The curve of lines on a prime Fano threefold of genus 8

TL;DR

This paper demonstrates that a general prime Fano threefold of genus 8 can be reconstructed from its Fano curve of lines and an associated theta-characteristic, revealing deep geometric relationships.

Contribution

It introduces a method to reconstruct prime Fano threefolds of genus 8 from their Fano curve and a natural embedding via a theta-characteristic.

Findings

Reconstruction of prime Fano threefolds from Fano curves of lines.

Identification of the theta-characteristic as key to embedding.

Establishment of a natural geometric correspondence.

Abstract

We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\Gamma,L)$, where $\Gamma$ is its Fano curve of lines and $L=O_{\Gamma}(1)$ is the theta-characteristic which gives a natural embedding $\Gamma \subset \matbb{P}^5$.