Morphisms To And From A Log Point

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Log points don’t behave like ordinary points in a few ways.

  • \(\Hom(W_{Q'}, W_Q)\) is large rather than a singleton
  • fs fiber products of log points can be disconnected
  • a morphism \(X\to W_Q\) is more than a point of \(\text{Log}\) with a lift, because it forces the structure map to vanish on the image of \(Q\setminus \{0\}\).

All of these phenomena can be seen from log morphisms to and from log points.

Morphisms between log points

Morphisms to a Log Point

Lemma

Let \((X, M_X)\) be an fs log scheme and \(W_Q = \Spec(Q\to k)\) a sharp fs log point (i.e. \(Q\) is a sharp fs monoid). Then

\begin{align*} \Hom(X, W) = \{\theta:Q\to \Gamma(X, M_X) ~ | ~ \alpha_X(\theta(q)) = 0 ~ \forall q\neq 0\} \end{align*}

Or equivalently, using \(\Hom(X, \Spec k[Q]) = \Hom(Q, \Gamma(X, M_X))\), a log map \(X\to W_Q\) is a morphism to the toric log scheme \(\Spec k[Q]\) whose underlying scheme map lands in the origin.

Equivalently still, using \(\Hom(X, \mathcal A_Q) = \Hom(Q, \Gamma(X,\overline{M}_X))\), a log morphism \(X\to W_Q\) is a morphism to \(\mathcal A_Q\) landing in \(BT_Q\) where \(T_Q = \Spec k[Q^{gp}]\) together with a trivialization of the induced \(T_Q\)-torsor.

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