Well, it could be because of jealousy. Percy might be jealous of Luke's past relationship with Annabeth or vice versa. Maybe they both have different views on how to protect Annabeth and that leads to the fight.
In fanfiction, Percy and Luke could fight over Annabeth due to their own insecurities. Percy may feel that Luke has an advantage because of his charm or experience. Luke, on the other hand, might see Percy as a threat to his connection with Annabeth. Also, their different loyalties - Percy to Camp Half - Blood and Luke to his own motives - could play a part in the fight over Annabeth as they try to influence her in different directions.
Misunderstandings about their future plans often lead to fights. Percy might want to focus on one type of adventure while Annabeth has other ideas. They argue about it, but in the end, the love between them overcomes the differences and they kiss. This shows how their relationship is complex and full of different opinions that need to be resolved.
Another reason might be the stress of their quests. They are constantly under pressure to save the world or their friends. This stress can build up and cause them to lash out at each other. For instance, if a quest isn't going well and they keep hitting dead ends, they might start blaming each other for mistakes made along the way.
Well, one reason could be that in the fanfiction, the writer might want to explore a different dynamic. Maybe Annabeth gets lured by Luke's charm that has been newly developed in the fanfic. Luke could be presented as more understanding or having a different kind of allure that attracts Annabeth away from Percy.
Well, it could be about different understandings of Percy. Annabeth knows Percy as a demigod, as a hero in their world. Rachel, on the other hand, might see Percy in a different light, more as an individual outside of the demigod stuff. They might fight because they both think they know what's best for Percy and how he should be. Also, it could be a fight for Percy's future. Annabeth might want Percy to stay in the demigod world and continue their adventures together, while Rachel could have different plans for him.
It could make their friends take sides. For example, some of Percy's closer friends at Camp Half - Blood might side with him, which could create a rift between those friends and Luke's supporters. It might also make Annabeth's friends more protective of her, and they might be more critical of both Percy and Luke.
Percy might be a source of comfort for Annabeth. School can be a tough place sometimes, with all the academic pressure and social drama. Percy is someone she can always rely on, and without him, she feels less secure. Maybe he has a unique way of making her laugh or feel better when she's down, and she misses that a great deal.
Perhaps jealousy plays a part. If Percy got a bit too friendly with another character, Annabeth might get jealous and they fight. However, they soon remember their strong bond and how much they mean to each other, so they make up. It's a common trope in fanfiction as it shows the depth of their relationship.
Well, it could be simply due to a different kind of chemistry. Sometimes in fanfiction, the author wants to explore an alternative relationship dynamic. Reyna has a certain allure and mystery about her that might attract Percy. And the story could be set up in such a way that Percy and Reyna have more shared experiences at a particular point in time, which makes him lean towards her instead of Annabeth.
It might be a test of strength. They could both be trying to prove themselves, either to their friends or to some higher power in the fanfiction world. They want to see who is truly the stronger hero.
Luke might become more enraged. Since Percy has thwarted his plan to do something to Annabeth, he could seek revenge. He might start plotting more devious schemes against Percy and Annabeth, or even their friends.