13,727 research outputs found
Geometric contextuality from the Maclachlan-Martin Kleinian groups
There are contextual sets of multiple qubits whose commutation is
parametrized thanks to the coset geometry of a subgroup of
the two-generator free group . One defines
geometric contextuality from the discrepancy between the commutativity of
cosets on and that of quantum observables.It is shown in this
paper that Kleinian subgroups that are
non-compact, arithmetic, and generated by two elliptic isometries and
(the Martin-Maclachlan classification), are appropriate contextuality filters.
Standard contextual geometries such as some thin generalized polygons (starting
with Mermin's grid) belong to this frame. The Bianchi groups
, defined over the imaginary quadratic field
play a special role
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
The multivariate arithmetic Tutte polynomial
We introduce an arithmetic version of the multivariate Tutte polynomial, and
(for representable arithmetic matroids) a quasi-polynomial that interpolates
between the two. A generalized Fortuin-Kasteleyn representation with
applications to arithmetic colorings and flows is obtained. We give a new and
more general proof of the positivity of the coefficients of the arithmetic
Tutte polynomial, and (in the representable case) a geometrical interpretation
of them.Comment: 21 page
Severi Varieties and Brill-Noether theory of curves on abelian surfaces
Severi varieties and Brill-Noether theory of curves on K3 surfaces are well
understood. Yet, quite little is known for curves on abelian surfaces. Given a
general abelian surface with polarization of type , we prove
nonemptiness and regularity of the Severi variety parametrizing -nodal
curves in the linear system for (here is
the arithmetic genus of any curve in ). We also show that a general genus
curve having as nodal model a hyperplane section of some -polarized
abelian surface admits only finitely many such models up to translation;
moreover, any such model lies on finitely many -polarized abelian
surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is
proved concerning the possibility of deforming a genus curve in
equigenerically to a nodal curve. The rest of the paper deals with the
Brill-Noether theory of curves in . It turns out that a general curve in
is Brill-Noether general. However, as soon as the Brill-Noether number is
negative and some other inequalities are satisfied, the locus of
smooth curves in possessing a is nonempty and has a component of
the expected dimension. As an application, we obtain the existence of a
component of the Brill-Noether locus having the expected
codimension in the moduli space of curves . For , the
results are generalized to nodal curves.Comment: 29 pages, 3 figures. Comments are welcome. 2nd version: added some
references in Rem. 7.1
- …