Flips for spaces of quadrics on del Pezzo varieties

TL;DR

This paper extends known birational flip constructions from cubic hypersurfaces to Hilbert schemes of quadrics on higher-dimensional del Pezzo varieties, leading to new semiorthogonal decompositions for orthogonal Grassmannians.

Contribution

It generalizes the standard flip construction for Hilbert schemes of quadrics to various del Pezzo varieties of degree at least 3, broadening the geometric and categorical understanding.

Findings

Standard flips constructed for Hilbert schemes of quadrics on del Pezzo varieties.

Derived category decompositions are achieved for these geometric objects.

Conjectural semiorthogonal decompositions proposed for orthogonal Grassmannians.

Abstract

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 $\mathrm{Gr}(2, 5)$. 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.