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For closed irreducible subvarieties $Y,Z$ of a smooth $d$-dimensional algebraic variety $X$, every irreducible component of their intersection has dimension at least $\dim Y+\dim Z-d$. Locally the diagonal of $X\times X$ is cut out by $d$ parameters. Intersecting it with $Y\times Z$ and applying the Krull height theorem gives the bound. The three-dimensional affine quadric cone shows that the same lower bound can fail when $X$ is singular.

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.