Smoothing cones over K3 surfaces

We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone is smoothable even though the projective cone is not.


1.1
In 1974, Pinkham [22] showed that the cone over a normal elliptic curve is smoothable if and only if the curve has degree ≤ 9. Schlessinger's 1973 criterion [26] can be used to show that the cone over an abelian variety of dimension ≥ 2 is never smoothable, and Mumford also used the same criterion to show that the cone over a curve of genus ≥ 2 embedded in sufficiently high degree is a non-smoothable singularity [21].
The cone over a K3 surface is a natural 3-dimensional generalisation of the cone over an elliptic curve, and our main result is the analogue of Pinkham's theorem. Theorem 1.2. Let S be a general K3 surface with primitive polarisation of genus g. Then the affine cone over S is smoothable if and only if g ≤ 10 or g = 12.
We make more precise statements about what "general" means in the article. This is connected with Brill-Noether loci as well as the rank of the Picard lattice.

1.3
A polarised K3 surface S, L of genus g is a K3 surface S with an ample line bundle L such that L 2 = 2g − 2 > 0. We define H 0 (S, nL), so that the affine cone over S, L is X = C a (S, L) = Spec R(S). Now X is normal ( [30, (3.1)]) and Gorenstein [10, §5], and if S is nonsingular, then X has an isolated singularity at the vertex.

1.4
The singularity at the vertex P is resolved by a single blow up f : X → X at P , with exceptional divisor E ⊂ X that is isomorphic to S with normal bundle O S (−1). By the adjunction formula, K X = f * K X − E, so P is a log canonical 3-fold singularity. Moreover, P is an elliptic singularity; that is, R 2 f * O X = C P and R 1 f * O X = 0. Such singularities are important in the study of 3-folds of general type and the boundary of their moduli spaces (see [7]).

1.5
By [26,22], the C * -action on X induces a grading on T 1 X , the space of isomorphism classes of infinitesimal deformations of X: Example 1.6. Suppose S is a quartic hypersurface in P 3 defined by 3 i=0 x 4 i = 0, and let X ⊂ A 4 be the affine cone over S. Then where the grading is shifted by −4. Thus T 1 X (k) is nonzero in degrees −4 ≤ k ≤ 4 and the graded pieces have dimensions 1, 4, 10, 16, 19, 16, 10, 4, 1. Since X is a hypersurface singularity, it is clearly smoothable.

1.7
The "only if" part of Theorem 1.2 proceeds by showing that for g = 11 or g ≥ 13, T 1 X is concentrated in degree zero; that is, T 1 X (k) vanishes for k = 0. It then follows from work of Schlessinger, that the only deformations of X are cones. In fact, we show that T 1 X (k) vanishes for |k| ≥ 2 by Green's conjecture for curves on K3 surfaces. For |k| ≥ 1, we interpret vanishing of T 1 X (k) in terms of ramification of the map of moduli stacks ϕ g,k : P g,k → M g k , where ϕ g,k maps a pair S, C ∈ |kL| to the stable curve C (see Section 4 for more precise statements). This extends work of Mori and Mukai on the uniruledness of the moduli space of curves [18].

1.8
The "if" part is proved by sweeping out the cone. By [3], the general K3 surface S of genus g ≤ 10 or g = 12 is the anticanonical section of a Fano 3-fold. Thus the affine cone X over S is realised as a hyperplane section through the vertex of the cone Y over the Fano. If we vary the hyperplane section so that it misses the vertex, then we obtain a smoothing of the vertex of X.

1.9
Another viewpoint, is that sweeping out the cone gives a smoothing of the projective cone C p (V, L) over a variety V polarised by L. That is, C p (V, L) = Proj R(V ) [x], where x is the adjoined cone variable. A naive guess would be that all smoothings of the affine cone over a variety C a (V, L) are induced by smoothings of the projective cone C p (V, L). This is not true: Pinkham [23] gave an example of a variety V whose affine cone C a (V ) is a smoothable singularity, even though the projective cone C p (V ) is not. At first sight, Pinkham's example seems to be quite special; V is 0-dimensional and C a (V ) is Cohen-Macaulay but not Gorenstein or normal. We exhibit a 3-dimensional, normal and Gorenstein singularity C a (V ), which is smoothable even though C p (V ) is not. Our example is the affine cone over a particular surface of general type in its canonical model. Smoothing affine cones is more general than smoothing projective cones. On the other hand, if V is a nonsingular Calabi-Yau variety, in light of Pinkham's theorem on elliptic curves and Theorem 1.2 above, we may reasonably ask: Are all smoothings of C a (V ) induced by deformations of C p (V )?

