Percy's Harmartia is his honesty and naivety. He thinks that if he just tells Hades the truth he will be allowed to leave with his mother, but Hades is not that kind.
Annabeth is Athena's daughter, so Annabeth thinks she will beat everyone when it comes to combat, but Percy beats her and she realizes that she isn't the best and stops being so full of pride.
There is a reversal of fortune when Percy goes from weird high school boy to cool hero demigod. This is catalysed by Percy being attacked by the fury.
Percy has his anagnorisis when he realises that he really is a Demi God because he can control water.
Percy's nemesis is Luke, Herme's son. Luke wants the old gods to destroy themselves in a huge war, so he makes it look like Percy stole Zeus's lightning bolt. A fate that can not be avoided is Percy and Luke fighting, so Percy can save the world.
There is a purging of the audience's emotions when Percy & co. are about to be fed to the souls. We feel fear and empathy for Percy's plight.