Let C be a smooth projective curve (resp. (S, L) a polarized K3 surface) of genus g 11, non-tetragonal, considered in its canonical embedding in P g−1 (resp. in its embedding in |L| ∨ ∼ = P g ). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in P g+r , not a cone, with dim(Y ) = r+2 and ωY = OY (−r), if the cokernel of the Gauss-Wahl map of C (resp. H 1 (TS L ∨ )) has dimension larger or equal than r + 1 (resp. r). This relies on previous work of Wahl and Arbarello-Bruno-Sernesi. We provide various applications.A central theme of this text is the extendability problem: Given a projective (irreducible) variety X ⊂ P n , when does there exist a projective variety Y ⊂ P n+1 , not a cone, of which X is a hyperplane section? Given a positive integer r, an r-extension of X ⊂ P n is a variety Y ⊂ P n+r having X as a section by a linear space. The variety X is r-extendable if it has an r-extension that is not a cone, and extendable if it is at least 1-extendable. The following result provides a necessary condition for extendability.(0.1) Theorem (Lvovski [25]). Let X ⊂ P n be a smooth, projective, irreducible, non-degenerate variety, not a quadric. SetIf X is r-extendable and α(X) < n, then r α(X).This introduced the idea, explicit in Voisin's article [38], that the elements of (coker(Φ C )) ∨ (or rather of ker( T Φ C )) should be interpreted as ribbons, or infinitesimal surfaces, embedded in P g and extending C: see Section 4.The following statement is a first converse to Theorem (0.1), and a central element of the proofs of our Theorems (2.1) and (2.18).(0.2) Theorem (Wahl [43], Arbarello-Bruno-Sernesi [3]). Let C be a smooth curve of genus g 11, and Clifford index Cliff(C) > 2. Every ribbon v ∈ ker( T Φ C ) may be integrated to (i.e. is contained in) a surface S in P g having the canonical model of C as a hyperplane section.Note that if v = 0, then the surface S is not a cone as only the trivial ribbon may be integrated to a cone. Conversely, we observe that actually unicity holds in Theorem (0.2) (see (2.2.2) and Remark (4.8)): up to isomorphisms, given a ribbon v ∈ ker( T Φ C ), the surface S integrating it in P g is unique. For v = 0, this is the content of the aforementioned theorem of Wahl and Beauville-Mérindol, see (2.3).We prove a statement for K3 surfaces analogous to Theorem (0.2) (Theorem (2.17)).Theorem (0.2) provides a characterization of those curves having non-surjective Wahl map in the range g 11 and Cliff > 2. Wahl [42, p. 80] suggested to study the stratification of the moduli space of curves by the corank of the map Φ C : This is done in our Theorem (2.1) to the effect that, in the same range, the curves with cork(Φ C ) r are those which are r-extendable.We give various applications of our results, in particular to the smoothness of the fibres of the forgetful map which to a pair (S, C) associates the modulus of C, where S ⊂ P g is a K3 surface and C is a canonical curve hyperplane section of S (Theorem (2.6)). The same result is proven for the analog...
