The Farey tree and embeddings of lens spaces and rational balls in ${\mathbb{C}\mathbb{P}}^2$

Motivated by a conjecture of Kollár, we study embeddings of multiple rational homology balls in ${\mathbb{C}\mathbb{P}}^2$. To each node of the Farey tree, we associate such an embedding of three rational homology balls with lens space boundary, extending earlier work of the second author and of Lisca and Parma, using a recursive Kirby calculus argument. We also give further explicit constructions of embeddings of triples of rational homology balls into homotopy ${\mathbb{C}\mathbb{P}}^2$s.