  Recently, Masuda-Sato and Precup-Sommers independently proved an LLT version
of the Shareshian-Wachs conjecture which says that the Frobenius
characteristics of the cohomology of the twin manifolds of regular semisimple
Hessenberg varieties are unicellular LLT polynomials. The purpose of this paper
is to study the geometry of twin manifolds and we prove that they are related
by explicit blowups and fiber bundle maps. Upon taking their cohomology, we
obtain a direct proof of the modular law which establishes the LLT
Shareshian-Wachs conjecture.
