13 problems

Let $H\subset\mathbb{A}^n$ be the hypersurface defined by a polynomial $F\in\mathbb{Z}[X_1,\ldots,X_n]$ of degree $d\geq 2$, with absolutely irreducible leading form. Let…

Let $V\subset\mathbb{P}^N$ be an irreducible projective variety, and let $C(V)$ and, where relevant, $\gamma(V)$ be the constants in the Projective Dimension Growth Conjecture. Uni…

Let $V\subset\mathbb{P}^N$ be an irreducible projective variety of degree $d\geq 2$ defined over $\mathbb{Q}$, and let $N_{\mathrm{proj}}(V,B)$ denote the number of rational points…

Dimension Growth conjecture. The number of such points should grow no faster than

Let $Z$ be an irreducible projective variety in $\mathbb{P}^{n-1}(\mathbb{Q})$ of degree $D\geq 2$, and let $Z_H(B)$ denote the number of rational points of $Z$ of height at most…

Let $X\subseteq\mathbb{P}_{\mathbb{Q}}^{M-1}$ be an irreducible projective variety of degree at least two defined over $\mathbb{Q}$. Let $N_X(B)$ be the number of rational points o…

Let $X\subseteq {\mathbb P}_{\mathbb Q}^n$ be an integral hypersurface of degree $d$, and let $N(X,B)$ denote the number of rational points on $X$ of height at most $B$. Uniform di…

Serre's dimension growth conjecture. There is a constant $\mathfrak{z}=\mathfrak{z}(\mathscr{V})>0$ such that

Serre's dimension growth conjecture. There is a constant $c>0$ such that

Let $X$ be an integral projective variety in projective space, let $d$ be its degree, and let $N(X;B)$ denote the number of rational points of $X$ of height at most $B$. Dimension…

Let $F_2$ be the free group on two generators, and let $\mathbb{Z} \wr F_2$ denote their wreath product. The exponential dimension growth conjecture asserts that the dimension grow…

Let $X$ be a smooth projective variety with non-negative Kodaira dimension. The paper compares its upper-bound theorem, which gives estimates in terms of $\delta(X,L)$, with the di…

Serre's dimension growth conjecture. There is a constant $c=c_V>0$ such that