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We now discuss intrinsics which apply to a scheme at a single point.
At the expense of increasing the number of intrinsics, we try to follow
the convention that an intrinsic may simply take a point as its argument
or it may take both a point and a scheme as its arguments.
In the former case, the implicit scheme argument is taken to be the scheme
associated to the point set of the point.
In the latter case, it is first checked that the point can be coerced into some
point set of the given scheme argument.
There are reasons for allowing both methods.
Of course, if one is confident about which scheme, X say, a point p lies on
then there is no ambiguity about writing, say, IsNonsingular(p)
rather than IsNonsingular(X,p).
On the other hand, the second expression is easier to read, and also guards
against the possibility of accidentally referring to the wrong scheme;
that is a particular risk here since the answer makes sense even if p
lies on some other scheme---imagine the confusion that could arise given
a point of a nonsingular curve lying on a singular surface inside
a nonsingular ambient space, for instance.
But also there are trivial cases when scheme arguments are necessary,
IntersectionNumber(C,D,p) for example.
In fact, that particular example exemplifies the value of points being
highly coercible---it is very convenient that the point p could
lie in a point set of either C or D or indeed neither of these
as long as it could be coerced to them if necessary.
Sometimes a function will require that the point argument is rational,
that is, has coordinates in the base ring.
Subsections
IsSingular(X,p) : Sch,Pt -> BoolElt
Returns true if and only if the point p is a singular point of the scheme X.
IsNonsingular(X,p) : Sch,Pt -> BoolElt
Returns true if and only if the point p is a nonsingular point of the scheme X.
IsOrdinarySingularity(X,p) : Sch,Pt -> BoolElt
Returns true if and only if the tangent cone to the scheme X at the point p
is reduced and X is singular at p.
Currently, the scheme X must be a hypersurface.
Multiplicity(X,p) : Sch,Pt -> RngIntElt
The multiplicity of the point p as a point of the scheme X.
Currently, the scheme X must be a hypersurface.
TangentSpace(X,p) : Sch,Pt -> Sch
The tangent space to the scheme X at the point p.
This linear space is embedded as a scheme in the same ambient space as X.
An error will be signalled if p is a singular point of X or is not a
rational point of X.
TangentCone(X,p) : Sch,Pt -> Sch
The tangent cone to X at the point
p embedded as a scheme in the same ambient space.
Currently, the scheme X must be a hypersurface
for the tangent cone to be defined. We anticipate a new implementation
of Mora's tangent cone algorithm in the near future and this intrinsic
will be updated to use it.
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