相关论文: Fake projective planes

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually…

This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…

This small note, without claim of originality, constructs the projective plane over the octonionic numbers and recalls how this can be used to rule out the existence of higher-dimensional real division algebras, using Adams' solution of the…

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 $\pi$…

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

Let $p$ and $q$ be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on…

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the…

The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.

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.…

Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology,…

A fake wedge is a diagram of spaces K <- A -> C whose double mapping cylinder is contractible. The terminology stems from the special case A = K v C with maps given by the projections. In this paper, we study the homotopy type of the moduli…

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the…

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with…

Finite projective planes are constructed using groups that satisfy simple-looking conditions. The resulting projective planes include many known planes and possibly new ones, and are precisely those having a collineation group fixing a flag…

A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are…

This report gives an overview of the history of finite projective planes and their properties before going on to outline the proof that no projective plane of order 10 exists. The report also investigates the search carried out by…

The number of apparent double points of an irreducible projective variety $X$ of dimension $n$ in $\mathbb{P}^{2n+1}$ is the number of secant lines to $X$ passing through a general point of $\mathbb{P}^{2n+1}$. This classical notion dates…

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with…

We present the complete list of all singularity types on Gorenstein $\mathbb{Q}$-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of $58$…