The fanfiction often showcases Percy's protectiveness as a form of flirting. He'll jump in to defend Annabeth from any threats, real or imagined, and then play it off as if it was no big deal, but Annabeth can see the care behind it. This creates a sort of dynamic where his actions speak louder than words in terms of his feelings for her.
In some fanfictions, Percy might flirt with Annabeth during a quest. For example, he could be constantly saving her from small perils and then making a joke about being her 'knight in shining armor', with a wink and a charming smile. This kind of flirting adds a light - hearted and romantic element to their adventure.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.
One reason could be the popularity of Percy as a character. Fans might be more inclined to write about him in a dominant role because they like him better. So, they create fanfictions where he dominates Artemis.
Perhaps a story where Percy is chosen by the Fates to mediate between Hestia and Artemis. There has been some sort of misunderstanding between the two goddesses. Percy, with his charm and unique perspective, tries to get them to see eye to eye. He arranges meetings in different places - sometimes in Hestia's warm and cozy domain, and other times in Artemis's wild and untamed forests. Through Percy's efforts, they not only resolve their differences but also realize that they can work together for the greater good.
In some fanfictions, Percy could be made Artemis' son for creative and often unique plot twists. Fanfiction allows for a lot of creative liberties, so it's not unheard of. However, in the original Percy Jackson lore, there's no such relation.
Their relationship might develop through facing external threats. For instance, if there are enemies that target them because of their forced union. They would have to rely on each other's skills. Percy's ability to control water and his combat skills, along with Artemis's archery and connection to the wild, could be used to fend off these threats. As they fight side by side, they start to see each other in a new light and develop a bond.
In the fanfiction, their relationship development could be complex. Since they are sharing a bed, there is a forced closeness. Percy might accidentally cross some of Artemis's boundaries at first, which would cause friction. But through talking it out and getting to know each other better, they could start to respect each other's boundaries more. As time passes, they might find common ground in their love for nature (in Percy's case, his connection to the sea which is also part of nature). This could lead to a slow but steady growth of a relationship that is based on mutual understanding and respect.
Annabeth is quite popular. Since she has a relationship with Percy in the original series, in fanfiction related to Percy and Artemis, she can be a source of conflict or support. For example, she might be jealous of the time Percy spends with Artemis. Another popular character is Thalia. Her connection to Artemis as a member of the Hunters makes her an important figure. She can add more depth to the stories, especially when it comes to the dynamics between the Hunters and Percy. And then there's Grover. His loyalty to Percy and his unique abilities as a satyr can bring a different flavor to the fanfiction. He might be the one who tries to mediate between Percy and those who are skeptical of his association with Artemis.