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(Edit) explained better the setup of the log deformation problem
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aiz89
aiz89
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(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?

(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 $(C,{\mathcal M}_C)$. According to a general theorem, 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?

(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?

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aiz89
aiz89
  • 49
  • 5

Comparison of logarithmic deformations and normal deformations

(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 $(C,{\mathcal M}_C)$. According to a general theorem, 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?