How to prove that any finite set in the affine plane is realizable as intersection of TWO plane curves?


Thanks.



Finite number of points as intersection of two plane curves
What you want to show is that a radical ideal $I$ in $k[X,Y]$ must be equal to the radical of a two-generated ideal.


The image of $I$ in $k(X)[Y]$ is certainly principal (because $k(X)[Y]$ is a PID). So we can write $s(X)I\subset (f)$ for some $f\in k[X,Y]$.


Let $A=k[X]/(s)$ and let $B$ be $A$ mod nilpotents. Let $J$ be the radical of the image of $I$ in $A[Y]$ and let $\overline{J}$ be the image of $J$ in $B[Y]$. Then $\overline{J}$ is principal because $B$ is a product of fields. It follows that $J$ is prinicipal (because it's a radical ideal and it's principal mod nilpotents). Lift a generator to $g\in I$. 


It's straightforward to check that $I$ is equal to the radical of $(f,g)$. 


