2,321 research outputs found
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Configurations of infinitely near points
We present a survey of some aspects and new results on configurations, i.e.
disjoint unions of constellations of infinitely near points, local and global
theory, with some applications and results on generalized Enriques diagrams,
singular foliations, and linear systems defined by clusters
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