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Let $k$ be an algebraic closed field. Let $n$ be a positive integer.

Let $X$ be an irreducible, proper and smooth scheme over $k$ with an immersion $i:X\hookrightarrow E:=\mathbb P^N_k$ with $N$ large enough. Let $x_0\in X$ be a closed point and $0\neq f_0\in \mathcal O_{X,x_0}/\mathfrak m_{x_0}^n$.

Does there exist a pencil $D\in Gr(1,\check{E})$, here $Gr(1,\check{E})$ is the variety of lines in $\check{E}=(\mathbb P^N_k)^\vee$, such that the following conditions are satisfied:

1) The axis of $D$ meets transversely with $X$.

2)There exist two points $F$ and $G$ on $D$(view $F$ and $G$ as hyperplanes in E) such that $x_0\notin F$, $x_0\in G$ and $\frac{G}{F}\equiv f_0 \mod \mathfrak m_{x_0}^N$.

3) There exist a open neighborhood $U\subset D$ of $G$ in $D$ such that for every $G_1\in U-\{G\}$, the hyperplane $G_1$ meets transversely with $X$.


Edit:due to abx's example, in the case $f_0=0$ the pencil may not exist. In the first time, $f_0\in \mathcal O_{X,x_0}/\mathfrak m_{x_0}^N$, but now I try to let the degree of $f_0$ different with $N$,i.e.,$f_0\in \mathcal O_{X,x_0}/\mathfrak m_{x_0}^n$ ($n<N$).

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1 Answer 1

Take $f_0=0$. Then you are asking for a hyperplane which has a contact of order $\geq N$ with $X$ at $x_0$, so that its intersection with $X$ has multiplicity $\geq N$ at $x_0$. This does not exist in general. For instance take $X=\mathbb{P}^2$, embedded in $\mathbb{P}^N$ by the complete linear system $|\mathcal{O}(d)|$. A hyperplane cuts out on $X$ a curve of degree $d$, which has multiplicity $\leq d\ll N$ at every point.

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Quit right! But how about the case $f_0\neq 0$? –  Ely Jan 18 at 9:14
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Same -- only a little more technical. Roughly, your pairs $(G,F)$ depend on $2N$ parameters, the dimension of $\mathcal O_{X,x_0}/\mathfrak m_{x_0}^N$ is much larger -- you get only a small subvariety of that space. –  abx Jan 18 at 9:53
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Thanks, now I want the degree of $f$ much smaller that $N$, how about this case? –  Ely Jan 18 at 15:17
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I think there is no uniform answer: it will depend on deg($f$) versus $N$, and also on the geometry of $X$. –  abx Jan 18 at 15:24
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After passing to a suitable $d$-fold embedding of $X$, I think we don't need to worry about the degree of $f_0$ versus N. Does there exist such a pencil for $X=\mathbb P^m_k$ with $i$ be the Sergre imbedding? –  Ely Jan 19 at 3:06

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