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# 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, , corresponding to a rational map of projective varieties
• phi, an instance of the type RationalMapping, a rational map between projective varieties
• Optional inputs:
• QuickRank => , 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 .