O'Grady's L conjecture for degree-2 polarized deformations of Hilbert schemes of K3 surfaces

Let $n\geq 2$, and let $(X,H)$ be a degree-$2$ polarized irreducible symplectic variety, meaning that $X$ is an irreducible symplectic manifold deformation equivalent to $(K3)^{[n]}$ and $H$ is an indivisible ample divisor with $(c_1(H),c_1(H))=2$. Let ${\cal Q}_n^0$ be the open parameter space of such polarized varieties in a fixed Hilbert scheme, and let $(X_t,H_t)$ denote the variety corresponding to $t\in {\cal Q}_n^0$. O'Grady's L conjecture. There exists an open dense subset $U_n\subset {\cal Q}_n^0$ such that, for every $t\in U_n$, the linear system $|H_t|$ has no base-locus and the map $f_t\colon X_t\to |H_t|^{\vee}$ has degree $2$ onto its image $Y_t$. More precisely, there is an involution $\phi_t\colon X_t\to X_t$ such that $f_t$ factors as

$$X_t\overset{\pi_t}{\longrightarrow}X_t/\langle\phi_t\rangle\overset{\nu_t}{\longrightarrow}Y_t,$$

where $\pi_t$ is the quotient map and $\nu_t$ is the normalization map; if $h_t:=c_1(H_t)$ and $R_{h_t}(v)=-v+(v,h_t)h_t$, then $H^2(\phi_t)=R_{h_t}$. The conjecture predicts the generic behavior of degree-$2$ polarizations on deformations of $(K3)^{[n]}$, extending the simple linear-system behavior of polarized K3 surfaces; the supplied text gives examples but no resolution.