1.10
We also give examples of K3 surfaces of genus 11 and ≥ 13 whose affine cone is smoothable. These are hyperplane sections of the anticanonical model of a Fano 3-fold with b 2 ≥ 2, from the classification of Mori-Mukai [17]. Moreover, we exhibit K3 surfaces of genus 7 whose affine cone has at least two topologically distinct smoothings, analogous to the affine cone over a del Pezzo surface of degree 6.

1.11
We describe the contents of this paper. In section 2 we review certain criteria related to vanishing of graded pieces of T 1 X , a formula for computing graded pieces of T 1 X , and our example (2.19) of a 3-dimensional singularity whose projective cone is not smoothable, from 1.9. We also define certain deformation functors and morphisms between them, which relax projective normality conditions (2.8, 2.10, 2.14). We then establish generalised criteria for formal smoothness of those morphisms of functors (2.11, 2.14). As applications we give a short proof of Pinkham's theorem about smoothability of the affine cone over an elliptic curve (2.13) and prove that the affine cone over any polarised abelian variety of dimension ≥ 2 is not smoothable (2.17). In Section 3 we study vanishing of T 1 X (k) using Wahl's criterion and Green's conjecture for curves on a K3 surface. This also has an interesting interpretation via the classification of Fano 3-folds of index > 1 in terms of Clifford index. The main technical part of the proof is in Section 4, where we prove vanishing of T 2 Review and properties of graded T 1 X 2.1 Criterion for negative gradedness Let Art C be the category of Artin local C-algebras with residue field C and (Sets) be the category of sets. For an algebraic scheme X, let Def X : Art C → (Sets) be the usual deformation functor (cf. [27, 2.4.1]). For a projective scheme X ֒→ P N , let Hilb X := Hilb X֒→P N : Art C → (Sets) be the Hilbert functor parametrizing embedded deformations of X ֒→ P N (cf. [27, 3.2

.1]).
Let (X, L) be a polarized manifold, that is, X is a smooth projective variety such that dim X ≥ 1 and L is an ample line bundle. Let C a (X, L) := Spec k≥0 H 0 (X, kL) be the affine cone over (X, L). By [13, 8.8.6], C a (X, L) is normal. We also have the following property.
Proposition 2.2. Let (X, L) be a polarized manifold. Assume that H i (X, kL) = 0 for all 0 < i < dim X and k ∈ Z. Then we have the following: (i) The cone C a (X, L) is Cohen-Macaulay; (ii) If ω X ≃ L ⊗m for some m ∈ Z, then C a (X, L) is Gorenstein.
Proof. For (i), it is enough to check the conditions (a) and (b) in [10, 5.1.6(ii)]. We can check (a) by the construction of C a (X, L). The condition (b) is nothing but our assumption. Part (ii) follows from [10, 5.1.9].
Write C a := C a (X, L) for the affine cone over X. For k ∈ Z ≥0 , let A k := C[t]/(t k+1 ) and T  [30, (3.2)], the C * -action on C a induces a grading T 1 Ca = k∈Z T 1 Ca (k). Let C a → C a be the blowup at the vertex P ∈ C a . Then C a ≃ Tot(L ∨ ) := Spec O X k≥0 L k is the total space of the dual line bundle L ∨ of L. The punctured cone C ′ a := C a P is isomorphic to C a X if we view X as the zero section in C a . Note that C ′ a ≃ Spec O X k∈Z L k . We have the inclusion map ι : C ′ a → C a , the bundle π : C a → X, and the C * -bundle C ′ a → X, which we also denote by π. Restricting the short exact sequence 0 → π * L → T Ca → π * T X → 0 of [26, §4] because π * L| C ′ a is the trivial line bundle. More precisely, by [30,Proposition 3.3], we have isomorphisms where E L is the extension . We use the following criterion about the vanishing of the graded pieces T 1 Ca (k) when L induces a projectively normal embedding in P n . and it is formally smooth. Every deformation of C a (X, L) is a cone.
We describe the eigencomponent T 1 Ca (k) for an arithmetically Gorenstein embedding X ⊂ P n . Proposition 2.4. Let X ⊂ P n be a smooth, arithmetically Gorenstein variety of dimension ≥ 2, and let C a = C a (X, O X (1)) be the affine cone over X. If dim X ≥ 3, then Since codim P C a ≥ 3 and C a is Cohen-Macaulay, we have . By the projection formula and the Leray spectral sequence, we know that If X is not a surface, then H 1 (O X (k)) = H 2 (O X (k)) = 0 for all k because X is arithmetically Gorenstein. Thus the long exact sequence associated to (1) above gives T 1 Ca (k) = H 1 (T X (k)). Now suppose that X is an arithmetically Gorenstein surface with ω X = O(c). Then H 1 (O X (c + k)) vanishes for all k, and this gives an inclusion of T 1 Ca (c + k) in H 1 (T X (c+k)) for all k. For k > 0, we have H 2 (O X (c+k)) = 0 by Kodaira vanishing, and thus T 1 Ca (c + k) = H 1 (T X (c + k)) for all k > 0. Now T 1 Ca (c + k) ∼ = T 1 Ca (c − k) by a theorem of Wahl [31, §2.3], and this completes the proof.

