Q: What does it mean to be log smooth over the standard log point?
Suppose we have a morphism \(f:X\to S=\Spec(\mathbb N\to k)\) which is log smooth. This means that, etale locally at least, we have charts \(X\to \Spec k [P]\) for \(X\)fitting into a commutative diagram
noting that \(S\to \Spec k[\mathbb N]\) is the global chart for \(S\). We know what \(\Spec k[\mathbb N]\) is, it’s \(\mathbb A^1\) with the divisorial log structure at a single point, and since the above horizontal arrows are charts and hence strict, the standard log point \(S\) must map to this point.
The second portion of log smoothness requires that the induced morphism \[X\to S\times_{\mathbb A^1}\Spec k[P]\] is smooth in the traditional sense. This fiber product is the log divisor of \(\Spec k[P]\), call it \(D\), so we must have a smooth map \(X\to D\).
We then must study two questions:
- Under what assumptions is the map \(X\to D\) smooth?
- What are the possible morphisms \(\Spec k[P]\to \Spec k[\mathbb N]\), and what do they look like over the nontrivial log point? That is, what can we obtain for \(D\)?