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

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

Let X be a del Pezzo surface of degree one over an algebraically closed field (of any characteristic), and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is…

Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also…

We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^3. This is the maximal possible length of such a sequence on this surface which has…

Classification of curves in a projective space occupies minds of many mathematicians. First step in doing so is classification of curves on a given surface. This brings us to consideration of the nonsingular Del Pezzo Surface in $P^4_k.$ We…

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general…

We characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type $\mathbf{A}_3$ over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these…

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$.…

Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal…

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of…

In this paper, we study the Gieseker moduli space $\mathcal{M}_{1,1}^{4,3}$ of minimal surfaces with $p_g=q=1, K^2=4$ and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the Albanese map is…

Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain…

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a…

We develop the method of inducing semiorthogonal decompositions of projective varieties with isolated rational singularities from those of small resolutions of singularities, which generalizes semiorthogonal decompositions for singular…

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.

We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify…

The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r…

We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c_2) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the…

This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general…

We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane f: X ---> P^1, i.e., that every Brauer class is obtained by pullback from an element of Br k(P^1). As a…