Related papers: On derived equivalence classes of algebraic variet…

Curved A-infinity algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A-infinity algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras…

Let X be a smooth, projective variety defined over a local field K. Following Manin, two K-points of X are called R-equivalent if they can be joined by a rational curve defined over K. The main result of this note shows that if there are…

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible…

Consider a complete abelian category which has an injective cogenerator. If its derived category is left--complete we show that the dual of this derived category satisfies Brown representability. In particular this is true for the derived…

We prove that a compact complex analytic variety is algebraizable if and only if its bounded derived dg-category of coherent sheaves is saturated.

In this article we describe indecomposable objects of the derived categories of a branch class of associative algebras. To this class belong such known classes of algebras as gentle algebras, skew-gentle algebras and certain degenerations…

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick…

Let $X$ be a topologically stratified space, $p$ be any perversity on $X$, and $k$ be a field. We show that the category of $p$-perverse sheaves on $X$, constructible with respect to the stratification and with coefficients in $k$, is…

In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This…

Let X,Y be projective schemes over a discrete valuation ring R, where Y is generically smooth and g:X \to Y a surjective R-morphism such that g_*\mathcal{O}_X = \mathcal{O}_Y. We show that if the family X \to Spec(R) is isotrivial, then the…

We assume that the existence and termination conjecture for flips holds. A complex projective manifold is said to be {\it of almost general type} if the intersection number of the canonical divisor with every very general curve is strictly…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $\mathscr{G}$ of homological dimension $\le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its…

We study the group of autoequivalences of the derived categories of coherent sheaves on smooth projective elliptic surfaces with non-zero Kodaira dimensions. We find a description of it when each reducible fiber is a cycle of $(-2)$-curves.

Following the work of Totaro and Pereira, we study sufficient conditions under which collections of pairwise-disjoint divisors on a variety over an algebraically closed field are contained in the fibers of a morphism to a curve. We prove…

We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…

We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud. The…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and…