3 problems

Picard-rank finiteness conjecture. The Picard number of $\mathcal{Y}_s$ is greater than or equal to $2$ for at most finitely many values of $s\in\mathbb{P}^1(\mathbb{C})$.

Fix an integer $d\geq 5$. Let $\operatorname{NL}_d$ be the Noether–Lefschetz locus, and let a component be special when its codimension is strictly less than the maximal value…

Let $V$ be the parameter space of smooth complex surfaces of degree $d$ in $\mathbb{P}^{3}$, and let $\operatorname{NL}_d$ be the Noether-Lefschetz locus, whose components of codim…