#11599: Wrap fan moriphism in toric morphism
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Reporter: vbraun | Owner: AlexGhitza
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.7.2
Component: algebraic geometry | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Comment(by novoselt):
We should. I have been using the following code so far:
{{{
def toric_morphism_list(phi, X, Y):
r"""
Given a fan morphism phi from X.fan() -> Y.fan(), return the list
representation of the corresponding toric morphism.
"""
x = [SR.var(x_i, domain="positive") for x_i in
X.coordinate_ring().variable_names()]
result = [1] * Y.fan().nrays()
for rho, x_rho in zip(X.fan(1), x):
sigma_prime = phi.image_cone(rho)
degrees = sigma_prime.ray_matrix() \ phi(rho.ray(0))
for i, d in zip(sigma_prime.ambient_ray_indices(), degrees):
result[i] *= x_rho^d
return result
def toric_morphism_dictionary(phi, X, Y):
r"""
Given a fan morphism phi from X.fan() -> Y.fan(), return the
dictionary representation of the corresponding toric morphism.
"""
x = [SR.var(x_i, domain="positive") for x_i in
X.coordinate_ring().variable_names()]
y = [SR.var(y_i, domain="positive") for y_i in
Y.coordinate_ring().variable_names()]
result = dict((y_i, 1) for y_i in y)
for rho, x_rho in zip(X.fan(1), x):
sigma_prime = phi.image_cone(rho)
degrees = sigma_prime.ray_matrix() \ phi(rho.ray(0))
for i, d in zip(sigma_prime.ambient_ray_indices(), degrees):
result[y[i]] *= x_rho^d
return result
}}}
Using the dictionary representation it is quite easy to compute pullbacks,
the problem here is that the underlying map of total coordinate rings is
not a ring homomorphism, since it is likely to involve roots. The
following paper may be useful for "correct and general" implementation:
http://arxiv.org/abs/1004.4924
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11599#comment:1>
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