= Intersection dimension bound on a smooth variety
{title2=$\dim(Y\cap Z)_j\ge\dim Y+\dim Z-\dim X$}
= Intersection dimension estimates
{synonym}
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.
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