6 problems

Let $V$ be a proper, smooth projective variety over $\mathbb{Q}$, let $n\geqslant 1$, and let $f:V\to \mathbb{P}_{\mathbb{Q}}^n$ be a dominant morphism with geometrically integral…

Let $X$ and $Y$ be smooth projective varieties over $\a0\mathbb{Q}$, and let $\phi:X\to Y$ be a dominant map with geometrically integral generic fibres. For $B>0$, let $H$ be a hei…

Let $f({\boldsymbol x})\in {\mathbb Z}[x_1,x_2,\ldots,x_n]$ be a homogeneous polynomial of degree $d$, and let $V\subset {\mathbb P}^{n-1}$ be the projective variety defined by…

Conjectural asymptotic. There is a constant $c>0$ such that

Loughran–Sofos conjecture. If the set being counted is non-empty and the fibre over every codimension $1$ point of $\mathbb{P}^n$ contains an irreducible component of multiplicity…

Let $X$ satisfy the stated assumptions, let $H$ be an anticanonical height function, and let $\pi:Y\to X$ be a proper surjective almost smooth morphism with geometrically integral…