# 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 $$x_i(t)$$.

Denote by $$H_0$$ the flat limit of the $$H_t$$ as $$t\mapsto 0$$, and assume that all the $$x_i(t)$$ collapse to the same point $$x_0\in X\cap H_0$$ for $$t\mapsto 0$$.

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

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

• What is the dimension of $X$? Jun 23 at 10:09
• You may assume that $\dim(X) + (k-2) < N$. While $k-2$ might be bigger or smaller of equal to $\dim(X)$. Jun 23 at 10:42
• The answer to your last question is "no." Let $N$ equal $4$. Let $H_0$ be a $2$-plane in $\mathbb{P}^N = \mathbb{P}^4$. Let $X\subset H_0$ be a plane quartic curve that contains a line $L_0$ whose intersection multiplicity with $X$ at one point $x_0$ equals $4$. Let $L_t$ be a family of general lines in $H_0$ that specialize to $L_0$. Let $H_t$ be a family of general $2$-planes in $\mathbb{P}^3$ that specialize to $H_0$ and such that each $H_t$ contains $L_t$. Then $H_0$ contains $X$ and the intersection multiplicity of $X$ and $H_0$ at $x_0$ is undefined. Jun 23 at 11:20
• I am assuming that $X$ is non degenerated in $\mathbb{P}^N$. Jun 23 at 11:45
• The answer is still “no.” If you have further hypotheses, please state them. Jun 23 at 22:14