19 problems

Let $\f:\mathbb{P}^2\dashrightarrow\mathbb{P}^2$ be a rational map, and let $\lambda_1=\lim_{n\to\infty}\operatorname{deg}(\f^n)^{1/n}$ be its first dynamical degree. The algebraic…

Let $T_{a,b,c,d}$ be a truly skew pentagram map, with parameters $(a,b,c,d)$ as in the paper, and let its dynamical degree be denoted by $\rho(T_{a,b,c,d})$. Khesin–Soloviev's conj…

Let $T$ be an irreducible variety, let $X/T$ be a family of smooth irreducible projective varieties, and let $f:X/T\dashrightarrow X/T$ be a family of dominant rational maps. Write…

Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map, and let … be its dynamical degree. Bellon–Viallet conjecture. The dynamical degree is an algebraic integ…

Let $X$ be a projective variety over a number field $K$, let $f:X\dashrightarrow X$ be a dominant rational self-map, and let $V$ be an irreducible subvariety of dimension $k$. Supp…

Let $F:\mathbb{C}^4\rightarrow\mathbb{C}^4$ be the birational map … let $\phi(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$, and for generic $c\in\mathbb{C}$ let $X_c\subset\mathbb{P}^4$ be the…

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a dominant rational self-map. Assume that…

Let $X$ be a smooth projective variety defined over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a birational self-map. Its first d…

Matsuzawa–Matsumoto–Sano–Zhang conjecture. The set $Z_f(d)$ is not Zariski dense in $X$. This concerns the distribution of algebraic points of bounded degree whose arithmetic degre…

Let $K$ be a number field, let $X$ be a projective variety over $K$, and let $f:X\to X$ be a surjective endomorphism. For a positive constant $d$, let $X(d)$ denote the degree-boun…

Let $X$ be a smooth projective variety defined over an algebraically closed field $\mathbf{k}$, and let $f\colon X \vdash X$ be a dominant self-correspondence. For a prime $\ell$ d…

Let $X$ be a smooth projective variety of dimension $d$, and let $f\colon X\dashrightarrow X$ be a dominant rational self-map. For each integer $1\leqslant k\leqslant d$, let…

Let $X$ be a smooth projective variety, let $f\colon X\dashrightarrow X$ be a dominant rational map, and let $V\subset X$ be a subvariety of dimension $k$. Let $\alpha_{k+1}(f;V)$…

Let $X$ be a smooth projective variety, let $f\colon X\dashrightarrow X$ be a dominant rational map, and let $V\subset X$ be a subvariety of dimension $k$. Let $\lambda_k(f;V)$,…

Let $K$) be an algebraically closed field with a suitable theory of height functions, let $X$ be a smooth projective variety over $K$, let $\varphi:X\dashrightarrow X$ be a dominan…

Let $X$ be a nonsingular irreducible algebraic variety defined over $\overline{\mathbb{Q}}$, and let $\varphi:X\dashrightarrow X$ be a dominant rational map defined over…

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined over $\overline{\mathbb{Q}}$, with dynamical degree $\delta_\varphi>1$. Let $\ell_\varphi$…

Polynomial correction conjecture. The infimum $\ell_\varphi$ exists and is an integer satisfying $0\leq\ell_\varphi\leq N$.

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined over $\overline{\mathbb{Q}}$. For a point $P\in\mathbb{P}^N(\overline{\mathbb{Q}})$ whose o…