# 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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