4
$\begingroup$

Let $X,Y$ be different irreducible projective varieties in $\mathbb P^n$ (over an algebraically closed field). Let $Z$ be an irreducible component of $X\cap Y$. Then the intersection multiplicity of $X$ and $Y$ along $Z$ is an (non-negative) integer defined in literatures such as Fulton's Intersection theory.

Let $m$ be the intersection multiplicity of $X$ and $Y$ along $Z$. Suppose also that $X$, $Y$ and $X\cap Y$ are non-trivial, $X$ and $Y$ are of dimensions $\ge 1$ and do not contain each other. Can we intuitively interpret the intersection multiplicity the following way:

For every $P\in Z$ and every curve on $Y$ passing $P$ and not contained in $X$, the intersection multiplicity of the curve with $X$ at $P$ is $\ge m$. Furthermore, there is such a curve such that equality holds.

$\endgroup$

0

You must log in to answer this question.