On the automorphisms of Mukai varieties

TL;DR

This paper investigates the automorphism groups of Mukai varieties, showing they are generally trivial with a few notable exceptions, and extends these results to certain Fano threefolds and higher genus cases.

Contribution

It provides a detailed description of the automorphism groups of Mukai varieties, identifying when they are trivial and highlighting unexpected exceptions.

Findings

Most Mukai varieties have trivial automorphism groups.

Exceptions occur in specific genera and codimensions.

A generic prime Fano threefold of genus 7 to 10 has no automorphisms.

Abstract

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for 7 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement is extended to g = 12.