Lower deformations of isolated hypersurface singularities

For a smooth real or complex algebraic variety $V$ and a hypersurface $W\subset V$ with isolated singular points, the deformations of these singular points are studied that can be realized by variation of $W$ in the linear system $|W|$. For isolated singularities of hypersurfaces, lower deformations are introduced, which have an explicit topologicalanalytic description. Under some conditions, it is shown that such deformations can be realized independently even if the linear system $|W|$ is not a joint versal deformation of all singular points of $W$. Several applications of lower deformations are given, including the construction of divisors with prescribed isolated singularities and the construction of real algebraic curves with prescribed topology.