#15239: Nondegeneracy for subschemes of toric varieties
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Reporter: vbraun | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.4
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: | Reviewers:
Authors: Volker Braun | Work issues:
Report Upstream: N/A | Commit:
Branch: | f885028d7ab463ff6e870ffa1ff61558304a5986
u/vbraun/is_nondegenerate | Stopgaps:
Dependencies: |
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Comment (by kedlaya):
There are also some subtleties surrounding the underlying mathematical
concept. In particular, I would also take issue with this doctest:
{{{
sage: P1 = toric_varieties.P1()
sage: X = toric_varieties.P2_112().cartesian_product(P1)
sage: X.inject_variables()
Defining z0, z1, z2, z3, z4
sage: Y = X.subscheme([z4]) # = P2_112 x {point}
sage: Y.is_nondegenerate()
True
}}}
As far as I understand, nondegeneracy is only defined when you have a
local complete intersection, and then the condition is that the
intersection with each stratifying torus should be transversal. That's
stronger than saying that the intersection should be lci, as in this
example: the intersection of a smooth variety with itself is smooth but
not transversal.
However, I struggle to find an "official" definition of nondegeneracy in
the literature which would clear up this ambiguity. I've only ever seen it
used for global complete intersections.
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Ticket URL: <http://trac.sagemath.org/ticket/15239#comment:12>
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