# Divisibilities among nodal curves

## TL;DR

This paper proves that certain divisibility properties do not occur among nodal curves on smooth algebraic surfaces, with implications for the study of Enriques and K3 surfaces, using lattice theory.

## Contribution

It establishes a general divisibility non-existence result for classes of square -1 or -2 from nodal curves, advancing understanding of lattice structures on algebraic surfaces.

## Key findings

- No effective or anti-effective classes of square -1 or -2 arise from nodal curves
- Results apply to Enriques and K3 surfaces
- Uses properties of root lattices and their duals

## Abstract

We prove that there are no effective or anti-effective classes of square -1 or -2 arising from nodal curves on smooth algebraic surfaces by way of divisibility. This general fact has interesting applications to Enriques and K3 surfaces. The proof relies on specific properties of root lattices and their duals.