Let $f:X\rightarrow S = \text{Spec}(k)$ be an n-marked stable curve of genus g, where $k$ is an algebraically closed field of characteristic $0$. Suppose that $X$ has $l$ double points $\{p_1, \ldots, p_l\}$.
I'm interested in whether it's possible to find a family of stable curves $F: \mathcal{X}\rightarrow \mathbb{A}^l = \text{Spec}(k[t_1, \ldots, t_l])$ such that the following hold:
- The fibre of $F$ over $0$ is $X$.
- We can find an \'{e}tale neighbourhood of $p_i$ in $\mathcal{X}$ of the form $\text{Spec}(k[t_1, \ldots, t_l][x,y]/(xy - t_i)$.
- If $D_i$ is the connected component of $Sing(\mathcal{X}/\mathbb{A}^l)$ containing $p_i$, then $D_i$ is closed in $\mathcal{X}$ and we have a Cartesian diagram
$\require{AMScd}$ \begin{CD} D_i @>{}>> \mathcal{X}\\ @VVV @VVV\\ Z(t_i) @>{}>> \mathbb{A}^l. \end{CD}
- $Sing(\mathcal{X}/\mathbb{A}^l) = \bigsqcup_i D_i$.
Is this possible?
I've been approaching this using deformation theory. Namely, by Theorem 21.3 of Hartshorne's deformation theory book, the universal deformation of $f$ is algebraisable in the sense that I can find a scheme $S$ of finite type over $k$, $s_o\in S$, and a flat finite type family $\mathcal{X}$ over $S$ whose fibre over $s_0$ is $X$ and with the completed \'{e}tale local ring of $s_0$ in $S$ being of the form $k[[t_1, \ldots, t_m]]$ for $m = 3g-3+n$. By Theorem 21.4 of the same book I can take $S$ to be \'{e}tale over $\mathbb{A}^m$. Unfortunately it's not clear to me that $\mathcal{X}\rightarrow S$ will satisfy what I want.
My motivation for this problem comes from logarithmic geometry. I start with a log curve over $k$ where $k$ has its trivial log structure, and I would like to know if there is some log curve over a log smooth base which has fibre over a closed point my original log curve.