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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?

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  • $\begingroup$ What is the dimension of $X$? $\endgroup$
    – Jason Starr
    Jun 23 at 10:09
  • $\begingroup$ You may assume that $\dim(X) + (k-2) < N$. While $k-2$ might be bigger or smaller of equal to $\dim(X)$. $\endgroup$
    – LaGra
    Jun 23 at 10:42
  • $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Jun 23 at 11:20
  • $\begingroup$ I am assuming that $X$ is non degenerated in $\mathbb{P}^N$. $\endgroup$
    – LaGra
    Jun 23 at 11:45
  • $\begingroup$ The answer is still “no.” If you have further hypotheses, please state them. $\endgroup$
    – Jason Starr
    Jun 23 at 22:14

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