18 problems

Let $X$ be a variety over a number field $F$, let $D$ be a divisor on $X$, and let $L$ be a big divisor. For a finite set of places $S$ and a model $(\mathcal U,\mathcal L)$ of a d…

Height comparison conjecture. There exist a family of $v$-adic metrics on $K_Y$ and constants $c_2>c_1>0$ such that

Strong saturation conjecture. The complex analytic variety $V({\bf C})$ is ${\cal L}$-primitive.

Asymptotic arithmetic fibration conjecture. The set of ${\cal L}$-targets of $V$ forms an asymptotic arithmetic ${\cal L}$-fibration.

Stacky Batyrev–Manin conjecture. There exists a thin set $T\subset\mathcal X\langle F\rangle$ and $C>0$ such that

Let $V$ be a smooth weak Fano variety over a number field $F$ such that $V(F)$ is Zariski dense, and let $H_F$ be a relative adelic height on the anticanonical line bundle. Manin–P…

Dimension growth conjecture. For every $\varepsilon>0$,

Let $F$ be a number field, and let $V$ be a smooth projective variety over $F$ that is strongly $\mathcal{L}$-saturated for a metrized line bundle $\mathcal{L}$. Let…

Wooley's conjecture. There exists a constant $c>0$ depending only on $X$ such that

Let $(X,D)$ be a klt Campana orbifold over $K$ with a good integral model $(\mathcal{X},\mathcal{D})$ over $\mathcal{O}_{K,S}$. Let $H_{\mathcal{L}}$ be the height associated with…

Batyrev–Tschinkel conjecture. Then

Batyrev–Manin conjecture. There exists an open subvariety $U \subseteq V$ such that

Let $(X,D_\epsilon)$ be a Campana orbifold over a number field $F$ such that $-(K_X+D_\epsilon)$ is ample, and let $(\mathcal X,\mathcal D_\epsilon)$ be a good integral model over…

Batyrev–Manin Conjecture C'. If $U$ is sufficiently small, then

Refinement of the Manin–Peyre conjectures. There exist a Zariski open subset $U$ of $V$, a polynomial $P_{U,H}$ of degree $\rho$, and $\delta\in(0,1)$ such that, for $B\geq 1$,

Let $X$ be a smooth projective variety over a number field $K$, with Néron–Severi space $\mathrm{NS}(X)_{\mathbb{R}}$ and pseudo-effective cone $\mathrm{PEf}(X)$. Assume…

Let $X$ be a canonical Fano variety over a number field $K$, with a metrized anti-canonical divisor and associated height function $H$. For a suitable subset $U\subset X(K)$, write…

Loughran's conjecture. There exist an open subset $U\subset X$ with $\mathscr{B}\subset\operatorname{Br}U$ and a constant $c_{U,\mathscr{B},H}>0$ such that