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Full exceptional collections on Fano threefolds and groups generated by spherical twists on K3 surfaces

Anya Nordskova, Michel Van den Bergh

TL;DR

The paper resolves the structure of full exceptional collections on Fano threefolds with four-vector-bundle presentations by reducing to a K3 anticanonical divisor and analyzing the group generated by spherical twists on its derived category. It proves that every full exceptional collection on such X is a shift of vector bundles, and that the action of the braid group $B_4$ with shifts, $B_4 \rightthreetimes \mathbb{Z}^4$, is transitive and free. A central technical achievement is proving that the subgroup of $\mathrm{Aut}\,\mathcal{D}^b(Y)$ generated by spherical twists is free, with explicit generators given by spherical twists along vector bundles corresponding to root data in the Mukai lattice of a Picard-rank-1 K3 surface $Y$ of degree $d=2\delta$. The authors compute concrete instances for $Y$ arising as anticanonical sections of ${\mathbb P}^3$, $Q_3$, $V_5$, and $V_{22}$, showing the generated groups are free on four generators, and they give a family of examples where the group is infinitely generated, illustrating the richness of the autoequivalences in this setting. The work connects Bridgeland stability, the Hurwitz action, and hyperbolic geometry (Fuchsian/group actions) to derive explicit presentations and transitivity results, with potential implications for understanding derived-category autoequivalences of Fano varieties. All mathematical notation is kept within $...$ as required, and the core results are presented in a way that supports both theoretical insight and computational applications in derived-category geometry.

Abstract

For a Fano threefold admitting a full exceptional collection of vector bundles of length four we show that all full exceptional collections consist of shifted vector bundles. We prove this via a detailed study of the group generated by spherical twists on an anticanonical divisor. For example, we prove that this group is free and provide explicit generators.

Full exceptional collections on Fano threefolds and groups generated by spherical twists on K3 surfaces

TL;DR

The paper resolves the structure of full exceptional collections on Fano threefolds with four-vector-bundle presentations by reducing to a K3 anticanonical divisor and analyzing the group generated by spherical twists on its derived category. It proves that every full exceptional collection on such X is a shift of vector bundles, and that the action of the braid group with shifts, , is transitive and free. A central technical achievement is proving that the subgroup of generated by spherical twists is free, with explicit generators given by spherical twists along vector bundles corresponding to root data in the Mukai lattice of a Picard-rank-1 K3 surface of degree . The authors compute concrete instances for arising as anticanonical sections of , , , and , showing the generated groups are free on four generators, and they give a family of examples where the group is infinitely generated, illustrating the richness of the autoequivalences in this setting. The work connects Bridgeland stability, the Hurwitz action, and hyperbolic geometry (Fuchsian/group actions) to derive explicit presentations and transitivity results, with potential implications for understanding derived-category autoequivalences of Fano varieties. All mathematical notation is kept within as required, and the core results are presented in a way that supports both theoretical insight and computational applications in derived-category geometry.

Abstract

For a Fano threefold admitting a full exceptional collection of vector bundles of length four we show that all full exceptional collections consist of shifted vector bundles. We prove this via a detailed study of the group generated by spherical twists on an anticanonical divisor. For example, we prove that this group is free and provide explicit generators.

Paper Structure

This paper contains 30 sections, 61 theorems, 156 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Generalities
  4. Spherical twists
  5. The Hurwitz action
  6. Stability conditions and derived autoequivalences of K3 surfaces
  7. Fuchsian groups
  8. Geometric considerations around ${\mathcal{P}}^+(Y)$, ${\mathcal{P}}_0^+(Y)$
  9. Introduction
  10. The hyperboloid model
  11. Application to K3 surfaces
  12. Transfer of the $\operatorname {PSL}_2({\mathbb R})$ action
  13. Groups generated by spherical reflections in $\mathop{\mathrm{Aut}}\nolimits^+\operatorname {H}^*(Y)$
  14. Free products generated by elements of order two
  15. Preliminaries on orbifold Riemann surfaces
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $X$ be a threefold with a full exceptional collection of four vector bundles. Then:

Figures (6)

  • Figure 1: A fundamental domain for $G \subset \Gamma_0^+(2)$ containing ${\frac{i}{\sqrt{2}} , 1+\frac{i}{\sqrt{2}},2+\frac{i}{\sqrt{2}}}$ and $3+\frac{i}{\sqrt{2}}$.
  • Figure 2: A fundamental domain for $G \subset \Gamma_0^+(3)$ containing $\frac{i}{\sqrt{3}} , \frac{1}{2}+\frac{i}{2\sqrt{3}}, 1+\frac{i}{\sqrt{3}}$ and $2+\frac{i}{\sqrt{3}}$.
  • Figure 3: A fundamental domain for $G \subseteq \Gamma_0^+(5)$ containing $\frac{-1}{2}+\frac{i}{2\sqrt{5}}, \frac{i}{\sqrt{5}} , \frac{1}{2}+\frac{i}{2\sqrt{5}}$ and $1+\frac{i}{\sqrt{5}}$.
  • Figure 4: A fundamental domain for $G = \Gamma_0^+(11)$ containing $\frac{-1}{3}+\frac{i}{3\sqrt{11}}, \frac{i}{\sqrt{11}}, \frac{1}{3}+\frac{i}{3\sqrt{11}},$ and $\frac{1}{2}+\frac{i}{2\sqrt{11}}$.
  • Figure 5: A fundamental domain for $G \subset \Gamma_0^+(1) \cong \operatorname {PSL}_2({\mathbb Z})$ containing $i, 1+i$ and $2+i$.
  • ...and 1 more figures

Theorems & Definitions (122)

  • Theorem 1.1: Theorem \ref{['thm:main']}, Corollary \ref{['cor:allsheaves']}
  • Remark 1.2
  • Proposition 1.4: Dolgachev, Kawatani14, see also Remark \ref{['rem:pm1']}, Proposition \ref{['prop:imHN']}
  • Theorem 1.6: Theorem \ref{['thm:sphtwistfree']}
  • Theorem 1.7: Theorem \ref{['thm:examples']}
  • Proposition 1.8: Proposition \ref{['pr:compserre']}
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3: Bondal, MR992977
  • Proposition 3.4: Bondal, MR992977
  • ...and 112 more