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Let's suppose that I am given a pencil generated by the vector fields $X$ and $Y$ in $\mathbb{C}^2$, $\{ Z_\lambda \}_{\lambda\in\mathbb{P}^1}$, that is, $$ Z_\lambda = X + \lambda Y $$

Assume that $X$ and $Y$ are non-singular except in the origin $(0,0)$.

By Bertini's Theorem, does this mean that for a generic $\lambda \neq 0,\infty $, the generic element of the pencil $Z_\lambda$ is non-singular except in the origin? If not, can someone please explain me what I am missing?

OhMyGod
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    I didn't know this version of Bertini's theorem for vector fields, but the missing hypothesis here is probably that $(0,0)$ should be the "base locus". With this assumption, your conclusion should be true. If $X$ and $Y$ are curves in $\mathbb C^2$, the base locus is simply their intersection. If they are vector fields, I don't know...! – Francesco Genovese Jul 22 '13 at 10:11
  • What do you mean by base locus? – OhMyGod Jul 23 '13 at 03:03
  • Well, in the case of a pencil of curves, it is simply the intersection of all curves in the pencil. In the case of vector fields, I don't know! – Francesco Genovese Jul 23 '13 at 10:19

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