 Three-dimensional affine quadric cone

The affine variety $X=V(xy-zw)\subset {\mathbb{A}}^4$ has dimension three and a unique singular point of an algebraic variety at the origin. The planes $V_X(x,z)$ and $V_X(y,w)$ have dimension two but intersect only at that origin. This is a basic failure of intersection dimension estimates at singular points.