Q:

meaning of scheme theoretic - defining equation

In algebraic geometry, an algebraic set $X$ is the common zero set of polynomials.
In this case, these polynomials is called the defining equation of $X$.
I know that any algebraic set regards as scheme by Hartshorne book (II.2.6).
My question is simple... what is the meaning of scheme theoretic - defining equation??

A:

It means that you keep track of the actual polynomials in the defining ideal, as opposed to just the reduced portion (the radical of the ideal).  For example, $x$ and $x^2$ define the same zero set in $k[x,y]$, but scheme-theoretically, the schemes corresponding to $k[x,y]/(x)$ and $k[x,y]/(x^2)$ are not the same.

