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Toric divisors from fans in sublattices #16334
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comment:2
Alright, here's a short patch that fixes the problem. I don't know whether it's the best way of working around this, but it's simple and does the trick. Volker, could you have a look? :) |
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Author: Jan Keitel |
Commit: |
Changed branch from u/jkeitel/toric_divisor_sublattice to u/jkeitel/toric_divisors_sublattice |
New commits:
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comment:4
I am confused - the polyhedron of the toric variety whose fan is in the lattice N lives in the dual lattice M, why is it intersected with the sublattice where the fan lives? |
Branch pushed to git repo; I updated commit sha1. New commits:
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comment:6
Hi Andrey, thanks for having a look at this. You're of course right. Best, |
comment:7
Looks good to me... Andrey, any further thoughts? |
Reviewer: Andrey Novoseltsev, Volker Braun |
comment:8
I think this polyhedron should live in the |
comment:9
Okay, if I understand you correctly, that's actually quite easy to realise. I have a patch that does this and I'll push it tomorrow when I'm back at the institute. |
comment:11
Sorry, it took a bit longer because something else came up. The newest change tries to do what you just suggested, but since polyhedra corresponding to divisors do not necessarily have to be lattice polytopes, it looks a bit clumsy. Could you have a look and check whether there's a nicer way of doing it? Best, Jan |
comment:12
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:
Usually, we have that As a consequence, we have the following behavior:
which leads directly to this (obviously incorrect, see the
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? |
comment:13
Just reordering may not be sufficient in more complicated examples. I think the assumption is that pairing between dual lattices is the "usual dot product", i.e. sum of products of corresponding components. If this is not the case for a particular representation, then everything will go wrong. Solutions are to either set generators of the dual upon construction correctly or set the pairing properly. Given that * is natural for pairing, it may be a bit counterintuitive to have |
Work Issues: fix inner product between N and M |
Stopgaps: wrongAnswerMarker |
Currently, there's a problem with toric divisors of toric varieties created from fans that live in a sublattice.
The following examples illustrates that:
However, the real polyhedron should be a two-dimensional compact polygon:
CC: @vbraun @novoselt @sagetrac-jakobkroeker
Component: algebraic geometry
Keywords: toric
Stopgaps: wrongAnswerMarker
Work Issues: fix inner product between N and M
Author: Jan Keitel
Branch/Commit: u/jkeitel/toric_divisors_sublattice @
108339b
Reviewer: Andrey Novoseltsev, Volker Braun
Issue created by migration from https://trac.sagemath.org/ticket/16334
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