#9972: Add fan morphisms
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Reporter: novoselt | Owner: mhampton
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-4.6.1
Component: geometry | Keywords:
Author: Andrey Novoseltsev, Volker Braun | Upstream: N/A
Reviewer: Volker Braun, Andrey Novoseltsev | Merged:
Work_issues: |
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Comment(by vbraun):
I think the geometric image/preimage of a cone is not particularly useful
for toric morphisms. A fan morphism maps cones to cones, but does *not*
define linear maps of the individual cones. One should think of it as a
map from the poset of domain cones to the poset of codomain cones. Or, in
geometric terms, maps of the poset of torus orbits to the poset of torus
orbits.
The usual blow-up example
{{{
sage: c1 = Cone([(1,0),(1,1)])
sage: c2 = Cone([(1,1),(0,1)])
sage: domain_fan = Fan([c1, c2])
sage: codomain_fan = Fan([Cone([(1,0),(0,1)])])
sage: f = FanMorphism(identity_matrix(ZZ,2),domain_fan,codomain_fan)
sage: ray = Cone([(1,1)])
sage: f.image_cone(ray)
2-d cone of Rational polyhedral fan in 2-d lattice N
}}}
means, geometrically, that the orbit closure `P^1` corresponding to the
cone `ray` (given by the relative star of `ray`) is mapped to the point
corresponding to the `f.image_cone(ray)`. Conversely, the preimage cones
of `f.image_cone(ray)` are `c1`, `c2`, and `ray`. Geometrically, this
means that the preimage of the point `f.image_cone(ray)` consists of the
torus orbits (north pole), (south pole), `C^*` making up the fiber `P^1`.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9972#comment:55>
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