44 problems

Let $X$ be a Fano variety and $C$ a smooth projective curve. A Manin component is an irreducible component of the relevant morphism space distinguished from exceptional components.…

Let $(X, D_\epsilon = \sum_{\alpha \in \mathcal A} \epsilon_\alpha D_\alpha)$ be a klt Fano Campana orbifold over a number field $F$, let $L$ be a big $\mathbb Q$-divisor on $X$, a…

Modern formulation of Manin's conjecture. There exists a finite set of thin maps $f_i:Y_i\to X$ such that

Let $(\underline{X},\Delta)$ be a log pair and set $\underline{U}=\underline{X}\setminus\operatorname{Supp}(\Delta)$. Assume that $\underline{U}$ is geometrically $\mathbb A^1$-con…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $\mathsf{S}_{\lambda,\mathbb Z}$ be the corresponding integral section classes…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $c(\mathcal X,\lambda)$ be the leading constant in the all-height Manin conjec…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, let $M_{\alpha}$ be the unique Manin component for sufficiently large…

Let $\pi:\mathcal X\to B$ be a good Fano fibration and let $\mathsf Z\subset\mathcal X_{\eta}(K(B))$ be the exceptional set. A properly constructible thin set is a finite union of…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $q$ be the cardinality of the constant field, and suppose that $\mathcal X_{\eta}(K(B))$ is not thin. With…

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration, let $\mathcal{X}_{\eta}$ be its generic fiber, and let $\mathsf Z$ be the union of the images of all breaking thin maps into…

Let $X$ be a smooth Fano variety over a number field $K$, and let … be a boundary divisor with weights $m_i\in\mathbb{N}$ and prime divisors $D_i$. Let $S$ be a finite set of place…

Let $A,B\in\mathbb{Z}$ satisfy $4A^3-27B^2\neq 0$, let $Q(u,v)$ be a positive-definite binary quadratic form, and define … For rational points represented by integral $(x,y,u,v)$ w…

Let $S$ be a split quartic del Pezzo surface over the finite field $k=\mathbb F_q$, let $B$ be the base curve, and let $\alpha$ be an ample curve class with invariants $a,a'$ and…

Let $(X,D_\epsilon)$ be the Campana orbifold, let $L$ be the underlying big divisor, and let $F$ and $S$ be as above. Let $\alpha((X,D_\epsilon),L)$, $a((X,D_\epsilon),L)$, and…

Peyre's refined conjecture. There exists a thin subset $Z\subset X(K)$ such that, for $U=X\setminus Z$, one has

The generalized Manin conjecture. If $\mathcal{X}(F)$ is Zariski dense in $\mathcal{X}$, then there exists a thin subset $Z\subset\mathcal{X}(F)$ and a positive constant $C$ such t…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration, and let $\operatorname{Sec}(\mathcal{X}/B)$ be its moduli space of sections. A component is accumulating when it is governed by the…

Let $\kappa$ be a field of characteristic $0$, let $X$ be a smooth Fano variety over $\kappa$, and let $f:Y\to X$ be a thin map from a smooth projective variety. Call $f$ a breakin…

Let $\kappa$ be a number field and let $X$ be a smooth Fano variety over $\kappa$. Let $\mathcal{L}=(L,\{\|\cdot\|_v\})$ be an adelically metrized big and nef $\mathbb{Q}$-divisor,…

Let $v$ be a non-archimedean place of a number field $F$, let $k_v$ be the residue field with $q_v=\#k_v$, and let $\pi_v:X_v\to B_v$ be the reduction of a Fano fibration. Let…

Let $B$ be a smooth projective curve over $\mathbb{F}_q$, let $\pi:\mathcal{X}\to B$ be a Fano fibration with an adelic metrization on its relative canonical bundle, and let…

Geometric Manin's conjecture. Sections of Fano fibrations over any ground field are governed by two key principles: pathological families of sections are controlled by the Fujita i…

Manin–Peyre conjecture. There should exist a thin subset $\Omega\subset V(\mathbb{Q})$ such that

Let $X$ be a smooth projective variety over a field of characteristic $0$, and let $L$ be a big and nef $Q$-Cartier divisor. Define the Fujita invariant by … When $L$ is nef but no…

Browning–Van Valckenborgh prediction. One has