Derived category of V_{12} Fano threefolds

TL;DR

This paper studies the derived category of V_{12} Fano threefolds, revealing a semiorthogonal decomposition involving exceptional bundles and a genus 7 curve, and relates the Fano surface to the symmetric square of this curve.

Contribution

It provides a semiorthogonal decomposition of the derived category for V_{12} Fano threefolds and links the Fano surface to a symmetric square of a genus 7 curve.

Findings

Derived category admits a semiorthogonal decomposition with two exceptional bundles and a genus 7 curve

Fano surface is isomorphic to the symmetric square of the genus 7 curve

Establishes a geometric relationship between the Fano surface and the associated curve

Abstract

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve.