The maximum cut problem on blow-ups of multiprojective spaces

TL;DR

This paper determines the maximum cut for certain blow-up varieties, providing bounds and optimal embeddings that connect algebraic geometry with combinatorial optimization, and highlights new optimality properties of Weyl group orbits.

Contribution

It introduces bounds for the maximum cut problem on minuscule varieties and constructs optimal embeddings for semidefinite relaxations, linking algebraic geometry with combinatorial optimization.

Findings

Maximum cut for ${ m Bl}_{4}( ext{P}^2)$ is 12.

Bounds for minuscule varieties are asymptotically sharp.

Optimal embeddings relate to Weyl orbits of root systems.

Abstract

The maximum cut problem for a quintic del Pezzo surface ${\rm Bl}_{4}(\mathbb{P}^2)$ asks: Among all partitions of the 10 exceptional curves into two disjoint sets, what is the largest possible number of pairwise intersections? In this article we show that the answer is twelve. More generally, we obtain bounds for the maximum cut problem for the minuscule varieties $X_{a,b,c}:={\rm Bl}_{b+c}(\mathbb{P}^{c-1})^{a-1}$ studied by Mukai and Castravet-Tevelev and show that these bounds are asymptotically sharp for infinite families. We prove our results by constructing embeddings of the classes of $(-1)$-divisors on these varieties which are optimal for the semidefinite relaxation of the maximum cut problem on graphs proposed by Goemans and Williamson. These results give a new optimality property of the Weyl orbits of root systems of type $A$,$D$ and $E$.