ag.algebraic geometry – Intersection multiplicity in flat families of linear spaces Answer

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ag.algebraic geometry – Intersection multiplicity in flat families of linear spaces

let $X\subset\mathbb{P}^N$ be an irreducible projective variety and $\{H_t\}_{t\in \mathbb{C}^{*}}$ a family of $(k-2)$-dimensional linear subspaces of $\mathbb{P}^N$ intersecting $X$ in $k$ distinct points $x_i

Denote by $H_0$ the flat limit of the $H_t$ like $t\mapsto 0$and assume that all the $x_i

In this situation we can get some information on the intersection multiplicity of $X$ other $H_0$ at $x_0$?

For instance, if $k-2 = 2$ is the intersection multiplicity of $X$ other $H_0$ at $x_0$ equal to 4?

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