The 2 and 3 representative projective planar embeddings

AbstractA graph embedded on a surface is n-representative if every nontrivial closed curve in the surface which does not intersect edges of the embedding must contain at least n vertices of the graph. The property of being n-representative on a surface is closed upward under minor inclusion; hence, by the results of N. Robertson and P. D. Seymour (Graph minors. VIII. A Kuratowski theorem for general surfaces, submitted for publication), the set of minor minimal n-representative embeddings on a surface is finite up to isomorphism. The property of being minor minimal n-representative is invariant under Y-Δ operations. The set of minor minimal 2 and 3 representative embeddings on the projective plane are found. These embeddings are used to produce the topologically minimal 2 and 3 representative projective embeddings

Cubic Partial Cubes from Simplicial Arrangements

We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure

Equivalent birational embeddings II: divisors

Two divisors in $\P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the special case of plane curves and rational hypersurfaces. For the latter we characterise surfaces Cremona equivalent to a plane.Comment: v2 Exposition improved, thanks to referee, unconditional characterization of surfaces Cremona equivalent to a plan