Another reason could be that Percy wants to start a conversation about Annabeth. By showing off the fanfiction, he can get people talking about her character, her strengths, and her role in their world. This could lead to more in - depth discussions about her and also about the whole story universe they are in.
Percy might show his admiration through his actions, like always looking at her with a soft gaze. For example, in a scene where they are in a group, his eyes would constantly find their way back to her. He could also be more protective of her, subconsciously wanting to keep her safe because he values her so much, which is also a way of showing how he sees her beauty not just physically but in a deeper sense.
Annabeth could show her determination by going out of her way to be in the same places as Percy. She'll be wherever he is, whether it's at his favorite spots or events he attends. And she'll always be there with a kind word or a helpful gesture, constantly reminding him of their connection. Also, she might face any danger or difficulty that comes her way while trying to get close to him again, showing that she's willing to do whatever it takes.
One great fanfiction is 'Together in Trials'. It shows how they stand by each other during various challenges in the demigod world. They face mythical monsters and tricky situations, always having each other's backs.
One specific moment could be when they are in a dangerous situation, like facing a powerful monster, and Percy steps in front of Annabeth to protect her. His confident stance and the way he holds his weapon makes her feel safe because she knows he'll do anything to keep her from harm.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
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.
It could be Chiron, the centaur. He's always keeping an eye on the campers and would likely be concerned if he caught them up to something. He might give them a stern talking - to and some extra chores as punishment.
Percy might be a bit flustered and embarrassed. He's protective of his relationship with Annabeth, so having Magnus walk in might make him feel like their privacy has been invaded. He could quickly try to shield Annabeth or just look really shocked.