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RationalMaps :: idealOfImageOfMap

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



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 :

For the programmer

The object idealOfImageOfMap is a method function with options.