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Let $f \in \mathbb{C}[x,y,z]$ be a polynomial which defines an isolated singularity $0 \in D:= (f=0) \subset \mathbb{C}^3$. Assume that $\mathcal{D}:= (f+tx =0) \subset \mathbb{C}^3 \times \mathbb{C}$ defines a smoothing of $D$ over a small disk. Let $g \in \mathbb{C}[x,y,z]$ be a polynomial which has a zero of order $\ge 2$ at $0$.

Question Let $\mathcal{D}' := (f+t(x+g)=0)$ be a deformation of $D$ induced by $x+g$. Is $\mathcal{D}'$ smoothing?

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Call $d$ the maximum between the degrees of $f$ and $g$. Let us homogenize the equations to homogeneous equations of degree $d$ in four variables. The polynomial $f + t(x + g)$ corresponds to a linear system of curves $C_t$ of degree $d$ in $\mathbb P^3$. By Bertini's theorem $C_t$ is smooth away from the base points of the linear system, for all but finitely many values of $t$. We don't care about the singularities at infinity. If $p \in \mathbb C^3$ is a base point of the linear system, then either $p$ is the origin, in which case $C_\infty$ is smooth at $p$, or is not, and in this case $C_0$ is smooth at $p$. Since being smooth at a base point is an open condition, we have have that $C_t$ is smooth at all points of $\mathbb C^3$ for all but finitely many values of $t$.

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Thank you very much for the clear answer. – tarosano Aug 5 '12 at 20:14

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