1
$\begingroup$

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?

$\endgroup$

1 Answer 1

3
$\begingroup$

Yes.

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 $t$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.