Packing Integral Tori in Del Pezzo Surfaces

TL;DR

This paper generalizes a packing result for integral Lagrangian tori from the 2-sphere product to Del Pezzo surfaces, demonstrating that a single Clifford torus can form a maximal integral packing.

Contribution

It extends the integral packing result to Del Pezzo surfaces, showing the Clifford torus suffices for maximal packing in these cases.

Findings

Existence of a maximal integral packing with a single Clifford torus

Extension of packing results from $ ext{S}^2 imes ext{S}^2$ to Del Pezzo surfaces

Any other integral Lagrangian torus intersects the Clifford torus

Abstract

We extend a packing result of R. Hind and E. Kerman for integral Lagrangian tori in $\mathbb{S}^{2} \times \mathbb{S}^{2}$ to the Del Pezzo surfaces $(\mathbb{D}_{n}, \omega_{\mathbb{D}_{n}})$ for $n = 1, \dots, 5$. An integral torus is one whose relative area homomorphism is integer-valued, and we seek a maximal integral packing. By definition, this is a disjoint collection $\{L_{i}\}$ of integral Lagrangian tori with the following property: any other integral Lagrangian torus not in this collection must intersect at least one of the $L_{i}$. We show that one can always find such a packing consisting of only the Clifford torus.