Perhaps Percy has to go to Tartarus alone because he believes it's the only way to save his friends. In Tartarus, he might encounter old foes who are now even more powerful. He has to rely on his wits and his skills with his sword, Riptide. He could also make unexpected alliances with some of the creatures there in order to survive and achieve his goal.
In the 'Percy goes to Hogwarts alone fanfiction', it's possible that Percy will discover new aspects of himself. Since he's alone, he has to rely solely on his own wits and skills. He could stumble upon some hidden secrets of Hogwarts that others haven't found. Also, he may have some unique encounters with the Hogwarts professors who take notice of his solitary situation and offer him special guidance or tasks.
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There's a chance that Percy meets the personification of Tartarus itself. It could be a shadowy, all - powerful being that tries to break Percy's spirit. Percy, being the brave hero he is, would have to resist and find a way to either outwit or overpower this entity in order to continue his journey in Tartarus.
In this fanfiction, Percy going through Tartarus alone could involve him exploring the darkest parts of the underworld. He could discover secrets about the origin of the monsters there. Maybe he stumbles upon an ancient power that he has to either resist or use to his advantage. Along the way, he might have visions of his loved ones which both give him strength and also make him homesick.
In a fanfic where Percy goes to Hogwarts alone, he could form some interesting friendships. He might befriend Neville, who is also a bit of an underdog. Percy could teach Neville some of his own courage -building techniques. And he would surely be a star in the Quidditch field with his athletic abilities, even if he has to learn the rules first.
Well, in fanfiction, when Harry goes alone to the ball, he could end up being the center of attention in an unexpected way. Since he's alone, people might be curious about him. He might also use this time alone to observe the other characters more closely, learning secrets or motives that he wouldn't have noticed if he was with friends. And who knows, he could start a new trend of going solo to the ball, making others think it's a cool and independent thing to do.