Well, perhaps Rachel has some unique talents or abilities that catch Percy's attention. In the fanfiction, this might make Annabeth feel insecure about her own relationship with Percy. For example, if Rachel can communicate with the gods in a way that Annabeth can't, and Percy is impressed by it, Annabeth could get jealous.
One possible reason could be Percy's natural leadership skills. He often takes charge in dangerous situations and gets a lot of attention for it. Annabeth might be jealous because she also wants that level of recognition.
One possible reason could be jealousy. Maybe Rachel has something that Annabeth wants, like a closer relationship with a particular character. For example, if they both like the same guy, Annabeth might hate Rachel out of envy.
One reason could be their different areas of expertise. Annabeth is highly skilled in architecture and battle strategies in the Percy Jackson world. Hermione, being a brilliant witch in the Harry Potter universe, might feel jealous if she perceives Annabeth as having more practical and hands - on skills. Another factor could be their relationships. If Annabeth has a really great relationship with someone Hermione admires or wishes to be closer to, it could stir up jealousy. For example, if they were in a shared universe and Annabeth was closer to Harry or Ron than Hermione was.
Another way it could end is that through the jealousy, Annabeth discovers new things about herself. She realizes that she has been taking Percy for granted in some ways. So, she works on herself and her relationship with Percy. Rachel may or may not still be in the picture, but Annabeth and Percy's bond becomes deeper and more understanding, with Annabeth no longer feeling jealous.
Annabeth could be under some kind of magical influence. In the Percy Jackson universe, there are many magical elements. She could be cursed or charmed without her knowledge. This would be a very interesting take in a fanfiction as it would add an element of mystery and also give her a chance for redemption once the truth is discovered. Also, it could be that she has some internal insecurities that she hasn't dealt with and they manifest in this way, making her act out of character.
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.
One example could be a situation where Annabeth sees Percy getting a lot of attention from a new girl in their group. She might start acting cold towards Percy, which is a classic sign of jealousy in fanfiction. In such a story, the author might explore her insecurities about her relationship with Percy.
Perhaps Percy forgets an important promise he made to Annabeth. In their relationship, promises are significant. If he fails to keep one, like a promise to help her with a project or be at a certain place at a certain time, Annabeth would surely be mad. It could also be that Percy accidentally breaks one of Annabeth's creations or possessions, which she holds dear.
Communication is key. They need to sit down and talk about their feelings. For example, Percy could tell Annabeth that he's only being friendly with the new demigod who seems to like him, but his heart is with Annabeth. Annabeth could then share her insecurities. Another way is through a shared adventure. If they go on a quest together where they have to rely on each other, they'll realize that their relationship is stronger than any jealousy - inducing situation. They could also be given a task by Chiron that forces them to work together and face their jealousy head - on.
One reason could be that she's scared. The idea of becoming a mother at that point in her life might be overwhelming, and she might think she can't handle it with Percy. Maybe she wants to protect Percy from the difficulties that come with the pregnancy as well.