10 problems

Let $q$ be a prime power with $q>25$, and let … be the unit sphere over ${\mathbb F}_q$. An ${\mathbb F}_q$-primitive point is a point on this sphere whose coordinates generate the…

Linear volume-bound conjecture. There is a constant $\delta_n$ such that, for every such $X$,

Browning–Heath-Brown–Salberger's conjecture. For every $\varepsilon>0$,

Differential-equivalence conjecture. Differential equivalence should imply that $X$ and $Y$ are isomorphic.

For $q=2$, let $X(m,i)$ denote the characteristic-zero variety $X_\infty(m,i)(0)_{c0}$. Its irreducible components are described using rooted weighted binary forests with $i$ nodes…

Let $K$ be a field, let $\bar{K}$ be an algebraic closure of $K$, and let $X_K\subset K^n$ be a nonsingular $K$-rational algebraic variety. Assume that…

Product-one variety conjecture. The theorem's conclusions hold for every $m\geq 3$ and $r\geq 2$: (i)…

Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic zero, and let $\varphi:X\to X$ be an étale endomorphism. Generalized Jacobian conject…

The Main Question. Every algebraic variety $X$ is birational to a variety $X^{\rm m}$ that is either of semi-negatively curved or Kodaira--Iitaka type, or of positive fiber type.

The Main Conjecture. Every variety can be built up from these special varieties in a rather clear process.