6 problems

Let $V$ be a smooth weak Fano variety over a number field $F$ such that $V(F)$ is Zariski dense, and let $H_F$ be a relative adelic height on the anticanonical line bundle. Manin–P…

Batyrev's conjecture. There exists a polynomial $P(d)$ in $d$ such that the number of components of $\operatorname{Mor}(\mathbb P^1,X)$ parametrizing rational curves of $L$-degree…

Let $X$ be a nonsingular toric weak Fano $d$-fold. Two such varieties are weakly-F-equivalent if they are connected by a sequence of equivariant blow-ups, blow-downs, and flops thr…

Let $V$ be a smooth weak Fano variety over a number field $F$, let $\Gamma_F$ be the absolute Galois group, and write…

Let $X$ be a smooth, projective, geometrically integral, weak Fano variety over $\mathbb{Q}$. Let $U \subseteq X$ be an open subset and let…

Batyrev's conjecture. Then $h(m\alpha)$, considered as a function of $m$, is bounded above by a polynomial $P(m)$ whose degree is the largest relative dimension of a map