Conics on Barth--Bauer octics

TL;DR

This paper studies conic configurations on special Kummer octic surfaces, establishing bounds, uniqueness, and symmetries, and explores conics on related Mukai surfaces with a large number of conics.

Contribution

It provides bounds on the number of conics on Kummer octic surfaces, identifies a unique surface with maximal conics, and connects conic configurations to Mukai groups.

Findings

Bound of 176 conics on certain Kummer octic surfaces

Existence of a unique surface with 176 irreducible conics

Discovery of a double plane with 8910 smooth conics

Abstract

We analyze the configurations of conics and lines on a special class of Kummer octic surfaces. In particular, we bound the number of conics by $176$ and show that there is a unique surface with $176$ conics, all irreducible: it admits a faithful action of one of the Mukai groups. Therefore, we also discuss conics and lines on Mukai surfaces: we discover a double plane (ramified at a smooth sextic curve) that contains $8910$ smooth conics.