Derived categories of Keum's fake projective planes

We conjecture that derived categories of coherent sheaves on fake projective n  -spaces have a semi-orthogonal decomposition into a collection of n+1$n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum's surface). Then by passing to equivariant categories we construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.     

Exotic structures arising from fake projective planes

We present examples of exotic structures on pPC2#qPC2 for some small p and q obtained naturally from quotients of fake projective planes in complex geometry,which are classified by the work of Prasad-Yeung and Cartwright-Steger.     

Pathological projective planes: Associate affine planes

In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distinguishing a line as the line at infinity) as expressable in the following theorem.     

Constructively presented projective planes

An attempt to define a finite presentation for projective planes by analogy to finitely presented groups has been made in [9]. In this article we suggest another definition for a presentation of projective planes: constructively presented planes. A few simple results about such planes and an example of constructively presented planes with nonsolvable word problem are given.This paper was written while the author was visiting the University of Bergen, Norway.     

Stable periodic projective planes

We find all stable projective planes with finite topology which are properly embedded in , where is a discrete subgroup of translations in . Here stable means second order minimum of the area. The surfaces we obtain are a quotient of the helicoid and quotients of the doubly periodic Scherk surfaces.

     

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.Dedicated to Professor Helmut Karzel on his 60^th birthday     

Smooth projective translation planes

A projective plane is called smooth if both the point space and the line space are smooth manifolds such that the geometric operations are smooth. We prove that every smooth projective translation plane is isomorphic to one of the classical planes over , , or .Dedicated to Professor Dr. H. Salzmann on the occasion of his 65th birthday     

Gorenstein Q-homology projective planes

We present the complete list of all singularity types on Gorenstein 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 possible singularity types,each except two types supported by an example.     

Enumeration of maps on the projective plane

1. IntroductionA lnap is rooted if an edge is distinguished togetl1er with an end and a side of the edge.An edge belo11ging to only one face is called double (or 8ingular by some author), al1 othersbelonging to exactly two faces are called s1ngle. The enumeration of rooted p1anar maps wasfirst introduced by Tutte['], Techniques originated by Tutte [2,3l for enumerating variousclasses of rooted Inaps on tIle sphere are here applied to the c1asses of alI rooted maps onthe projective plane. Th…     

Subplanes of finite projective planes

We study questions of the hereditariness of the property of (, )-transitivity (the property of being a plane of translations) of a finite protective plane for its subplanes. In particular it is shown that hereditariness holds for subplanes containing the point .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1101–1102, July–August, 1991.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation