Flips for spaces of quadrics on del Pezzo varieties

For a cubic hypersurface $X$, work of Galkin–Shinder and Voisin shows the existence of a birational map relating the Hilbert scheme of two points $X^{[2]}$ with a certain projective bundle over $X$. Belmans–Fu–Raedschelders show that this is a standard flip, a particularly nice type of birational map inducing decompositions of derived categories. We show that this geometric construction extends to produce standard flips for Hilbert schemes of quadrics on various higher-dimensional del Pezzo varieties of degree at least 3, including cubics, intersections of two quadrics, and linear sections of $\mathrm{G}\mathrm{r}(2,5)$. The resulting construction also generalizes results of Chung–Hong–Lee for quintic del Pezzo varieties. As an application, we produce a conjectural semiorthogonal decompositions for orthogonal Grassmannians of lines.