idealOfImageOfMap -- finds defining equations for the image of a rational map between varieties or schemes

Synopsis

Usage:

im = idealOfImageOfMap(p)

im = idealOfImageOfMap(phi)
Inputs:
- p, a ring map, corresponding to a rational map of projective varieties
- phi, an instance of the type RationalMapping, a rational map between projective varieties
Optional inputs:
- QuickRank => a Boolean value, default value true, whether to compute rank via the package FastMinors
- Verbosity => ..., default value 0
Outputs:
- im, an ideal, defining equations for the image

Description

Given a rational map $f : X \to Y \subset P^N$, idealOfImageOfMap returns the defining ideal of the image of $f$ in $P^N$. The rings provided implicitly in the inputs should be polynomial rings or quotients of polynomial rings. In particular, idealOfImageOfMap function returns an ideal defining a subset of the ambient projective space of the image. In the following example we consider the image of $P^1$ inside $P^1 \times P^1$.

i1 : S = QQ[x,y,z,w];

i2 : b = ideal(x*y-z*w);

o2 : Ideal of S

i3 : R = QQ[u,v];

i4 : a = ideal(sub(0,R));

o4 : Ideal of R

i5 : f = matrix {{u,0,v,0}};

             1      4
o5 : Matrix R  <-- R

i6 : phi = rationalMapping(R/a, S/b, f)

                             S
o6 = Proj R - - - > Proj(---------)   {u, 0, v, 0}
                         x*y - z*w

o6 : RationalMapping

i7 : idealOfImageOfMap(phi)

o7 = ideal (w, y)

                  S
o7 : Ideal of ---------
              x*y - z*w

i8 : psi = rationalMapping(Proj(S/b), Proj(R/a), f)

                             S
o8 = Proj R - - - > Proj(---------)   {u, 0, v, 0}
                         x*y - z*w

o8 : RationalMapping

i9 : idealOfImageOfMap(psi)

o9 = ideal (w, y)

                  S
o9 : Ideal of ---------
              x*y - z*w

This function frequently just calls ker from Macaulay2. However, if the target of the ring map is a polynomial ring, then it first tries to verify whether the ring map is injective. This is done by computing the rank of an appropriate Jacobian matrix.

Ways to use idealOfImageOfMap :

idealOfImageOfMap(RationalMapping)
idealOfImageOfMap(RingMap)

For the programmer

The object idealOfImageOfMap is a method function with options.