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Stahl
Stahl
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Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.)

In this case, $\operatorname{Spec}\Bbb{C}[\Bbb{N}] \simeq \operatorname{Spec}\Bbb{C}[t],$ so the pictured disk is the complex affine line, represented as a subset of the complex plane. We can also compute $\mathcal{M}_x$ for each $x\in\operatorname{Spec}\Bbb{C}[\Bbb{N}].$ At the points other than the origin, the log structure on $\operatorname{Spec}\Bbb{C}[\Bbb{N}]$ is trivial in the sense that $\mathcal{M}_x \simeq\mathcal{O}_{X,x}^\times.$ However, at the origin, we have $\mathcal{M}_x\simeq \Bbb{C}^\times\times\Bbb{N},$$\mathcal{M}_x\simeq \mathcal{O}_{X,x}^\times\times\Bbb{N},$ and the origin gets embellished with a picture of $\operatorname{Spec}\Bbb{N}.$

It seems to me that the picture really decorates each point $x\in X$ with a picture of $\operatorname{Spec}(\overline{\mathcal{M}}_x),$ where $\overline{\mathcal{M}} = \mathcal{M}/\mathcal{O}_X^\times$ is the characteristic of the log structure.

Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.)

In this case, $\operatorname{Spec}\Bbb{C}[\Bbb{N}] \simeq \operatorname{Spec}\Bbb{C}[t],$ so the pictured disk is the complex affine line, represented as a subset of the complex plane. We can also compute $\mathcal{M}_x$ for each $x\in\operatorname{Spec}\Bbb{C}[\Bbb{N}].$ At the points other than the origin, the log structure on $\operatorname{Spec}\Bbb{C}[\Bbb{N}]$ is trivial in the sense that $\mathcal{M}_x \simeq\mathcal{O}_{X,x}^\times.$ However, at the origin, we have $\mathcal{M}_x\simeq \Bbb{C}^\times\times\Bbb{N},$ and the origin gets embellished with a picture of $\operatorname{Spec}\Bbb{N}.$

It seems to me that the picture really decorates each point $x\in X$ with a picture of $\operatorname{Spec}(\overline{\mathcal{M}}_x),$ where $\overline{\mathcal{M}} = \mathcal{M}/\mathcal{O}_X^\times$ is the characteristic of the log structure.

Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.)

In this case, $\operatorname{Spec}\Bbb{C}[\Bbb{N}] \simeq \operatorname{Spec}\Bbb{C}[t],$ so the pictured disk is the complex affine line, represented as a subset of the complex plane. We can also compute $\mathcal{M}_x$ for each $x\in\operatorname{Spec}\Bbb{C}[\Bbb{N}].$ At the points other than the origin, the log structure on $\operatorname{Spec}\Bbb{C}[\Bbb{N}]$ is trivial in the sense that $\mathcal{M}_x \simeq\mathcal{O}_{X,x}^\times.$ However, at the origin, we have $\mathcal{M}_x\simeq \mathcal{O}_{X,x}^\times\times\Bbb{N},$ and the origin gets embellished with a picture of $\operatorname{Spec}\Bbb{N}.$

It seems to me that the picture really decorates each point $x\in X$ with a picture of $\operatorname{Spec}(\overline{\mathcal{M}}_x),$ where $\overline{\mathcal{M}} = \mathcal{M}/\mathcal{O}_X^\times$ is the characteristic of the log structure.

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Stahl
Stahl
  • 1.3k
  • 8
  • 19

Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.)

In this case, $\operatorname{Spec}\Bbb{C}[\Bbb{N}] \simeq \operatorname{Spec}\Bbb{C}[t],$ so the pictured disk is the complex affine line, represented as a subset of the complex plane. We can also compute $\mathcal{M}_x$ for each $x\in\operatorname{Spec}\Bbb{C}[\Bbb{N}].$ At the points other than the origin, the log structure on $\operatorname{Spec}\Bbb{C}[\Bbb{N}]$ is trivial in the sense that $\mathcal{M}_x \simeq\mathcal{O}_{X,x}^\times.$ However, at the origin, we have $\mathcal{M}_x\simeq \Bbb{C}^\times\times\Bbb{N},$ and the origin gets embellished with a picture of $\operatorname{Spec}\Bbb{N}.$

It seems to me that the picture really decorates each point $x\in X$ with a picture of $\operatorname{Spec}(\overline{\mathcal{M}}_x),$ where $\overline{\mathcal{M}} = \mathcal{M}/\mathcal{O}_X^\times$ is the characteristic of the log structure.