Arianne Martell is Prince Doran's eldest child and, under Dornish law's absolute primogeniture — eldest child inherits regardless of sex, a rule the rest of the Seven Kingdoms treats as an eccentricity rather than the plain justice Dorne considers it — heir to Sunspear in her own right, no brothers required to stand ahead of her in the line. That right made her father's apparent favoring of her younger brother Quentyn, sent abroad on missions of real consequence while she was kept at court and told little, read to her as something close to betrayal.
She answered it by conspiring, without her father's knowledge and against his stated interests, to crown Myrcella Baratheon queen with Dornish backing — a plot recruiting allies from among her household guard and companions and unraveling badly when Gerold Dayne, called Darkstar, took the scheme somewhere Arianne had never intended it to go, maiming Myrcella in the attempt. Doran had her arrested for it himself, and it was only then, confronting his own daughter under guard, that he finally told her the truth: that her exclusion from his confidence had never been distrust, but protection, while he finished laying a much longer plan than the one she'd tried to run around him.
What role she plays in that longer plan now that she knows its shape is, as of the latest word reaching the Citadel, still being written. What is already clear is that Dorne's next ruler, whenever that succession comes, has already shown she will act on her own judgment when she believes her house's cause demands it — for good or for the considerable trouble that judgment can cause.