Related papers: A categorical Torelli theorem for quartic del Pezz…

We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived…

We show that the general Enriques surface can be recovered from the Kuznetsov component of its bounded derived category of coherent sheaves.

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…

We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof…

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…

Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$),, and asked whether such surfaces always have a closed point of degree…

We classify del Pezzo non-commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the…

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric…

We study the categorical Torelli theorem for smooth (weighted) hypersurfaces in (weighted) projective spaces via the Hochschild--Serre algebra of its Kuznetsov component. In the first part of the paper, we show that a natural graded…

We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a…

Let $Y$ admit a rectangular Lefschetz decomposition of its derived category, and consider a cyclic cover $X\to Y$ ramified over a divisor $Z$. In a setting not considered by Kuznetsov and Perry, we define a subcategory $\mathcal{A}_Z$ of…

Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a…

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…

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism…

We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the…

We study the algebraic Brauer classes on open del Pezzo surfaces of degree $4$. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is generated by…

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible…

We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…

We prove a relative version of the fact that semiorthogonal decompositions of the bounded derived category of coherent sheaves are strongly constrained by the base locus of the canonical linear system. As an application we prove that the…

We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and…