Related papers: Toric log del Pezzo surfaces with one singularity

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant…

We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting minimally on them. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.

We determine the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. We prove that many of these admit a K\"ahler-Einstein metric and most of them do not have tigers.

The intersection matrix of a finite simplicial complex has as each of its entries the rank of the intersection of its respective simplices. We prove that such matrix defines the triangulation of a closed connected surface up to isomorphism.

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…

We show that the variable cohomology of a general complete intersection of quadrics can be identified with the intersection cohomology of a double covering. As a consequence, we show that the middle cohomology of a general complete…

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we…

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…

There are several variations of the definition of log del Pezzo pairs in the literature. We define their suitable smooth models, and we show that they are the same. In particular, we obtain a characterization of smooth log del Pezzo pairs…

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable…

We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant…

We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove…

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

We study the biregular and birational geometry of degree 6 del Pezzo surfaces with Picard number 1, defined over an arbitrary perfect field. Using Galois cohomology techniques, we obtain an explicit description of cocycles for such surfaces…

Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let \Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that…

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's…

We give an optimal upper bound for the anti-canonical volume of an $\epsilon$-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface.…

We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric…

Affine cylinders (genus zero surfaces with two singularities) and affine tori (genus one surfaces without singularities) are among the simplest examples of surfaces endowed with a complex affine structure. Their geodesic flows are…

We consider a real del Pezzo surface without points. We prove that the same surface over complex numbers field $\mathbb{C}$ has Picard number is at least two.