Related papers: Explicit equations of a fake projective plane

We consider numerical semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links. We say that a semigroup is representable if it can be realized in this way. In this article, we study…

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of…

The quotient of a finite-dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety. Under appropriate assumptions, the McKay correspondence relates the geometry of nice resolutions of…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding.…

Let B_{2t} be a bipartite planar graph with an even number of regions. We are able to find bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph. We also will investigate the minimal…

The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new connections between projection bodies, centroid bodies, and…

In this paper, we classify the configurations of the singular points which appear on the quotients of the projective plane by the $1$-foliations of degree $-1$ in characteristic $2$.

A $K$-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple $K$-equivalent maps, which reduces…

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of flat projective structures having positive dimensional Lie algebra of projective…

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$…

The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of…

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem,…

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes…

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This…

Rosenfeld postulated ``generalized'' projective planes, which exploit a correspondence between rank-one idempotents of Jordan algebras $\mathfrak{J}_3(\mathbb{A})$ and points of projective planes $\mathbb{A}P^2$. The isometry groups of the…

We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in several cases. Moreover applications to the…

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the…

Let $\mathbb{k}$ be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of graded algebras, quadratic graded twisted tensor products of $\mathbb{k}[x,y]$ and $\mathbb{k}[z]$. When such an algebra is…

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20].…