Rationality of Euler-Chow series and finite generation of Cox rings

In this paper we work with a series whose coefficients are the Euler characteristic of Chow varieties of a given projective variety. For varieties where the Cox ring is defined, it is easy to see that in this case the ring associated to the series is the Cox ring. If this ring is noetherian then the series is rational. It is an open question whether the converse holds. In this paper we give an example showing the converse fails. However we conjecture that it holds when the variety is rationally connected. As an evidence of this conjecture, It is proved that the series is not rational, and in a sense defined, not algebraic, in the case of the blowup of the projective plane at nine or more points in general position. Furthermore, we also treat some other examples of varieties with infinitely generated Cox ring, studied by Mukai and Hassett-Tschinkel. These are the first examples known where the series is not rational. We also compute the series for Del Pezzo surfaces.Comment: 26 pages. In this last version we correct many typos and add a cite of a work of Artebani and Laface in Theorem 1.6 which was brought to our attention. More typo correction

A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard

A New World Map on an Irregular Heptahedron

Using polyhedral approximations of the globe for the purpose of creating map projections is not a new concept. The implementation of regular and semi-regular polyhedra has been a popular method for reducing distortion. However, regular and semi-regular polyhedra provide limited control over the placement of the projective centers. This paper presents a method for using irregular polyhedra to gain more control over the placement of the projective centers while maintaining the reduced distortion quality found in polyhedral projections. The method presented here uses irregular polyhedra based on gnomonically projected Voronoi partitions of the sphere