"Shall we cut to it? I know everyone is waiting to know who you’re going to ask because, you know, you need to ask someone and I’ve been asked by three people already and I know I’m not alone in refusing them all. In case, you know, you were to ask me."
-- Polly Chapman to Scorpius Malfoy (CC3.2)
The Blood Ball, in the alternate timeline where Lord Voldemort killed Harry Potter (due to the use of an Experimental Time Turner), is a formal dance held at Hogwarts School, much like the Yule Ball. From its name, however, the implication is that only pure-blood students are allowed to attend (CC3.2).
Polly Chapman, a fellow Slytherin, who participates in the torture of Muggle-borns at Hogwarts, tries to pressure Scorpius Malfoy, the school's "The Scorpion King", to ask her to be his date for the Blood Ball (CC3.2).
Scorpius has no idea what Polly is talking about at first. Her mention of torture in the dungeons confirms his suspicions about the kind of world he is now in. Muggle-borns and half-bloods are not safe at Hogwarts and Muggles are not safe anywhere (CC3.2, CC3.3, CC3.16).
This event plants the idea in his mind that someone might fancy him and, when the timelines have gone back into place, Scorpius feels able to ask Rose Weasley out, although it is not stated whether he is asking her to a formal ball like this one (CC4.14).
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About the play: http://www.harrypottertheplay.com