Generalised criteria
In the following, we generalise Proposition 2.3 (i) and (ii) for a general ample line bundle L. That is, we do not assume that X is projectively normal. For that purpose, we prepare several notions on projective cones and two deformation functors in Definitions 2.6 and 2.8. We do not use these directly in the proof of Theorem 1.2, but we give some other applications, and the functors may be useful in studying some of the questions raised in Section 5.
to be the projective cone over (X, L) and H t ⊂ C p (X, L) to be the divisor defined by the variable t. There is a natural C * -action on R(X, L)[t] which is trivial on t, and this induces a C * -action on C p (X, L).  (1) is not necessarily very ample. We can also check that Next we define two deformation functors associated to an algebraic scheme with a closed subscheme and a line bundle.
) Let X be an algebraic scheme, D ⊂ X be its closed subscheme, and L be a line bundle on X.
, and a closed subscheme Then we obtain a deformation functor Def (X,D) : Art C → (Sets) of the pair (X, D).
be the set of isomorphism classes of deformations of (X, L) over A. Then we obtain a deformation functor Def (X,L) : Art C → (Sets) of the pair (X, L).
Remark 2.9. Let (X, L) be a polarized manifold. Let T 1 (X,L) := Def (X,L) (A 1 ) be the tangent space for Def (X,L) . It is known that T 1 (X,L) ≃ H 1 (X, E L ) and we can take H 2 (X, E L ) as an obstruction space, where E L is the sheaf as in (3)   Using this morphism of functors, we can generalize Proposition 2.3(i) as follows.
Proposition 2.11. Let (X, L) be a polarized manifold and let C p := C p (X, L), C a := C a (X, L) be the projective (respectively affine) cones over X. Assume that T 1 Ca (k) vanishes for all k > 0. Then the restriction morphism Φ : Def (Cp,Ht) → Def Ca is formally smooth.
Proof. We follow the approach of [2, Theorem 12.1], that is, we shall prove the following: (a) The tangent map dΦ : The vertical homomorphisms are induced by open immersions ι p : C ′ p ֒→ C p and ι a : C ′ a ֒→ C a and they are injective since C a and C p are normal. We will show that . This is sufficient since we can construct a deformation of be the eigendecompositions with respect to the C * -actions on C ′ p and C ′ a . Since all the homomorphisms in the above diagram are C * -equivariant, we have a homomorphism dΦ ′ (k) : Since π p is an affine morphism, we obtain an isomorphism by the projection formula and (π p ) by the projection formula for the affine morphism π a and (π a ) * O C ′ a ≃ k∈Z O X (kL). Hence, for k ∈ Z, we obtain Next consider the situation in (b). We shall construct a lift (C p, between the obstruction spaces of the functors. This is injective since we have a commutative diagram Thus we obtain a lift of (C p,A , H t,A ) over A ′ as required.
We define smoothability as follows.
Definition 2.12. Let X be an algebraic scheme. We say that X is smoothable if there exists a deformation φ : X → S of X over some quasi-projective curve S whose fiber X s := φ −1 (s) is smooth for general s ∈ S.
As a corollary of Proposition 2.11, we obtain a quick proof of Pinkham's theorem on cones over elliptic curves. Proof. Note that T 1 (4). This vanishes for all k > 0 by the exact sequence (3) and Ca (k) = 0 for all k > 0 and, by Proposition 2.11, the restriction morphism Φ : Def (Cp(E,L),Ht) → Def Ca(E,L) is formally smooth.
Assume that C a (E, L) is smoothable. By the formal smoothness of Φ, the projective cone C p (E, L) also has a smoothing C p → S over some quasi-projective curve S. Since −K Cp(E,L) is ample, the general fibre C p,s is a smooth del Pezzo surface of degree d. Hence we obtain d ≤ 9 by the classification of del Pezzo surfaces.
Conversely assume that d ≤ 9. For a given (E, L), we can explicitly construct a smooth del Pezzo surface S such that E ∈ |−K S | and L ≃ O S (−K S )| C . For example, first choose an embedding ι E : E ֒→ P 2 as a cubic curve and take p 1 , . . . , p 9−d ∈ E such that L ≃ O E (3) ⊗ O E (−p 1 − · · · − p 9−d ). Let S → P 2 be the blow-up at p 1 , . . . , p 9−d . Note that we can choose the embedding ι E such that p 1 , . . . , p 9−d are in general position and S is del Pezzo. Then we see that L ≃ O S (−K S )| C by the construction. For this S, we see that C a (E, L) is a hypersurface in C a (S, −K S ) since H 0 (S, −mK S ) → H 0 (E, mL) is surjective by H 1 (S, −(m − 1)K S ) = 0 for all m ≥ 1. Thus we can construct a smoothing by sweeping out the cone.
Using the functor Def (X,L) as in Definition 2.8, we have the following analogue of Proposition 2.3 (ii) for a general polarization.
Proposition 2.14. Let X be a smooth projective variety and L an ample line bundle on X. Assume that dim X ≥ 1 and H 1 (X, kL) = 0 for all k > 0. (ii) Suppose that T 1 Ca (k) = 0 for k = 0. Then the morphism Γ is formally smooth. Thus C a (X, L) has only conical deformations.
Remark 2.15. The morphism Γ of Proposition 2.14 can also be formulated as a morphism of functors Γ w : Hilb w X → Def Ca(X,L) , where Hilb w X denotes the Hilbert functor of embedded deformations of X in weighted projective space wP n via the Proj-construction X = Proj k≥0 H 0 (X, kL). . A general projective variety X has non-algebraic deformations. By considering deformations of (X, L), we can avoid this problem.
We see that H 0 (X A , kL A ) is flat over A by [30,Corollary 0.4.4] and H 1 (X, kL) = 0 for all k > 0. Hence C a (X A , L A ) is a deformation of C a (X, L) and we can define Γ by letting Γ A ((X A , L A )) := C a (X A , L A ).
(ii) We follow the approach of [2, Theorem 12.1] as in Proposition 2.11. That is, we shall prove the following: Let ι : C ′ a ֒→ C a be the open immersion of the punctured neighborhood. Then ι * : Def Ca → Def C ′ a is the restriction by ι, and we define Γ ′ := ι * •Γ : Def (X,L) → Def C ′ a to be the composition. Then the tangent map dΓ ′ is decomposed as and ι * is injective since the cone C a is normal. We shall prove dΓ is surjective. We can describe dΓ ′ as the natural homomorphism where π a : C ′ a → X is the C * -bundle projection. Note that we have H 1 (C ′ a , π * a E L ) ≃ k∈Z H 1 (X, E L (kL)) and dΓ ′ is induced by the adjunction morphism E L → (π a ) * π * a E L . Thus we see that dΓ ′ is an isomorphism onto the degree 0 part of the target space. Hence dΓ is an isomorphism onto T 1 Ca (0). By this and the assumption T 1 Ca (k) = 0 for k = 0, we see that dΓ is surjective. This proves (a).
Next we shall prove (b). The obstruction class to lifting between the obstruction spaces. By the construction of the obstruction map and π * a E L ≃ T C ′ a , we have a commutative diagram Note that o ′′ is induced by the adjunction homomorphism E L → (π a ) * π * a E L which is a split injection. Hence o ′′ is injective and o Γ ′ is also injective.
We see that o Γ ′ (o(ξ A )) = 0 sinceξ A ∈ Def Ca (A) and its imageξ ′ A ∈ Def C ′ a (A) can be extended over A ′ . Hence we have o(ξ A ) = 0 and obtain (b). This concludes the proof of Proposition 2.14.
We think that the following result is known to experts, but we could not find it in the literature. We prove it as an application of Proposition 2.14.
Corollary 2.17. Let X be an abelian variety of dimension n ≥ 2 and L an ample line bundle on X. Then the affine cone C a = C a (X, L) has only conical deformations.

