13 problems

Let $E$ be an elliptic curve over a number field. Its essential dimension is denoted by $\operatorname{ed}E$. Infinite essential-dimension conjecture for elliptic curves. Then … Th…

Let $k$ be a field, let $G$ be a finite group, and let $X$ be a $k$-variety with a faithful $G$-action. For every prime $p$, let $G_p$ be a Sylow $p$-subgroup of $G$. Duncan's conj…

Let $A$ be an abelian variety over a field $L$ of characteristic zero, and let $[p]:A\to A$ denote multiplication by a prime $p$. Brosnan's conjecture. For every prime $p$, the map…

Let $k$ be an algebraically closed field, let $G$ be a connected reductive linear algebraic group over $k$, let $R$ be a regular local ring containing $k$, and let … be a $G$-torso…

Let $k$ be a field, let $d\geq 1$, and let $D$ be a central division algebra of degree $d$ over $k$. For a positive rational number $r$, write $\operatorname{Mod}_{D,r}$ for the fu…

Let $k$ be a field of characteristic $p>0$, let $n$ be a natural number, and let $\mathrm{C}_{p^n}$ be a cyclic group of order $p^n$. Ledet's conjecture. … This conjecture concerns…

Verschiebung lower-bound conjecture. The essential dimension of $G$ over $k$ satisfies

Let $k$ be a field of positive characteristic, and let $G$ be a finite commutative unipotent group scheme over $k$. Let…

Generalized fixed-point conjecture. If $\operatorname{rdim}_k(H_i)\leq d$ for all $i=1,\dots,r$, then there exists a $d$-dimensional $k$-variety $X$ with a faithful $G$-action and…

Dolgachev's inequality.

Sylow-versality conjecture. If $X$ is $G_p$-versal for every prime $p$, then $X$ is $G$-versal.

Let $\mathcal{X}$ be an amenable stack over $k$. Let $L$ be an extension of $k$, and let $\xi$ be an object of $\mathcal{X}(\operatorname{Spec} L)$. Here $\operatorname{ed}_k\xi$ d…

Let $G$ be a finite group acting faithfully on a variety $X$. For each prime $p$, let $G_p$ be a Sylow $p$-subgroup of $G$. Sylow versality conjecture. The variety $X$ is $G$-versa…