Derived symmetries for crepant contractions to hypersurfaces

TL;DR

This work constructs derived symmetries for crepant contractions to hypersurfaces by using ambient-space pullbacks that are spherical, producing a twist autoequivalence that sits in a natural triangle and is compatible with base change. It extends the framework to non-Cartier divisors and general codimension by introducing families of spherical twists G_m and relating them to the contraction, via Serre-based factorization and Orlov blowup techniques. The results yield explicit relations among twists, including T_{g^*} ⊗ N_{tilde{X}} ≅ (T_{G_1} ... T_{G_n})^{-1} [2], and provide a canonicity perspective through a hypersurface twist hyperT with connections to fat spherical twists and other known constructions. The paper also provides base-change compatibility, local models, and concrete geometric examples (e.g., conifolds, Pagoda, quartic K3 degenerations) to illustrate how the approach produces new derived autoequivalences and clarifies their interplay with ambient geometry. Overall, it offers a robust, dimension-agnostic method to derive and relate autoequivalences associated to crepant contractions via spherical-categorical techniques.

Abstract

Given a crepant contraction f to a singularity X, we may expect a derived symmetry of the source of f. Under easily-checked geometric assumptions, I construct such a symmetry when X is a hypersurface in a smooth ambient S, using a spherical functor from the derived category of S. I describe this symmetry, relate it to other symmetries, and establish its compatibility with base change.

Figures (4)

• Figure 1: ams Sketch of the singular geometry of $X$ along $Y$ in the setting of Theorem \ref{['keythm.blowupB']}. The vertical planes are fibres of $\mathcal{N}_{Y} {{S}}$. This bundle contains the normal cone $\mathcal{C}_{Y} {X}$, shown by thickened lines. Recall that $Z \subset Y$ is the codimension $2$ locus where $\theta$ vanishes: over $z$ in $Z$, $\mathcal{C}_{Y} {X}$ coincides with $\mathcal{N}_{Y} {{S}}$; over $y$ not in $Z$, $\mathcal{C}_{Y} {X}$ is cut out of $\mathcal{N}_{Y} {{S}}$ by a non-zero linear function $\theta(y)$.
• Figure 2: ams Sketch of the singular geometry of $X$ along $Y$ in the setting of Theorem \ref{['keythm.blowupB']}. The vertical planes are fibres of $\mathcal{N}_{Y} {{S}}$. This bundle contains the normal cone $\mathcal{C}_{Y} {X}$, shown by thickened lines. Recall that $Z \subset Y$ is the codimension $2$ locus where $\theta$ vanishes: over $z$ in $Z$, $\mathcal{C}_{Y} {X}$ coincides with $\mathcal{N}_{Y} {{S}}$; over $y$ not in $Z$, $\mathcal{C}_{Y} {X}$ is cut out of $\mathcal{N}_{Y} {{S}}$ by a non-zero linear function $\theta(y)$.
• Figure 3: ams Local sketch of quartic K3 surface $X$ from Example \ref{['eg.quarticK3']} with node $x$, in smooth ambient $3$-fold ${S}= \mathbb{P}^3$. Blowing up $X$ at $Y=\{x\}$ gives an exceptional $\mathrm{E}_{Y} X \cong \mathbb P^1$ which is conic in $\mathrm{E}_{Y} {S}\cong \mathbb P^2$.
• Figure 4: ams Sketch of the blowup of $X$ along $Y$ in the setting of Theorem \ref{['keythm.blowupB']}. Compare Figure \ref{['fig.singn1']} which shows the geometry before the blowup. The locus $\mathrm{E}=\mathrm{E}_{Y} {S}$ is a $\mathbb{P}^1$-bundle over $Y$. I illustrate fibres $\mathrm{E}_z$ over $z$ in $Z$, and $\mathrm{E}_y$ over $y$ not in $Z$. The normal bundle $\mathcal{N}_{\mathrm{E}} {{\tilde{{S}}}}$ is the total space of $\mathcal{O}_{{r}}(-1)$. This bundle contains the strict transform ofams comp $\mathcal{C}_{Y} {X}$, shown by thickened lines: its intersection with $\mathrm{E}$ is the locus $\mathrm{E}_{Y} X$.

Theorems & Definitions (130)

• Remark 1.1
• Theorem 1: amsTheorem \ref{['thm.main']}comp(Theorem \ref{['thm.main']})
• Remark 1.2
• Theorem 2: amsTheorem \ref{['thm.mainn1']}comp(Theorem \ref{['thm.mainn1']})
• Example 1.3
• Theorem 3: amsCorollary \ref{['cor.blowupB']}comp(Corollary \ref{['cor.blowupB']})
• Remark 1.4
• Example 1.5
• Remark 1.6
• Theorem 4: amsTheorem \ref{['thm.mainB']}comp(Theorem \ref{['thm.mainB']})