1-nodal prime Fano threefolds parametrized by Bridgeland stable objects in Kuznetsov component of del Pezzo threefolds

BIMSA-YMSC Geometry and Dynamics Seminar

Organizer：

Yu-Wei Fan (YMSC)

Speaker：

Shizhuo Zhang (IBS-CGP)

Time：

Wed., 1:30-2:30 pm, Sept. 25, 2024

Venue：

Zoom Meeting ID: 815 762 8413

Passcode: BIMSA

Title:

1-nodal prime Fano threefolds parametrized by Bridgeland stable objects in Kuznetsov component of del Pezzo threefolds

Abstract:

Let $X$ be a smooth Fano threefold of index one and genus 8, a classical result tells us that $X$ is uniquely determined by a smooth cubic threefold $Y$ and a rank two instanton bundle on it. First, I will show that in the modern categorical language, $X$ is uniquely determined by its Kuznetsov component \Ku(X) and a distinguished object inside it. Then I will describe a conjectural picture for prime Fano threefolds of other genus. Second, I extend the conjectural picture from smooth cases to nodal prime Fano threefold cases and prove part of the conjecture. Namely, a 1-nodal maximally non-factorial prime Fano threefold of genus g=2d+2 coming from the so-called bridge construction is uniquely determined by a smooth del Pezzo threefold of degree d and an (acyclic extension) of a stable non-locally free instanton sheaf of rank two and charge d-1. Equivalently, each X is determined by \Ku(X) and a distinguished object inside the Kuznetsov component. All these facts support a conjecture that those Fano threefolds at most 1-nodal maximally non-factorial are parametrized by a Bridgeland moduli space of stable objects of character (d-1) multiple ideal sheaves of line in Kuznetsov component of degree d del Pezzo threefold. This talk is based on a joint work with Daniele Faenzi and Xun Lin.