On the k-normality of projected algebraic varieties

We give a necessary and sufficient condition for the isomorphic projection of a k-normal variety to remain k-normal, k = 2; the condition is based on a scheme Z k naturally associated to degree k forms vanishing on the variety. We furnish many applications and examples especially in the case of varieties defined by quadratic equations. A non-vanishing theorem for the Koszul cohomology of projected varieties allows us to construct interesting examples in the last sections. All the results are effective and also interesting from the computational point of view