4 problems

Yasuda's smoothness conjecture. If $Z\nsubseteq Y$, then ${{\mathcal N}}_n(X)$ is smooth at $[Z]$.

Eventual smoothness conjecture. There is an integer $n_0$ such that ${\mathcal N}_n(X)$ is nonsingular for every $n\ge n_0$.

Let $K$ be an algebraically closed field with a non-archimedean valuation $v\colon K\to\Gamma$, let $X$ be as above, and let $\Delta_n$ and $\Gamma^\times$ denote the domains used…

Let $K$ be an algebraically closed field of characteristic $0$, and let $X$ be a reduced separated closed subscheme of finite type over $K$, of pure dimension $r$. Let $X^*$ be the…