Intersection multiplicity of limit linear spaces

Question

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.

Now let $\Lambda_{\xi, q}$, with $\xi \in C$, be a flat family of linear spaces parametrized by $C$. Assume that for a general $\xi\in C$ the intertsection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $\xi$ is greater or equal than $m_1$, and the intertsection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $q$ is greater or equal than $m_2$.

Finally, let $\Lambda_{q, q}$ be the limit linear space of the family $\Lambda_{\xi, q}$ when $\xi\mapsto q$ along $C$.

May we then conclude that the intertsection multiplicity of $\Lambda_{q, q}$ and $X$ at $q$ is greater or equal than $m_1+m_2$ ?

Answer 1

The answer is no. Take $X \subset \mathbb{P}^4$ be a hyperplane section of $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$. So $X$ is a cubic ruled surface, I denote by $L$ the ruling.

Let $H$ be a generic hyperplane containing $L$. Then $H \cap X = L \cup D_1 \cup D_2$, where $D_1$ and $D_2$ are two disjoint lines of $X$ transverse to the ruling. Hence, if $x_1 = D_1 \cap L$ and $x_2 = D_2 \cap L$, then $H$ and $X$ have intersection multiplicity $2$ at $x_1$ and $x_2$.

Now, let $H_t$ be a family of such generic hyperplanes for $t \neq 0$ and such that $D_1$ colides to $D_2$ when $t \rightarrow 0$. Then, By Bezout, we have $H_0 \cap X = 2.D + L$.

Hence, the tangent cone of $X \cap H_0$ at $x_1 = x_2 =x$ is $2.D + L$. It has degree $3$, so that the multiplicity of $H_0 \cap X$ at $x$ is $3$.

The subtle phenomenon here is that the tangency locus of $H_0$ along $X$ is non-reduced! It is equal to $D$ with an embedded point at $x$.

This paper : http://msp.org/pjm/2011/253-1/pjm-v253-n1-p01-p.pdf studies in details some phenomenon related to your question.

EDIT : I made a mistake in the computation of the tangent cone to $H_0 \cap X$ at $x_1 = x_2$. Now it is corrected.