Here is a small paragraph of a course of "Anwar Alameddin" that i found on the net accidently, and that i try to understand clearly, about intersection theory :
Let $ k $ be an algebraically closed field of characteristic different than $2$ and $ 3 $, such that $ f = y^2 - x^3 + 2x - 2 $ and $ g_c = y-c $ be two irreducible polynomials in $ k[x,y] $, for $ x \in k $.
Then, $ f $ and $ g_c $ give rise to two closed subschemes $ F $ and $ G_c $ of $ \mathbb{A}_{k}^2 $, induced by the surjections of ring homomorphisms : $ \varphi_f : k[x,y] \to k[x,y] / (f) $ and $ \varphi_{g_{c}} : k[x,y] \to k[x,y] / ( g_c ) $, respectively.
Let $ F.G_c $ denote the scheme-theoretic intersection of $ F $ and $ G_c $, i.e : the pull-back of the closed subschemes diagram in $ \mathrm{Sch} / k $ : $$ \xymatrix{ F.G_c \ar[r] \ar[d] & G_c \ar[d] \\ F \ar[r] & \mathbb{A}_{k}^1 } $$ Hence, $ F.G_c = \mathrm{Spec} k[x,y] / (f,g_c ) $. In contrast to set-theoretic intersection, the irreducible components of scheme-theoretic intersection of two reduced schemes are not necessary reduced.
For example, for $ c = \pm \sqrt{ 2 - \dfrac{4}{3} \sqrt{ \dfrac{2}{3} } } $, then straightforward calculations show that :
$$ F.G_c = \mathrm{Spec} k[x,y] / \Big( y^2 - x^3 + 2 x - 2 , y - \sqrt{ 2 - \dfrac{4}{3} \sqrt{ \dfrac{2}{3} } } \Big) $$ $$ \simeq \mathrm{Spec} > k[x,y] / \Big( (x - \sqrt{ \dfrac{2}{3} } )^2 ( x + 2 \sqrt{ \dfrac{2}{3} } ) \Big) $$
$$ \simeq \mathrm{Spec} k[x] / \Big( x - \sqrt{ \dfrac{2}{3} } \Big)^2 \coprod \mathrm{Spec} k[x] / \Big( x + 2 \sqrt{ \dfrac{2}{3} } \Big) $$ Hence, the intesection $ F.G_c $ consists of two rational points, with underlying topological points $ |P| = \Big( \sqrt{ \dfrac{2}{3} } , c \Big) $ and $ |Q| = \Big( -2 \sqrt{ \dfrac{2}{3} } , c \Big) $. These points are intrinsically different from each other, that $Q$ is reduced, whereas $ P $ is not.
My question is :
Why does $ |P| $ reduced, but $ |Q| $ is not reduced ?
Thank you in advance for your help.