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

Synopsis

Usage:

im = idealOfImageOfMap(a,b,f)

im = idealOfImageOfMap(a,b,g)

im = idealOfImageOfMap(R,S,f)

im = idealOfImageOfMap(R,S,g)

im = idealOfImageOfMap(p)
Inputs:
- a, an ideal, defining equations for X
- b, an ideal, defining equations for Y
- f, a matrix, projective rational map given by polynomial representatives
- g, a basic list, projective rational map given by polynomial representatives
- R, a ring, coordinate ring of X
- S, a ring, coordinate ring of Y
- p, a ring map, projective rational map given by polynomial representatives
Optional inputs:
- QuickRank => ..., default value true, An option for computing how rank is computed
- Verbose => ..., default value false
Outputs:
- im, an ideal, defining equations for the image of f

Description

Given $f : X \to Y \subset P^N$, this returns the defining ideal of $f(x) \subseteq P^N$. It should be noted for inputs that all rings are quotients of polynomial rings, and all ideals and ring maps are of these. In particular, this function returns an ideal defining a subset of the 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 : idealOfImageOfMap(a,b,f)

o6 = ideal (w, y)

o6 : Ideal of S

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 if the ring map is injective. This is done by computing the rank of an appropriate jacobian matrix.

Ways to use `idealOfImageOfMap` :

"idealOfImageOfMap(Ideal,Ideal,BasicList)"
"idealOfImageOfMap(Ideal,Ideal,Matrix)"
"idealOfImageOfMap(Ring,Ring,BasicList)"
"idealOfImageOfMap(Ring,Ring,Matrix)"
"idealOfImageOfMap(RingMap)"

For the programmer

The object idealOfImageOfMap is a method function with options.

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

Synopsis

Description

Ways to use idealOfImageOfMap :

For the programmer

Ways to use `idealOfImageOfMap` :