4 problems

Singular-locus density conjecture. There exists a constant $c>0$ such that, whenever $d$ and $p$ are sufficiently large,

Let $X$ be a smooth quasiprojective variety of positive dimension over $k=\mathbb F_q$. For sufficiently large $n$, consider embeddings … whose connected components are not mapped…

Let $X$ be a quasiprojective subscheme of ${\mathbb P}^n_{\mathbb Z}$ that is regular of dimension $m$, and let $\zeta_X$ denote the zeta function of $X$. Poonen's conjecture. The…

Toric Bertini conjecture. There is a finite union $\mathcal{E}$ of proper subtori of $\mathbb{G}_{\rm m}^n$ such that, for every subtorus $T$ with