Homeobox genes

What are homeobox genes? How do patterns in the evolution of these genes show shared ancestry among all animals? How do the patterns show divergence? As you know, arthropods possess a variety of different mouth parts; since homeobox genes control the development of the head region, scientists have argued that they were instrumental in driving the diversification of arthropods. Imagine a single group of arthropods with the same mouth parts and explain the steps that might have followed to give rise to the diversity we see today.