Affine cone

For a projective algebraic set $X\subseteq {\mathbb{P}}^n$, its affine cone is the union in ${\mathbb{A}}^{n+1}$ of the lines represented by the points of $X$, together with the origin.

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.

On $Y=\{xy=zw\}\setminus \{0\}$, the formulasglue to a morphism $Y\to {\mathbb{P}}^1$. The inverse image of $[0:1]$ is the ruling plane $V(x,z)\cap Y$, so its divisor line bundle is the pullback of ${\mathcal{O}}_{{\mathbb{P}}^1}(1)$.