BIRATIONAL GEOMETRY OF CALABI-YAU PAIRS AND 3-DIMENSIONAL CREMONA TRANSFORMATIONS

A Calabi-Yau (CY) pair is a pair (X, D) of a normal variety X and a reduced Z-Weil divisor D⊂X such that KX+D∼0 is a Cartier divisor linearly equivalent to zero. A Mori fibered (Mf ) CY pair is a Q-factorial (t, lc) CY pair (X, D) together with a Mori fibration f:X→S. Let (X, DX) and (Y, DY) be CY pairs. A birational map f:X to Y is volume preserving if there exists a common log resolution p:W→X, q:W→Y such that p∗(KX+DX) =q∗(KY+DY). Let (X, DX)→SXbe (Mf) CY pair. We define the special birational group of (X, DX) as the group SBir(X, DX) of volume preserving birational maps f:X to X. Our aim is to produce interesting subgroups of the Cremona group of birational self-maps of projective spaces as groups of the type SBir(Pn, DPn), where DPn is a hypersurface of degree n+ 1. Even in seemingly simple cases these groups could be quite hard to compute. We give an explicit presentation of SBir(X, DX) when X=P3 and DX is a quartic surface with divisor class group generated by the hyperplane section and whose singular locus is either empty or an A1-point. In general it seems that the worse are the singularities of the pair (X, DX) the more complicated is the group SBir(X, DX).