Families of hypersurfaces with noncancellation property

Abstract:

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