Singular hypersurfaces, versality, and Gorenstein algebras

We give precise upper bounds on the sum of the Tjurina numbers of the singu larities of a projective hypersurface to ensure that these singularities ar e simultaneously versally deformed by the family of all projective hypersur faces of the given degree, or, more strictly, by the family of all hypersur faces agreeing with the given one on a transverse hyperplane. We do this by relating the situation to a conjecture recently made by Eisenbud, Green, a nd Harris, and by proving some relevant cases of this conjecture. The argum ents exploit properties of Gorenstein algebras.