4 problems

Harris's conjecture. The locus $\operatorname{NL}_d$ has only finitely many exceptional components.

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 $s\in\mathbb{P}^1(\mathbb{C})$.

Let $(a,b)$ be an admissible pair of degree $d$ and length $t$, and let $(a^{min}_{d,t},b^{min}_{d,t})$ and $(a^{max}_{d,t},b^{max}_{d,t})$ be the pairs defined in the preceding co…

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…