On the normal sheaf of Gorenstein curves

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective 3-fold rational normal scroll S, implying that the normal sheaf on C embedded in S, and in Pg−1 as well, is unstable for g≥5, provided that S is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs.