Acyclic fake surfaces

On Acyclic Colorings of Graphs on Surfaces

A proper k-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic χ = −γ is at most O(γ). This is nearly tight; for every γ > 0 there are graphs ...

Real Fake Real Fake

Arithmetic of a fake projective plane and related elliptic surfaces

The purpose of the present paper is to explain the fake projective plane constructed by J. H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J. H. Keum as well as the ...

Fake!

Received: 3 October 2005 Accepted: 3 October 2005 One of my most memorable experiences with Claudio occured in Cleveland about 10 years ago. I was serving as department chair, and had organized a semester long research seminar, bringing one great scholar in for a full week, every week of the term. During their week in residence, the visitors would give two seminars and meet informally with facu...

Acyclic curves and group actions on affine toric surfaces

We show that every irreducible, simply connected curve on a toric affine surface X over C is an orbit closure of a Gm-action on X . It follows that up to the action of the automorphism group Aut(X) there are only finitely many non-equivalent embeddings of the affine line A in X . A similar description is given for simply connected curves in the quotients of the affine plane by small finite line...