Summary: How can we compute the ideal of the twisted cubic?
Chris Peterson and Hirotachi Abo
Let k = Q. Consider a polynomial map from A1
k to A3
k defined by
(t) = (t3
Here we describe V = (A1
k). A natural question is maybe:
Question 1. Is V A3
k an affine variety?
To answer to this question, it suffices to find an ideal I in k[x, y, z] such that
V = V (I). Recall that the map induces a ring homomorphism ~ from
k[x, y, z] to k[t]. It is obvious to see that the homomorphism is surjective.
Consider the kernel J := Ker() of this homomorphism. The kernel is,
needless to say, an ideal of k[x, y, z]. Thus ~ induces a ring homomorphism
from k[x, y, z]/J to k[t]. (This homomorphism is an isomorphism, which
implies that the corresponding varieties V (J) and A1
k are isomorphic.) The