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Flips for spaces of quadrics on del Pezzo varieties

Saket Shah

TL;DR

This work extends the standard flip observed for cubic hypersurfaces to flips for Hilbert schemes of quadrics on del Pezzo varieties of degree at least 3, unifying cases such as cubics, complete intersections of two quadrics, and linear sections of Gr(2,5). The authors develop a uniform geometric framework using Nakano contraction and linear spans of quadrics to realize these flips and identify their centers, then derive semiorthogonal decompositions of derived categories associated to the Hilbert schemes and related quadrics varieties. They apply these flips to obtain explicit SODs in several settings, including orthogonal Grassmannian fibrations, and propose conjectural decompositions guided by categorical heuristics. A sharp obstruction is also established in low degree (d = 2), showing that the flip-induced SOD phenomenon fails in quartic double solids, thereby delineating the degree range where the construction is applicable and informative for derived categories.

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.

Flips for spaces of quadrics on del Pezzo varieties

TL;DR

This work extends the standard flip observed for cubic hypersurfaces to flips for Hilbert schemes of quadrics on del Pezzo varieties of degree at least 3, unifying cases such as cubics, complete intersections of two quadrics, and linear sections of Gr(2,5). The authors develop a uniform geometric framework using Nakano contraction and linear spans of quadrics to realize these flips and identify their centers, then derive semiorthogonal decompositions of derived categories associated to the Hilbert schemes and related quadrics varieties. They apply these flips to obtain explicit SODs in several settings, including orthogonal Grassmannian fibrations, and propose conjectural decompositions guided by categorical heuristics. A sharp obstruction is also established in low degree (d = 2), showing that the flip-induced SOD phenomenon fails in quartic double solids, thereby delineating the degree range where the construction is applicable and informative for derived categories.

Abstract

For a cubic hypersurface , work of Galkin--Shinder and Voisin shows the existence of a birational map relating the Hilbert scheme of two points with a certain projective bundle over . 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 . 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.
Paper Structure (18 sections, 29 theorems, 110 equations)

This paper contains 18 sections, 29 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The flips
  3. Derived categories
  4. Acknowledgments
  5. Background
  6. Del Pezzo varieties
  7. Hilbert schemes
  8. Vector bundles on Hilbert squares
  9. Standard flips and birational geometry
  10. Flips via the Nakano contraction theorem
  11. Flips via linear spans of quadrics
  12. The case of cubics
  13. The case of the complete intersection of two quadrics
  14. The case of linear sections of $\operatorname{Gr}(2,V_5)$
  15. Linear spaces on linear sections of $\operatorname{Gr}(2,V_5)$
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $(X,H)$ is an $n$-dimensional del Pezzo variety with $3 \leq d(X) \leq 9$. Then the polarization $H$ is very ample, and there exists a standard flip diagram \begin{tikzcd} \bbP(\Sym^2 \cU_2^\vee) \arrow[r, hook, "\iota"] \arrow[d, "\pi"] & X^{\left[2\right]} \arrow[leftrightarrow, r

Theorems & Definitions (64)

  • Theorem 1
  • Theorem 2: Theorems \ref{['geomflipcubics']}, \ref{['geomflipquartic']} and \ref{['geomflipquintic']}
  • Remark
  • Corollary 3
  • proof
  • Definition 4
  • Proposition 5: Corollary 3.4 of kuzdelpezzo
  • Proposition 6: Proposition 2.2 of kuzdelpezzo
  • Theorem 7: Theorem 1.2 of kuzdelpezzo
  • Proposition 8
  • ...and 54 more