There's a possibility that Annabeth accidentally damaged something very important to Percy. It could be a sentimental item, like a gift from his mother. In the heat of the moment, Percy's anger towards her would be understandable as the item held great emotional value for him.
It could be because of a misunderstanding. In the fanfic, perhaps Annabeth was seen talking to an old acquaintance who Percy doesn't trust. He might jump to conclusions and get angry with her without really knowing the whole story. Like, if he thought she was plotting something behind his back with that person.
One possible reason could be a misunderstanding. Maybe Annabeth made a decision without consulting Percy, like choosing a quest path that Percy thought was too dangerous. Percy, being protective and having his own ideas about how things should be done, might get mad.
Well, in some fanfictions, Percy might be mad at Annabeth because of a misunderstanding. Maybe she made a decision without consulting him in a crucial situation regarding their adventures. For example, if she chose a path that put their friends in danger without his input, Percy could get angry.
It could strain their relationship a great deal. Their friendship and possible romantic relationship is based on trust. If Percy is mad, that trust is shaken. They might start avoiding each other, not sharing secrets like they used to.
Maybe in some fanfics, the author wants to create a new and controversial plot. They might make Percy hate Annabeth due to some misunderstandings, like a miscommunication about a quest or Percy believing Annabeth has betrayed him in some way.
She could start by being honest with Percy about her feelings. Just come clean and say why she wants to get back together. For example, she might say she realized she can't live without him. This straightforward approach might help Percy see that she is sincere.
One possible reason could be a misunderstanding. Maybe Annabeth made a decision without consulting Percy, like choosing a quest that she knew was dangerous but didn't tell him until it was too late. Percy might feel left out and mad about it.
When Annabeth is mad, Percy will likely try to make it up to her. He might start doing all kinds of things like bringing her favorite foods, trying to help her with her projects. But Annabeth might not be so easily swayed. She'll be looking for a real apology and for Percy to show that he's learned his lesson. So the relationship becomes a bit of a power struggle, with Percy trying to regain her favor and Annabeth deciding whether he's worthy of it.
It could be that Annabeth has a misunderstanding about Percy. She might think he's a threat to her position or status at the camp, so she bullies him to try and keep him in check.
Well, it could be that Annabeth constantly corrects Percy in front of others. For example, when Percy is making a plan to defeat a monster, Annabeth keeps interrupting and saying his plan is all wrong. This public humiliation might make Percy mad.
Another way Percy could react is by giving Annabeth the silent treatment. He wouldn't talk to her for a while, which would be really difficult for Annabeth as they are usually close. Percy would just go about his business, ignoring her attempts to communicate until he feels she has truly understood how much she has angered him.