(I'm trying to learn logarithmic geometry, and I'm extremely confused about something very basic in log deformation theory. It's very likely my question is nonsense, but I don't get it.)
Let's pick for instance a reducible nodal curve $C$ over an algebraically closed field (characteristic 0, if it matters), $C = C_1 \cup C_2$ with $C_i$ smooth and irreducible meeting at a point $x$. Let $Def^1(C) = \mathrm{Ext}^1(\Omega_C,{\mathscr O}_C)$ be the usual space of first order deformations. (When I write $\Omega_C$ I always mean the non-logarithmic one.)
If I understand correctly, there is a canonical way to think of $C$ as a log curve $(C,{\mathcal M}_C)$ (I'm talking about the "unmarked" version with characteristic $\overline{\mathcal M}_C = {\mathbb Z}_x$) over the standard logarithmic point. Let $Def^1_{log}(C)$ be the space of first order deformations of $(C,{\mathcal M}_C)$,
EDIT: by which I mean the following: let $B_0$ be the standard logarithmic point, and $B$ given by $\underline{B} = \mathrm{Spec} \mathop{} \mathbb{C}[\epsilon]/(\epsilon^2)$, with its logarithmic structure obtained by pulling back the log structure on ${\mathbb A}^1$ relative to the divisor $0$ along $\underline{B} \to {\mathbb A}^1 = \mathrm{Spec} \mathop{} {\mathbb C}[t]$ corresponding to $t \mapsto \epsilon$; then $Def^1_{log}(C)$ is the space of (integral, log smooth) deformations of $(C,{\mathcal M}_C)$ over $B$.
According to a general theorem (EDIT: which applies in the setup of the edit above), this should be an affine space for $H^1(C,T^{log}_C) = H^1(C,\omega_C^\vee)$, where $\omega_C$ is the dualizing sheaf of the usual curve.
The question. I'm confused about the relation between these spaces of deformations. I think there should be a map $$ \psi:Def^1_{log}(C) \to Def^1(C) $$ which just forgets the logarithmic structure, but I can't begin to comprehend what this map looks like. (The image of this map doesn't contain $0$, right?) Is it an affine map? If yes, which one?
Some thoughts. There is a canonical map $\Omega_C \to \omega_C$, which induces $\phi: H^1(C,\omega_C^\vee) = \mathrm{Ext}^1(\omega_C,{\mathcal O}_C) \to {\mathrm{Ext}}^1(\Omega_C,{\mathcal O}_C)$. A wild guess: maybe $\psi$ is a "translate" of $\phi$? Then what's the distinguished element of $Coker(\phi)$?
Bonus: What happens for more complicated nodal curves?
