Consider the ideals $I = \left< x\right> \cap \left< x,y\right>^2 = \left
What is the geometric difference between $I$ and $J$?
I know that the zero set for $J$ is a triple point at the origin on $k^2$ while the zero set for $I$ is the $y$-axis together with a double point at the origin on $k^2$.
But aren't they both colength $3$ ideals in $k[x,y]$?