De Volder–Laface conjecture for linear systems on K3 surfaces

Let $X$ be a K3 surface with $H^2=n\in2{\mathbb{Z}}$, and let ${\mathcal{L}}={\mathcal{L}}^n(d,m_1,\dots,m_h)$ be the system of curves in $|dH|$ through general points $P_1,\dots,P_h$ with multiplicities at least $m_1,\dots,m_h$. De Volder–Laface conjecture. The following assertions hold: (i) ${\mathcal{L}}$ is special if and only if ${\mathcal{L}}={\mathcal{L}}^4(d,2d)$ or ${\mathcal{L}}={\mathcal{L}}^2(d,d^2)$ with $d\geq2$; (ii) if ${\mathcal{L}}$ is non-empty, its general divisor has exactly the imposed multiplicities at the points; (iii) if ${\mathcal{L}}$ is non-special and has a fixed irreducible component $C$, then either ${\mathcal{L}}={\mathcal{L}}^2(m+1,m+1,m)=mC+{\mathcal{L}}^2(1,1)$ with $C={\mathcal{L}}^2(1,1^2)$, or ${\mathcal{L}}=2C$ with $C\in\{{\mathcal{L}}^4(1,1^3),{\mathcal{L}}^6(1,1,2),{\mathcal{L}}^{10}(1,3)\}$, or ${\mathcal{L}}=C$; (iv) if ${\mathcal{L}}$ has no fixed component, then either its general element is irreducible or ${\mathcal{L}}={\mathcal{L}}^2(2,2)$. The conjecture gives a detailed classification of special and reducible systems on these K3 surfaces.