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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/RationalMaps.m2:1847:0.