A smoothable affine cone with a non-smoothable projectivization
In [23, Ex. 2.11], Pinkham exhibited a Cohen-Macaulay curve singularity as the affine cone C a (X) over a certain projective 0-dimensional scheme X such that C a (X) is smoothable, but there is no smoothing induced by a deformation of the projective cone C p (X). Below, we construct a Gorenstein normal 3-fold singularity, the cone over a certain surface X of general type, such that C a (X) is smoothable but C p (X) is not. . The general and X = C a (S, K S ) is the cone over a divisor S of bidegree (3,4) in P 1 × P 2 under the Segre embedding in P 5 (S is the canonical model of a surface of general type with p g = 6 and K 2 = 11). Indeed, the 4 × 4 Pfaffians of M are the first three define the Segre embedding, and the last two cut out the divisor S. All deformations of X are obtained by varying the entries of M [15,32]. Thus after coordinate changes, the general fibre X ′ of any deformation of X is defined by the Pfaffians of where f ′ i = f i + h i for some polynomials h i . We first show that X is smoothable. Let λ be a nonzero constant, and choose h i sufficiently general with some terms of degree ≤ 1. Since λ is constant, Pfaffians 1 and 2 are redundant, and X ′ is a nonsingular complete intersection for suitably chosen h i . Now suppose that we restrict ourselves to deformations X ′ that are induced by a deformation of the projective cone C p (S) over S. Then λ ≡ 0 for degree reasons, and h i must have degree ≤ 3 -in particular, we see that the above smoothing is not induced by C p (S). Since λ = 0, X ′ passes through the origin, and a computation of the partial derivatives of Pfaffians 3, 4 and 5 shows that the Jacobian matrix of X ′ must have rank ≤ 2 there. Thus X ′ must be singular at the origin.
Remark 2.20. The above example is quite flexible. For example, we get 3-fold singularities with similar properties by taking a divisor S k in P 1 × P 2 of bidegree (k, k + 1) for any k ≥ 3.
3 Vanishing of T 1 X (k) for |k| ≥ 2 Theorem 3.1. Let S be a K3 surface with primitive polarisation L of Clifford index > 2. Let X be the affine cone over S, then T 1 X (k) vanishes for |k| ≥ 2.

