There is some disagreement as to the exact themes that Fitzgerald meant to represent in the story, but given the context and some of his other writings, there are a few that you can count on. One constant is the fact that people living in the "high" society are generally characterized as vapid and somewhat empty of real emotion and feeling. This is a theme present in many of Fitzgerald's works, perhaps most notably in The Great Gatsby.
Another theme is that of the incredible danger represented by inherited wealth. If you look at Anthony's actions and consider that the guarantee of great wealth likely encouraged many of the most imprudent, etc., you can see that Fitzgerald felt that it was a very dangerous thing indeed.