#19512: is_morphism for maps of products of projective spaces
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Reporter: gjorgenson | Owner:
Type: enhancement | Status: needs_work
Priority: minor | Milestone:
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: | Reviewers: Ben Hutz
Authors: Grayson Jorgenson | Work issues:
Report Upstream: N/A | Commit:
Branch: | bca35ddbb8d6518535b43e32ec34514fcd0ab8c8
u/gjorgenson/ticket/19512 | Stopgaps:
Dependencies: |
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Comment (by bhutz):
Actually, that example should be a morphism and does not fail the segre
embedding check (it just isn't dominant). So the 0 dimensional check is
(at least) not iff.
One solution that has come to mind is do something like define the
subscheme of the productprojectivespaces for each set of coordinates of
the map and check that there are no points on each such subscheme. Then we
have a set of polynomials defined at every point in the product (the
definition of a morphism)
For products of 2 projective spaces, the .dimension function works fine
via segre embeddings, but calling dimension on the defining ideal for
empty subschemes does not give the 'right' answer in products:
{{{
P.<x,y,z,u,v,w>=ProductProjectiveSpaces(QQ,[2,2])
H=End(P)
f=H([u,v,w,u^2,v^2,w^2])
m=0
for i in range(P.num_components()):
t=P[i].dimension_relative()+1
X=P.subscheme(list(f)[m:m+t])
print X.defining_ideal().dimension(), X.defining_ideal().dimension() -
P.num_components()
print "dim:",X.dimension()
m+=t
}}}
at the moment segre embedding (and hence dimension) is only implemented
for products of 2 projective spaces. It would be nice to get that to work
in general, but maybe the solution for this particular function is to be
dependent on the dimension function for subschemes and then improve the
dimension function as part of a separate ticket.
What do you think?
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Ticket URL: <http://trac.sagemath.org/ticket/19512#comment:8>
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