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Take the three-dimensional affine quadric coneThe polynomial $xy-zw$ is an irreducible polynomial: as a polynomial in $x$ over $k[y,z,w]$, it is primitive because $y$ and $zw$ are coprime elements of a unique factorization domain, and it is a linear irreducible polynomial over $k(y,z,w)$. Gauss lemma for polynomials then applies. Thus $X$ is an irreducible variety that is an affine variety, of algebraic dimension $3$ by the principal hypersurface dimension lemma in affine space ${\mathbb{A}}^4$.

The coordinate rings of $Y$ and $Z$ are respectively $k[y,w]$ and $k[x,z]$. Both are affine planes, hence irreducible closed subsets of algebraic dimension $2$. Their intersection is precisely the origin. ThereforeThe origin is the singular point of an algebraic variety of this three-dimensional affine quadric cone. The example shows why a smoothness of an algebraic variety hypothesis matters in intersection dimension estimates.

The Weierstrass elliptic function and its derivative give an isomorphism $z\mapsto (\wp (z),{\wp }^{\prime }(z))$ from a one-dimensional complex torus to a smooth plane cubic, with $0$ sent to its identity at infinity. The three nonzero two-torsion points of a complex torus are zeros of ${\wp }^{\prime }$. Its triple pole implies that these three zeros are simple. If two half-period values of the Weierstrass elliptic function coincided, $\wp -e$ would have at least four zeros counted with multiplicity, contradicting its double pole. The three cubic roots are therefore distinct, proving smoothness of an algebraic variety. The degree-two map $\wp$ has fibers $\{z,-z\}$, distinguished by ${\wp }^{\prime }$ away from the half-periods, proving bijectivity. A line pulls back to an elliptic function with a triple pole, and the zero-pole sum of an elliptic function says that its three intersections sum to zero on the torus. This gives exactly the chord-and-tangent group law, including repeated intersections interpreted by intersection multiplicity.