Extensions of curves with high degree with respect to the genus

We classify linearly normal surfaces S P^r+1 of degree d such that 4g-4 d 4g+4, where g1 is the sectional genus (it is a classical result that for larger d there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus 3 curves, whenever they verify Property N₂, using and slightly expanding the theory of integration of ribbons of the authors and E. ~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree d 2g+3. We classify surfaces having such a curve C as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over C are integrable if and only if d=2g+3. In the latter case we obtain the existence of a universal extension. Comment: v2: various complements with respect to v1; v3: correction in the statement of Hartshorne's Theorem 2. 5: v4: final version