Derived categories of Fano threefolds and degenerations

TL;DR

This paper proves that certain derived categories of Fano threefolds can be smoothly deformed from those of del Pezzo threefolds, confirming a conjecture and enabling new geometric insights.

Contribution

It establishes the deformation of derived categories from del Pezzo and nodal threefolds to prime Fano threefolds, confirming the Fano threefolds conjecture.

Findings

Derived categories of del Pezzo threefolds deform to those of prime Fano threefolds.

Corrects and proves the Fano threefolds conjecture from [Kuz09].

Describes a compactification of the moduli stack of prime Fano threefolds.

Abstract

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree $d = 1$ can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus $g = 2d + 2 \in \{4,6,8,10,12\}$. This corrects and proves the Fano threefolds conjecture of the first author from [Kuz09], and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds.