18 problems

Let $M\subset \mathbf{P}^{n-1}(\mathbf{Q})$ be a projective thin set of type II. For \text{\boldmath{x}}=[x_1:\hdots:x_n]\in \mathbf{P}^{n-1}(\mathbf{Q}), define the height … H…

Let $M\subset\mathbb{P}^{n-1}(\mathbb{Q})$ be a thin set of type II, and let $N_{\mathbb{P}^{n-1}}(M,B)$ denote the number of points of $M$ of height at most $B$. Projective type I…

Let $M$ be a thin set in $\mathbb{P}^{n-1}(\mathbb{Q})$. Write $N_{\mathbb{P}^{n-1}}(M,B)$ for the number of points of $M$ of height at most $B$. Serre's conjecture. For some…

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 $\mathcal{X}_{\eta}$ be its generic fiber, and let $\mathsf Z$ be the union of the images of all breaking thin maps into…

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 a variety be equipped with the setting of Manin's conjecture, and let its thin set be the exceptional subset occurring in the counting problem. A subset is non-Zariski dense if…

The source discusses an orbifold of interest and its set of Campana points. A set is thin if it belongs to the class of exceptional sets in the relevant Hilbert-property framework.…

Let $(X,D_\epsilon)$ be a Fano orbifold over a number field $F$, and let $(\mathcal X,\mathcal D_\epsilon)$ be a good integral model over $\operatorname{Spec}\mathcal O_{F,S}$, whe…

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 the covers referred to in Step (6) be the covers arising in the construction of the exceptional set for Manin's conjecture. Lehmann–Sengupta–Tanimoto conjecture. These covers m…

Let $X$ be a geometrically rationally connected, geometrically integral variety over a number field $F$, with $L$ a big and nef divisor. Let $Z_0$ be the initial exceptional subset…

Geometric exceptional-set conjecture. The exceptional set in Manin's Conjecture should coincide with the geometrically defined subset $Z\subset X(F)$ constructed above; equivalentl…

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

Let $X$ be a smooth uniruled variety over a number field $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. For every $F$-morphism $f:Y\to X$ from a smooth projective var…

Let $X$ be a smooth projective variety over a number field $F$ with ample anticanonical class $-K_X$. Let $\mathcal{L}=(L,\|\cdot\|)$ be a big and nef adelically metrized line bund…

Let $G\subset S_n$ be a transitive subgroup, let $X=(\mathbb{P}^m)^n/G$, and assume $m\cdot\operatorname{ind}(G)\ge 2$. Let $X(K)^{\operatorname{prim}}$ denote the primitive $K$-po…

Let $S$ be a variety over a number field $K$. A subset of $S(K)$ is thin if it is contained in the image of $T(K)\to S(K)$ for a generically finite morphism without a rational sect…