#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 | Resolution:
Keywords: schemes, Jacobian | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Description changed by kedlaya:
Old description:
> 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.
New description:
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
equal the ideal (1). This is 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. And anyway, the Jacobian ideal of the full ambient space
should cut out the empty subscheme.
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Ticket URL: <http://trac.sagemath.org/ticket/20512#comment:1>
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