A C-algebra is a ring that's also a complex vector space, and a C-algebra homomorphism is a ring homomorphism that's C-linear.
If R is a C-algebra, then C-Spec(R) := {the C-algebra homs R->C}.
If R->S is a C-algebra homomorphism, there's a natural map C-Spec(S) -> C-Spec(R).
If R is a polynomial ring, C-Spec(R) is the corresponding vector space. If R is a polynomial ring modulo an ideal I, C-Spec(R) is V(I).
The inclusion of a polynomial subring corresponds, under taking C-Spec, to linear projection.
Intersecting an ideal I with a polynomial subring, called elimination, corresponds to taking the projection of V(I) and then taking the closure.
Theorem. If I,J are ideals in C[x_1..x_n], so tI, (1-t)J are in C[t,x_1..x_n], then
I intersect J = ((tI) + (1-t)J) intersect C[x_1..x_n].
