#20676: Projective closure and affine patches for algebraic curves
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Reporter: gjorgenson | Owner:
Type: enhancement | Status: new
Priority: minor | Milestone: sage-7.3
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: | Reviewers:
Authors: Grayson Jorgenson | Work issues:
Report Upstream: N/A | Commit:
Branch: | b3002ef28b0b1f9ae39b1222c446625037420db9
u/gjorgenson/ticket/20676 | Stopgaps:
Dependencies: |
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Comment (by gjorgenson):
Replying to [comment:3 mmarco]:
Thanks for looking at this. I think it's not always the case that a given
set of generators is sufficient for finding generators for the ideal of
the projective closure. One example I have seen referenced in some texts
is the twisted cubic in A3 defined by the ideal {{{(y-x^2, z-x^3)}}}. Its
projective closure is {{{(x^2 − wy, xy − wz, y^2 − xz)}}}, but if the
given generators of the first ideal are homogenized we get {{{(wy-x^2,
zw^2-x^3)}}} instead.
A proof is given in the book ideals, varieties, algorithms that if the
generators of a groebner basis (with respect to a graded monomial
ordering) for the defining ideal of an affine variety are homogenized,
then the result is a set of generators for the ideal of the projective
closure, so I tried to emulate that here.
I'll work on getting some flexibility with the variable names.
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Ticket URL: <http://trac.sagemath.org/ticket/20676#comment:4>
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