Related papers: Addendum to `Fake Projective Planes'

This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of…

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and…

This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which…

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.…

For every fake projective plane $X$ with automorphism group of order 21, we prove that $H^i(X, 2L)=0$ for all $i$ and for every ample line bundle $L$ with $L^2=1$. For every fake projective plane with automorphism group of order 9, we prove…

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…

We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is…

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice…

We define fake weighted projective spaces as a generalisation of weighted projective spaces. We introduce the notions of fundamental group in codimension 1 and of universal covering in codimension 1. We prove that for every fake weighted…

let f be an endomorphism of a complex projective space, of degree bigger than one. Let us call an algebraic subset exceptional for f, if its inverse image is set-theoretically equal to itself. J.-Y. Briend, S. Cantat and M. Shishikura…

We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$,…

We classify all closed 1-connected manifolds $M$ which look like projective planes, i.e. with integral homology $H_*(M)=Z^3$. Furthermore, we give an explicit construction of these manifolds as Thom spaces of open disk bundles.

We construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order 3 or less; Theorems 2.1,…

We construct a new class of affine complements ${\mathbb P}^M\setminus S$ with the trivial group of automorphisms, where $S\subset {\mathbb P}^M$ is a rational hypersurface, $M$ is odd and $M\geqslant 5$.

Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes,…

Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.

In this study we performed a computer search for unitals in planes of order 16. Some new unitals were found and we show that some unitals can be embedded in two or more different planes.

This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the…

We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs…