29,600 research outputs found
Canonical bases in excellent classes
We show that any (atomic) excellent class can be expanded with hyperimaginaries to form an (atomic) excellent class which has canonical bases. When is, in addition, of finite U-rank, then is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class gives rise to an excellent class, which is simple if is of finite U-ran
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
A monodromy criterion for extending curves
A family of proper smooth curves of genus , parametrised by an open
dense subset of a normal variety , extends to if the natural map
on fundamental groups is an isomorphism. The criterion
of this note is actually more precise.Comment: 12 pages; revised version including a section on outer
representations for families of curve
On an analogue of the James conjecture
We give a counterexample to the most optimistic analogue (due to Kleshchev
and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras
associated to simply-laced Dynkin diagrams. The first counterexample occurs in
type A_5 for p = 2 and involves the same singularity used by Kashiwara and
Saito to show the reducibility of the characteristic variety of an intersection
cohomology D-module on a quiver variety. Using recent results of Polo one can
give counterexamples in type A in all characteristics.Comment: 12 pages. v2: final versio
- …