Orbits of smooth rational curves on Enriques surfaces

TL;DR

This paper provides a formula to count the orbits of smooth rational curves on Enriques surfaces using their Nikulin root invariant and Vinberg group, advancing understanding of their automorphism groups.

Contribution

It introduces a closed formula linking automorphism group actions to lattice invariants of Enriques surfaces, a novel approach in algebraic geometry.

Findings

Derived a formula for orbit counts of rational curves

Connected automorphism groups with lattice invariants

Enhanced classification methods for Enriques surfaces

Abstract

We give a closed formula for the number of orbits of smooth rational curves under the automorphism group of an Enriques surface in terms of its Nikulin root invariant and its Vinberg group.