The irreducibility conjecture for Severi varieties on general polarized K3 surfaces

Let $(Y,L)$ be a general polarized $K3$ surface of genus $g\geq 2$. For a fixed integer $0\leq \delta\leq g$, let $V_{\delta}(Y,L)$ denote the Severi variety of integral curves in $|L|$ having exactly $\delta$ nodes as singularities.

The K3 Severi irreducibility conjecture. For every fixed $0\leq \delta\leq g-1$, the Severi variety $V_{\delta}(Y,L)$ is irreducible.

The bound $\delta\leq g-1$ is necessary: for $\delta=g$, the finitely many rational curves in $|L|$ give a reducible Severi variety. The conjecture was open in the source; the paper proves irreducibility for general primitively polarized $K3$ surfaces when $g\geq 5$ and $0\leq \delta\leq g-4$, while connectedness is proved for all $0\leq \delta\leq g-1$.