weak del Pezzo surfaces.

math.AG↗
arXiv2015

On equivariant triangulated categories

Consider a finite group $G$ acting on a triangulated category $\mathcal T$. In this paper we investigate triangulated structure on the category $\mathcal T^G$ of $G$-equivariant objects in $\mathcal T$. We prove (under some technical conditions) that such structure exists. Supposed that an action on $\mathcal T$ is induced by a DG-action on some DG-enhancement of $\mathcal T$, we construct a DG-enhancement of $\mathcal T^G$. Also, we show that the relation "to be an equivariant category with respect to a finite abelian group action" is symmetric on idempotent complete additive categories.

math.AG↗
arXiv2012

Descent theory for semiorthogonal decompositions

In this paper a method of constructing a semiorthogonal decomposition of the derived category of $G$-equivariant sheaves on a variety $X$ is described, provided that the derived category of sheaves on $X$ admits a semiorthogonal decomposition, whose components are preserved by the action of the group $G$ on $X$. Using this method, semiorthogonal decompositions of equivariant derived categories were obtained for projective bundles and for blow-ups with a smooth center, and also for varieties with a full exceptional collection, preserved by the action of the group. As a main technical instrument, descent theory for derived categories is used.

math.AG↗
arXiv2011

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.

math.AG↗