144 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
The use of blocking sets in Galois geometries and in related research areas
Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
Prym varieties of genus four curves
Double covers of a generic genus four curve C are in bijection with Cayley
cubics containing the canonical model of C. The Prym variety associated to a
double cover is a quadratic twist of the Jacobian of a genus three curve X. The
curve X can be obtained by intersecting the dual of the corresponding Cayley
cubic with the dual of the quadric containing C. We take this construction to
its limit, studying all smooth degenerations and proving that the construction,
with appropriate modifications, extends to the complement of a specific divisor
in moduli. We work over an arbitrary field of characteristic different from two
in order to facilitate arithmetic applications.Comment: 30 pages; Some expository changes; removed erroneous (old) Thm 4.11
and changed (old) Thm 4.23 into (new) Thm 4.1
Successive minima of toric height functions
Given a toric metrized R-divisor on a toric variety over a global field, we
give a formula for the essential minimum of the associated height function.
Under suitable positivity conditions, we also give formulae for all the
successive minima. We apply these results to the study, in the toric setting,
of the relation between the successive minima and other arithmetic invariants
like the height and the arithmetic volume. We also apply our formulae to
compute the successive minima for several families of examples, including
weighted projective spaces, toric bundles and translates of subtori.Comment: To appear in Annales de l'Institut Fourier (Grenoble), 40 pages, 5
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