Families of hypersurfaces with noncancellation property

Abstract

Let k k be a field of characteristic zero and R R a factorial affine k k -domain. Let B B be an affine R R -domain. In an earlier work of the author, the criteria are given in terms of locally nilpotent derivations for B B to be R R -isomorphic to the residue ring of the form R [ X , Y , Z ] / ( X Y − φ ( Z ) ) R[X,Y,Z]/(XY-\varphi (Z)) for some φ ( Z ) ∈ R [ Z ] ∖ R \varphi (Z)\in R[Z]\setminus R . We give a criterion for B B to be R R -isomorphic to R [ X , Y , Z ] / ( X m Y − F ( X , Z ) ) R[X,Y,Z]/(X^mY-F(X,Z)) where m ≥ 1 m \ge 1 and F ( X , Z ) ∈ R [ X , Z ] F(X,Z)\in R[X,Z] is such that F ( 0 , Z ) ∈ R [ Z ] ∖ R F(0,Z)\in R[Z]\setminus R when B B is factorial. We also show that for m ≥ 1 m \ge 1 , the hypersurfaces defined by x m y − F ( x , z ) = 0 x^my-F(x,z)=0 have noncancellation property under some conditions on F ( X , Z ) ∈ R [ X , Z ] F(X,Z)\in R[X,Z] .