#20512: Jacobian of the tautologous subscheme of a toric variety is broken
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Reporter: kedlaya | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-7.2
Component: algebraic geometry | Keywords: schemes, Jacobian
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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This shouldn't return an error:
{{{
sage: P.<x,y,z> = ProjectiveSpace(2, QQ)
sage: X = P.subscheme([])
sage: X.Jacobian_matrix() # This works
[]
sage: X.Jacobian() #This doesn't
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute
'reduce'
}}}
I think the mathematically correct answer is that X.Jacobian() should
again equal X (i.e., P viewed as a closed subscheme of itself). However,
this is not consistent with the definition in the docstring:
{{{
* the `d\times d` minors of the Jacobian matrix, where `d` is
the :meth:`codimension` of the algebraic scheme, and
* the defining polynomials of the algebraic scheme. Note that
some authors do not include these in the definition of the
Jacobian ideal. An example of a reference that does include
the defining equations is [LazarsfeldJacobian].
}}}
In this case d=0, and the unique 0 by 0 minor of any matrix (empty or not)
is equal to 1.
--
Ticket URL: <http://trac.sagemath.org/ticket/20512>
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