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

Equivalent Birational Embeddings III: cones

Two divisors in $\mathbb P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. In this paper I study irreducible cones in $\mathbb P^n$ and prove that two cones are Cremona equivalent if their general hyperplane sections are birational. In particular I produce examples of cones in $\mathbb P^3$ Cremona equivalent to a plane whose plane section is not Cremona equivalent to a line in $\mathbb P^2$

On the geometry of normal horospherical G-varieties of complexity one

Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a G-variety. Using the combinatorial description of Timashev, we describe the class group of X by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for X and a criterion to determine whether the singularities of X are rational or log-terminal respectively.Comment: 29 pages, final version, to appear in J. Lie Theor

Embedding into the rectilinear plane in optimal O*(n^2)

We present an optimal O*(n^2) time algorithm for deciding if a metric space (X,d) on n points can be isometrically embedded into the plane endowed with the l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by J. Edmonds, our algorithm uses the concept of tight span and the injectivity of the l_1-plane. A different O*(n^2) time algorithm was recently proposed by D. Eppstein (2009).Comment: 12 pages, 13 figure