7 problems

Zariski-density conjecture. If $C$ contains an irreducible one-dimensional point, then the set of modular points contained in $C$ is Zariski dense.

Let $\mathcal{X}(\Gamma,\mathsf{SO}(p,p+1))$ be the character variety of representations of the surface group $\Gamma$ into $\mathsf{SO}(p,p+1)$. In addition to the components defo…

Let $K$) be a number field, let $X/K$ be a smooth projective variety, and let $f:X\to X$ be an endomorphism defined over $K$. For $P\in X$, write…

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic zero, and let $f:X\dasharrow X$ be a dominant rational self-map. Write $k(X)^f$ for the fie…

Zariski-density conjecture. Every irreducible non-affine Tits–Vinberg representation is Zariski dense in the real algebraic group corresponding to its quadratic form, as in the kno…

Zariski dense orbit conjecture. Either there exists $\alpha\in X(K)$ whose orbit under $\Phi$ is well-defined and Zariski dense in $X$, or there exists a non-constant rational func…

Matsuzawa–Meng–Zhang–Sano conjecture. The set $Z_f(d)$ is not Zariski dense in $X$. This conjecture predicts that, among points of bounded degree, points whose arithmetic degree is…