3 problems

Let $X\in\mathbbmss{Sch}_{\kappa}$ be affine. Suppose that $X\to\mathfrak{n}$ and $X\to\mathfrak{m}$ are smooth morphisms, where $\mathfrak{n}\hookrightarrow\mathfrak{m}$ are eleme…

Let $\mathfrak{n}$ be a fat point over $\kappa$, and let $X$ be a scheme smooth over $\mathfrak{n}$. Let $\mathfrak{x}$ be an admissible arc in $\operatorname{\bf Arc}_{\kappa}$ co…

Strictness conjecture. If $\mathfrak{x}$ has positive dimension, each inclusion in this chain is strict. The smooth schemes form the basic class of $\mathfrak{x}$-stable schemes, a…