Central Weyl involutions on Fano-Mukai fourfolds of genus 10

TL;DR

This paper investigates the fixed points and invariant lines of a specific involution on Fano-Mukai fourfolds of genus 10, revealing geometric structures related to the Weyl group of type G2.

Contribution

It provides a detailed description of the fixed point set and invariant surface scrolls of the central Weyl involution on these fourfolds, enhancing understanding of their symmetry properties.

Findings

Identification of fixed points of the involution

Description of the invariant surface scrolls

Connection to the Weyl group of type G2

Abstract

It is known that every Fano-Mukai fourfold X of genus 10 is acted upon by an involution $\tau$ which comes from the center of the Weyl group of the simple algebraic group of type ${\rm G}_2$. This involution is uniquely defined up to conjugation in the group Aut(X). In this note we describe the set of fixed points of $\tau$ and the surface scroll swept out by the $\tau$-invariant lines.