3.2
The Clifford index of a smooth curve C is computed over all special linear systems g r d on C. Clifford index is a refinement of gonality. The general curve has maximal Clifford index g−1 2 , and using this terminology, Clifford's theorem states that Cliff C ≥ 0 with equality if and only if C has a g 1 2 . It follows from work of Green-Lazarsfeld [12] (see also Reid [24]), that the Clifford index is constant for all curves in a linear system |C| on a K3 surface. Thus we define the Clifford index of a K3 surface (S, L) to be Cliff C for any C in |L|.
The generic polarized K3 surface of genus g has maximal Clifford index g−1 2 , and so the hypothesis of Theorem 3.1 holds for general K3 surfaces of genus g ≥ 7.
Curves of Clifford index 0 are hyperelliptic, index 1 means trigonal or a plane quintic, and index 2 means tetragonal or a plane sextic [8, §0].
Example 3.3. The K3 surface of genus 6 is a complete intersection H 1 ∩ H 2 ∩ H 3 ∩ Q inside the Plücker embedding of Gr (2,5) in P 9 , where H i are hyperplanes and Q is a hyperquadric. We compute T 1 X has nonzero graded pieces in degrees −2, −1, 0, 1, 2 with dimensions 1, 10, 19, 10, 1. The generic curve of genus 6 has Clifford index 2, while the generic curve of genus 7 has Clifford index 3. Thus the theorem is sharp.

