Annabeth's jealousy of Percy in fanfiction might be because of how others perceive them. Percy is sometimes seen as the 'chosen one' type, and Annabeth, despite her own great achievements, may feel overshadowed. This could be exacerbated if in the fanfic, Percy is constantly getting special treatment or rewards. For instance, if Percy is given a powerful weapon just for being who he is, while Annabeth has to earn everything she gets, it could make her jealous. Also, any sort of prophecy that seems to favor Percy more than her could be a source of envy in the fanfiction world.
Often, in fanfiction, Annabeth may be jealous of Percy's charisma. He has a way of attracting people to him, and that can make Annabeth, who is just as heroic but perhaps more reserved, feel jealous.
One reason Natsu gets jealous of Lucy in fanfiction could be that he sees her as his special person. He's used to having her attention, so when someone else gets it, he gets jealous. For instance, if she goes on a mission with someone else instead of him.
Lily's jealousy in such a fanfiction might be because of the way Severus treats Hermione. If he is kind, protective, or shows a level of respect towards Hermione that he didn't always show Lily, it could trigger her jealousy. For example, if in the story Severus helps Hermione with her studies or defends her, and he didn't do the same for Lily in the past, Lily would likely be jealous.
Often, she becomes overly possessive. If she sees Percy talking to another girl, she'll quickly insert herself into the conversation. She might put her arm around Percy or give the other girl a look that clearly says 'he's mine'. This possessiveness is a clear sign of her jealousy in the Percy Jackson fanfiction world.
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 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.
The jealousy can initially strain their relationship. Percy's jealousy might make Annabeth feel smothered or like he doesn't trust her. For example, if Percy is jealous of Annabeth's friendship with a male demigod, he might act coldly towards her, causing her to be confused and a bit hurt.
It's possible that Annabeth surprises Percy at high school by setting up a whole new section in the school library dedicated to Greek mythology. Percy loves Greek mythology, and when he discovers it, he's thrilled. Annabeth reveals that she worked with the school librarian for weeks to make it happen. This not only surprises Percy but also enriches the knowledge of the whole school about their demigod heritage.
The jealousy in Pucca - Garu fanfictions can also be a sign of Garu's deeper feelings for Pucca. He might not be very good at expressing his love in a normal way, so jealousy could be his way of showing that he cares. In some fanfictions, when Garu gets jealous, it forces him to confront his feelings for Pucca. He might realize that he doesn't want to see her with anyone else because he has strong romantic feelings for her. This can be a turning point in the fanfiction, leading to a more open and honest relationship between them.
Often, it starts with small incidents. For example, Percy might get praised for a simple task that Annabeth thinks she could have done better. This builds up her jealousy gradually.
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.