29 problems

Weak Manin's conjecture. There exists a non-empty Zariski open subset $U\subset X$ such that, for every $\epsilon>0$,

Let $S$ be a Del Pezzo surface over $\mathbb{Q}$ with at most rational double points, and let $S^\circ\subset S$ be a dense open subset in the Zariski topology. Let $-K_S$ denote t…

Let $X$ be a smooth Fano variety over $\mathbb C$ with Brauer group $Br(X)$. Let $C$ be a smooth connected genus $g$ curve. A component of $\operatorname{Mor}(C,X,\alpha)$ is under…

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 $k$ be a number field and let $X$ be a smooth Fano variety over $k$, so that $-K_X$ is ample. Fix an adelic metrization of the anticanonical bundle $comathcal O(-K_X)$, with as…

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 $A,B\in\mathbb{Z}$ 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 $\Delta\in\mathbb{Q}^\times$ satisfy $\sqrt{-\Delta}\notin\mathbb{Q}$, and let $f(z)\in\mathbb{Z}[z]$ be a separable polynomial of degree $3$ or $4$ that decomposes as a produc…

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…

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 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,…

Browning–Van Valckenborgh prediction. One has

Let $K$ be a number field, let $(X,D_{\epsilon})$ be a smooth klt Fano Campana orbifold over $K$, and let $(\mathcal{X},\mathcal{D}_{\epsilon})$ be a regular $\mathcal{O}_{K,S}$-mo…

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

Let $X$ be a geometrically uniruled, geometrically integral, smooth projective variety over a number field $F$, and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Assume that…

Let $X$ be a geometrically uniruled smooth projective variety defined over a number field $k$, let $L$ be a big and nef divisor on $X$, and let … be a height function. For a suitab…

Let $X$ be a smooth projective geometrically integral Fano variety over a number field, and set $L=-K_X$. For a thin map $f:Y\to X$ from a smooth projective geometrically integral…

Lehmann–Tanimoto's conjecture. Considering all such morphisms $f$ for which $f^*L$ is not big or

Strong Manin's conjecture. For any $\mathbb{Z}$-curve class $\alpha$ contained in the relative interior of $F(X,L)$, there is at most one Manin component parametrizing curves of cl…

Let $X$ be a Fano variety over $\mathds{Q}$ with at worst log terminal singularities and a Zariski dense set of rational points. Let $H\colon X(\mathds{Q}) \to \mathds{R}_{>0}$ be…