12 problems

Let $X$ be a smooth projective variety and let $D$ be a reduced effective anticanonical divisor on $X$ with at most normal crossings singularities. Anticanonical complement conject…

Let $(Y/\boldsymbol{\Delta})$ be a smooth projective Campana constellation supported by a firmament $\boldsymbol{\Gamma}$. Campana's conjecture. Points on $Y$ integral with respect…

Generalized potential-density conjecture. Under these assumptions, rational points on $V$ are potentially dense.

Let $f:X\to C$ be a fibration over a function field with $K$-core factorization $f=g_f\circ c_f$, and suppose the orbifold fibration … is not split. Conjecture IV{FF}. There should…

Let $f:X\to C$ be a fibration over a function field, with $K$-core factorization $f=g_f\circ c_f$ and base orbifold $(C(f)/\Delta(c(f)))$. Conjecture III{FF}. The $K'$-rational poi…

Let $c_X:X\to C(X)$ be an admissible representative of the core of a projective manifold $X$, defined over a field $K$ finitely generated over $\mathbb Q$. Conjecture IVA. The sour…

Let $X$ be a projective manifold defined over a finitely generated field $K\subset\mathbb C$. Call $X$ arithmetically special when $X(K')$ is Zariski dense for some finite extensio…

Let $M_B(X_s,r)$ be the Betti moduli space associated with a fiber in the paper, and call its $\overline{\mathbb{Z}}$-valued points integral points. Potential-density conjecture. T…

Let $X$ be a smooth projective variety over a number field $K$. A variety is weakly special if no finite étale cover of $X_{\overline{K}}$ rationally dominates a positive-dimension…

Let $K$ be a number field, let $X$ be a smooth (quasi-)projective variety over $K$, and let $Z$ be a subvariety of codimension at least $2$. Assume that the rational points on $X$…

Let $X$ be a projective variety defined over a number field. A variety is weakly special if some (equivalently, any) desingularization admits no dominant rational map to a positive…

Let $X$ be a $\mathbf{b}{Q}$-Gorenstein variety with a canonical divisor $K_{X}$ big. A closed subvariety $Y\subset X$ is potentially dense if, after a finite extension of the base…