5 problems

Moduli conjecture. For every $m\geq 3$ and every $n\geq 0$, there exists such an $S$ and $f$ for which the fibers over the $k$-points of $S$ are all non-isomorphic.

The conjecture. For every field extension $F$ of $k$,

Asok–Dorant's conjecture. If $k$ is an infinite field and $X$ is a smooth $\mathbb A^1$-connected scheme, then $U$ is also $\mathbb A^1$-connected. This conjecture was answered aff…

Codimension-two ${\mathbb A}^1$-connectedness conjecture. The scheme $U$ is ${\mathbb A}^1$-connected.

Let $F$ be the base field, let $X$ be a smooth variety over $F$, and write … to $X$ in this category. Define an equivalence relation on $X(F)$ generated by elementary $^1$-homotopi…