Central Weyl involutions on Fano-Mukai fourfolds of genus 10

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 ${\mathrm{G}}_2$. 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.

arXiv comment26 pages. Corrections due to the referee remarks; main changes are in (the proofs of) Lemmas 2.3, 2.5 and Proposition 2.7