Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations

We develop a framework that allows one to describe the birational geometry of Calabi-Yau pairs $(X,D)$. After establishing some general results for Calabi-Yau pairs $(X,D)$ with mild singularities, we focus on the special case when $X={\mathbb{P}}^3$ and $D\subset {\mathbb{P}}^3$ is a quartic surface. We investigate how the appearance of increasingly worse singularities on $D$ enriches the birational geometry of the pair $({\mathbb{P}}^3,D)$.