Wahl's criterion Theorem 3.1 is proved by using Koszul cohomology and
Green's conjecture for curves on a K3 surface, to show that S satisfies Wahl's criterion for vanishing of T 1 (k) for k ≤ −2.
Let (S, L) be a polarized K3 surface. By [25], we can choose C ∈ |L| a nonsingular irreducible curve. Since C is a hyperplane section of S ⊂ P g and the coordinate ring of S is Gorenstein, the Betti numbers of O C are the same as those of O S . Moreover, by adjunction C ⊂ P g−1 is a canonical curve, so we are reduced to studying the equations and syzygies of canonical curves.
3.6 Green's conjecture We refer to [11] for details on Koszul cohomology and Green's conjecture. For simplicity, we formulate everything in terms of Betti numbers. For a nonhyperelliptic canonical curve C ⊂ P g−1 of genus g, the free resolution of O C as an O P g−1 -module is 3 . This data is represented in a Betti table as follows: Thus for Wahl's criterion (5) to be verified, we need β 1,2 = β 2,2 = 0. This is equivalent to β g−3,1 = β g−4,1 = 0 by Koszul duality. Now, Green's conjecture relates nonvanishing of certain Betti numbers with existence of special linear systems on C: Conjecture 3.7 (Green [11]). Let C be a canonical curve in P g−1 . Then Proof of Theorem 3.1. Green's conjecture holds for curves on any K3 surface by Voisin [28,29] and Aprodu-Farkas [1]. Thus S satisfies Wahl's criterion if and only if C does not have a g r d with d − 2r ≤ g − 2 − (g − 4) = 2, which means that the Clifford index of S must be > 2.

Higher index Fano 3-folds, imprimitive embeddings and smoothings
Let S be a general K3 surface of genus ≤ 6, and write X = C a (S, O(1)). Now S has Clifford index ≤ 2, and in fact, T 1 X (k) does not vanish for some |k| ≥ 2. Here, we study the connection between this nonvanishing, imprimitive embeddings of K3 surfaces and Fano 3-folds of higher Fano index.
Fix I > 1 and take the affine cone Y = C a (S, O S (I)) over the nonprimitive embedding of S by O(I). By Proposition 2.4, we have In the table below, we list all nonsingular Fano 3-folds W, A of Fano index I > 1 with Pic W ≃ Z and ample generator A satisfying −K W = IA. Each entry of the table has two interpretations in terms of smoothings of general K3 surfaces of genus ≤ 6. Firstly, as a special subspace k≤0 T 1 X (kI) of T 1 X corresponding to a deformation of C a (S, O(1)) with total space C a (W, A). Secondly, as a deformation of C a (S, O S (I)) with total space C a (W, −K W ). We work this out in a series of examples below.
Example 3.10. Consider the cone X = C a (S, O(1)) over the quartic K3 surface from Example 1.6. Now, T 1 X contains an 11-dimensional subspace T 1 X (−2) ⊕ T 1 X (−4), corresponding to the deformation X → ∆ defined by where ∆ = C 11 (t ij , u). Let λ, µ : C → ∆ be maps λ : x → (x 2 , . . . , x 2 , x 4 ) and µ : y → (y, . . . , y, y 2 ), where for simplicity, we assume all coefficients are 1. Performing base change with respect to λ or µ induces one parameter smoothings of X, which we denote by X λ and X µ . The total space of X λ is the affine cone C a (V, O(1)) over a quartic Fano 3-fold V 4 ⊂ P 4 , and λ sweeps out the hyperplane section in C a (V ). On the other hand, the total space of X µ is the affine cone C a (W, O(1)) over W 4 ⊂ P (1, 1, 1, 1, 2), and µ sweeps out the weighted hyperplane section of weight 2 inside C a (W ). The subspace T 1 X (−2) ⊕ T 1 X (−4) was chosen so that X µ admits a weighted C *action. The subspaces T 1 X (−3) and T 1 X (−4) have similar properties, giving rise to smoothings of X that sweep out weighted hyperplanes in the cone C a (W, O(1)), where W is the 3-fold W 4 ⊂ P 4 (1, 1, 1, 1, k) for k = 3, 4. When k = 3, W has a 1 3 (1, 1, 1) quotient singularity, while k = 4 gives W ≃ P 3 . Example 3.11. Continuing with the quartic K3 surface S, we now take I = 4 and consider the affine cone Y = C a (S, O(4)). We see that Y is smoothable, because it is a hyperplane section of C a (P 3 , −K P 3 ). The smoothing given by sweeping out this hyperplane corresponds to the 1-dimensional vector space T 1 X (−4) ∼ = T 1 Y (−1). Example 3.12. Consider the affine cone X = C a (S, O(1)) over the K3 surface S of genus 6 from Example 3.3. The subspace T 1 X (−2) in T 1 X corresponds to a one parameter deformation of X, whose total space is the affine cone C a (W, O(1)) over the del Pezzo 3-fold of index 2: W = H 1 ∩ H 2 ∩ H 3 ∩ Gr(2, 5) in P 6 . The deformation is realised by varying the hyperquadric section Q = 0 cutting out X in C a (W ), to Q = t, where t is the deformation parameter.

