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

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical…

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived…

We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.

We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…

We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental…

Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \le d \le 9$ over $K$ with good reduction away from $S$. Let…

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical…

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that…

We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic…

For each integer d=2,3,4, there exists a field F with cohomological dimension 1 and a del Pezzo surface of degree d over F having no rational point. Proofs use the theorem of Merkur'ev and Suslin, the Riemann-Roch theorem on a surface and…

Let $G$ be a finite group and $H\subseteq G$ be its subgroup. We prove that if a smooth del Pezzo surface over an algebraically closed field is $H$-birationally rigid then it is also $G$-birationally rigid, answering a geometric version of…

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…

We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger than one.

In this article we study Ceva's theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining…

For an irreducible variety $X$ over a field $k$, the degree of irrationality $\operatorname{irr}_k X$ is the minimal degree of a dominant rational map $X \dashrightarrow \mathbb{P}_k^{\operatorname{\dim} X}$. When $X$ is a curve, this is…

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.

We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most $A_n$-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on…

We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type $2\Aone$ and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics,…

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…