4 problems

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

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…

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…

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.