# Compactifications of log schemes

Let $S$ be a quasi-separated, quasi-compact scheme. It is a well known theorem going back to Nagata that any separated, finite type morphism $f:X\to S$ factors as $X \to \overline{X}\to S$, where the first map is an open immersion and the second one is proper.

In this generality this is Thm 4.1 in B. Conrad - "Deligne's notes on Nagata compactifications".

In "Crystalline fundamental groups II" (Conjecture 2.2.21) Shiho conjectures that a similar statement should hold in the log setting, at least in the case where $S=(Speck, N)$ is a log point ($k$ is a perfect field). More precisely he conjectures that for any fine log scheme $X\to S$ there exists a compactification $X'\to S$, which is also a fine log scheme.

I was wondering whether there are any results in this direction by now?

• Related: would this follow from being able to find a compactification where $\overline{X} \setminus X$ gives a Cartier divisor? – Joe Berner Apr 6 '18 at 13:18