Torsion of differentials of hypersurfaces with isolated singularities
Let \(K\) be a field of characteristic 0. Let \(F\in R =K[X_1,\dots,X_N]\) be a reduced polynomial defining an affine hypersurface \(A\) with only isolated singularities. Let \(J=\bigl(\frac{\partial F}{\partial X_1},\dots,\frac{\partial F}{\partial X_N}\bigr)\) be the Jacobian ideal, \(I=(J:F)\), \(T(\Omega^{N-1}_{A/K})=\) torsion submodule of \(\Omega^{N-1}_{A/K}\). The author proves that there is an isomorphism of \(A\)-modules \(T(\Omega^{N-1}_{A/K})\approx I/J\). Under the additional assumption that \(K\) is algebraically closed, it is proved that \(T(\Omega^{N-1}_{A/K})\) and \((\Omega^N_{A/K})\) have the same (finite) dimension over \(K\). If \(F\) is quasi-homogeneous, \(A\) has isolated singularity if and only if \(\frac{\partial F}{\partial X_1},\dots,\frac{\partial F}{\partial X_N}\) is an \(R\)-sequence and in that case \(T(\Omega^{N-1}_{A/K}) \approx \Omega^N_{A/K}\) is a cyclic \(A\)-module [\textit{R. I. Michler}, Rocky Mt. J. Math. 26, No. 1, 229-236 (1996; see the preceding review)]. The author shows that for the reduced non-quasi homogeneous plane curve \(F:X^3Y^2+Y^5+X^7=0\) with isolated singularity at the origin \(\frac{\partial F}{\partial X}\), \(\frac{\partial F}{\partial Y}\) is an \(R\)-sequence and in that case \(T(\Omega^{N-1}_{A/K})\) is not cyclic. It is also proved that for the family of reduced plane curves defined by \(F=(c+X)^2Y^2/2+cX^2/2+X^3/3\), \(c\neq 0\), with a single isolated singularity at the origin, \(\frac{\partial F}{\partial X}\), \(\frac{\partial F}{\partial Y}\) do not form an \(R\)-sequence.
scientific article; zbMATH DE number 825326
scientific article