NONLIFTABLE ABELIAN AUTOMORPHISM GROUPS OF SMOOTH SURFACES IN ℙ3

Let X be a smooth hypersurface of degree d≥3 in the projective space ℙn+1. If (n,d)≠(1,3),(2,4) then a finite group G that acts on X faithfully is a subgroup of the projective linear group PGL(n+2,ℂ). Sufficient conditions are known for G to be lifted to a subgroup of the general linear group GL(n+2,ℂ). In this paper, we assume that G is a finite abelian subgroup of PGL(4,ℂ) such that G cannot be lifted to GL(4,ℂ). We determine G by using the action of G on linear subspaces of ℙ3. As an application, we determine nonliftable abelian groups of PGL(4,ℂ) acting faithfully on smooth hypersurfaces of degree d in ℙ3. We give examples of a smooth hypersurface X⊂ℙ3 and a nonliftable abelian group G such that G acts on X faithfully.