#16334: Toric divisors from fans in sublattices
-------------------------------------+-------------------------------------
Reporter: jkeitel | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: toric | Reviewers: Andrey Novoseltsev,
Authors: Jan Keitel | Volker Braun
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jkeitel/toric_divisors_sublattice| 108339b90651ed0adbd422d1b9fc4dc3ba926aca
Dependencies: | Stopgaps:
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Comment (by jkeitel):
Unfortunately, the diff on trac seems to be broken. Something else has
come up and I'm not sure what to do about it.
Consider the following:
{{{
sage: N = ToricLattice(4)
sage: S = N.submodule([(1,0,0,0), (0,1,0,0)])
sage: B = S.basis()
sage: S.dual()
2-d lattice, quotient of 4-d lattice M by Sublattice <M(0, 0, 1, 0), M(0,
0, 0, 1)>
sage: S.dual().gens()
(M[0, 1, 0, 0], M[1, 0, 0, 0])
sage: S.gens()
(N(1, 0, 0, 0), N(0, 1, 0, 0))
}}}
Usually, we have that {{{i}}}th generator of a lattice is 'dual' (don't
know whether that's the right adjective) to the {{{i}}}th generator of the
dual lattice. Here, however, there is an additional reordering going on.
As a consequence, we have the following behavior:
{{{
sage: cones = [Cone([B[0], B[1]]), Cone([B[1], -3*B[0]-2*B[1]]),
Cone([-3*B[0]-2*B[1], B[0]])]
sage: X = ToricVariety(Fan(cones))
sage: p = (-X.K()).polyhedron(); p
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 3
vertices
sage: p.integral_points()
((-1, -1), (-1, 0), (-1, 1), (-1, 2), (0, -1), (0, 0), (1, -1))
sage: (-X.K()).sections()
(M[-1, -1, 0, 0],
M[0, -1, 0, 0],
M[1, -1, 0, 0],
M[2, -1, 0, 0],
M[-1, 0, 0, 0],
M[0, 0, 0, 0],
M[-1, 1, 0, 0])
}}}
which leads directly to this (obviously incorrect, see the {{{z1^-3}}})
output:
{{{
(z1^6, z0*z1^3, z0^2, z0^3/z1^3, z1^4*z2, z0*z1*z2, z1^2*z2^2)
}}}
What should be changed? Finding the generators of a dual of a lattice? The
polyhedron method of the divisor? The monomial method of the divisor?
--
Ticket URL: <http://trac.sagemath.org/ticket/16334#comment:12>
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