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

Abstract In this paper, we develop a framework that allows one to describe the birational geometry of Calabi–Yau pairs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>D</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (X,D) . After establishing some general results for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>D</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (X,D) with mild singularities, we focus on the special case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> X=\mathbb{P}^{3} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> D\subset\mathbb{P}^{3} is a quartic surface. We investigate how the appearance of increasingly worse singularities in 𝐷 enriches the birational geometry of the pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi>D</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (\mathbb{P}^{3},D) , and leads to interesting subgroups of the Cremona group of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> \mathbb{P}^{3} .