Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

Abstract

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...

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

“…A famous theorem of Zak-Lvovski [47,24] states that if X is not a quadric, and h 0 (N X/P n (−1)) < min{n + 1 + r, 2n + 1}, then X is not r-extendable. Quite remarkably, a converse of the theorem of Zak-Lvovski was recently obtained in [6,Thms. 2.1 and 2.19] in the case of X a canonical curve or a K3 surface, to the effect that h 0 (N X/P n (−1)) ≥ n+1+r is a sufficient condition for r-extendability, provided that the curve (respectively, any smooth hyperplane section of the surface) has genus at least 11 and Clifford index at least 3.…”

“…The study of h 0 (N C/P g−1 (−1)), or equivalently, of the corank of the gaussian map Φ ω C ,ω C , is a tricky question and a history of its own. We refer for instance to the works [46,44,42,7,10,11] and the very recent works on K3 surfaces [2,6]. It is still an open question to determine the possible values of this for all curves, although the value is known for general curves and for a general curve of any fixed gonality.…”

Bott vanishing for algebraic surfaces

“…4 Failure of Bott vanishing on a K3 surface in terms of elliptic curves of low degree Theorem 4.1 clarifies the meaning of Bott vanishing for a K3 surface X. Namely, if H 1 (X, Ω 1 X ⊗ B) = 0 for an ample line bundle B, then one of three conditions must hold: B 2 is less than 20, there is an elliptic curve of low degree with respect to B, or X is an anticanonical divisor in a singular Fano 3-fold Y with B = −K Y | X . The proof is based on recent work of Ciliberto, Dedieu, and Sernesi, which in turn buids on the work of Arbarello, Bruno, and Sernesi [3,9].…”

“…This cohomology group has a direct geometric meaning, related to the map from the moduli space of curves on K3 surfaces to the moduli space of curves (section 3).Roughly speaking, the failure of this vanishing for a K3 surface is caused either by elliptic curves of low degree on the surface, or by the existence of a (possibly singular) Fano 3-fold in which the K3 surface is a hyperplane section. The proofs build on a long development, starting with the work of Beauville, Mori, and Mukai about moduli spaces of K3 surfaces, and leading up to recent advances by 26,27,3,9]. We give a complete description of all K3 surfaces X with an ample line bundle B of high degree such that H 1 (X, Ω 1 X ⊗B) is not zero.…”

Moduli of curves on Enriques surfaces

“…Proof. Thanks to the bijective map δ in (3) and α g,φ being generically injective, the fact that χ −1 g,φ ((C, K S ⊗ O C )) is a point is equivalent to the fact that (c ι g,φ ) −1 ( C) is a point, where c ι g,φ is as in (6). The latter will follow if c −1 2g−1 ( C) is a point.…”