# 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, , projective rational map given by polynomial representatives
• g, , projective rational map given by polynomial representatives
• R, a ring, coordinate ring of X
• S, a ring, coordinate ring of Y
• p, , 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 .