Lefschetz Properties of complete intersesctions and maximal variation of hyperplane sections

For an artinian complete intersection of forms of degree $d$, we prove that the multiplication by a general linear form is injective up to degree $\frac{3d-1}{2}$, independently of the number of variables. This is accomplished using algebraic and geometric properties of the union of two complete intersection curves and their general hyperplane sections. As a consequence, we solve, for any $d\ge 4$, the problem about the maximal variation of the hyperplane sections of a smooth hypersurface of degree $d$ posed by A. Beauville.