Vanishing of T 1 (k) for |k| = 1
The following is a key result of this section.  4.2 Some moduli spaces (stacks) of K3 surfaces Before proving Theorem 4.1, we introduce the necessary material on moduli spaces of K3 surfaces. Good references for this are [3] and [6, §6]. The main point is that the spaces we consider are smooth as stacks, and so we can use infinitesimal methods to study morphisms between them.
Let F g be the moduli stack of polarized K3 surfaces (S, L) such that L is primitive and L 2 = 2g − 2. We define P g,k to be the moduli stack of pairs (S, C), where C is a stable curve in |kL| for some line bundle L such that (S, L) ∈ F g . Finally, we write M g for the moduli stack of stable curves of genus g. Then we have a forgetful map ϕ g,k : P g,k → M g k (6) sending (S, C) to C, where g k is the genus of C ∈ |kL|. Following the construction of [3, §3], we can show that P g,k is a smooth irreducible Deligne-Mumford stack. When k = 1, Beauville [3, (5.1)] shows that the vanishing in Theorem 4.1 is equivalent to generic finiteness of the morphism of stacks ϕ g : P g → M g , and Mukai [19,Theorem 7] proves that ϕ g is generically finite when g = 11 and g ≥ 13 (see below for definitions and notation). Beauville's proof actually shows that for each k > 0, the vanishing of H 1 (S, Ω 1 S (kL)) is equivalent to generic finiteness of the forgetful map ϕ g,k defined in (6) above.
Remark 4.3. After the first version of this article appeared, Angelo Lopez and Shigeru Mukai informed us that ϕ g is actually birational onto its image for g = 11 and g ≥ 13, cf. [5], [20].

4.4
When k ≥ 2, the following lemma is a bit stronger than the generic vanishing proved using Mori-Mukai. Thus it is enough to show the vanishing for k = 1, 2 in Theorem 4.1. S (k)) = 0. Then H 1 (Ω 1 S (k + 1)) = 0. Proof. Let C ∈ |L| be a smooth member. We compute H 1 (T S (k)). First use the long exact sequence associated to to show that h 1 (Ω 1 S | C (k + 1)) = 0 for k ≥ 2. Then for k ≥ 2, the lemma follows from the long exact sequence associated to Remark 4.6. By the Riemann-Roch formula, we can estimate the dimension of T 1 X (1) using h 1 (T S (1)) ≈ −χ(T S (1)) = 20 − (2g − 2). This shows that Fano 3-folds of genus g > 10 are superabundant: they are "not expected" to exist.

Mukai's construction
In order to prove that ϕ g,k is generically finite, we use the following theorem of Mori-Mukai [18]: Assumption ( * ) Let S ⊂ P m be a smooth K3 surface with m ≥ 5 such that S is set-theoretically an intersection of quadrics, and the map is an isomorphism. Suppose that S contains an irreducible smooth curve C such that (1)) is an isomorphism and deg C ≥ m + 1, so that p a (C) > 0. Let H = S ∩ L be a smooth transverse hyperplane section of S such that C ∩ H is in general position in L. . Let S, Γ = C +H be a pair satisfying assumption ( * ). Then for every embedding i : Γ → S ′ into a K3 surface S ′ , there exists an isomorphism I : S → S ′ whose restriction to Γ coincides with i.
Proposition 4.9. The map ϕ g,k is generically finite for g = 11 or g ≥ 13.
Proof. It is enough to construct a pair (S, Γ g,k ) ∈ P g,k such that In [19, §4], Mukai constructed such pairs for g = 11, g ≥ 13 when k = 1. We summarise the construction here. Let E ⊂ P 5 be a sextic normal elliptic curve. Let S := Q 1 ∩ Q 2 ∩ Q 3 be a smooth complete intersection of three quadrics Q i which contain E. Since S contains an elliptic curve, there is an elliptic fibration S → P 1 . Let H ∈ |O S (1)| be a general hyperplane section. We can assume that S contains the following singular fibres: where E i ≃ P 1 and E i · H = i for all i = 1, . . . , 4. Let Γ := E ∪ H. Then Γ is a stable curve of genus 11. We can check that (S, Γ) satisfies Assumption ( * ). Let We construct Γ g for g ≥ 19 by adding smooth fibres to Γ i for 13 ≤ i ≤ 18. Then since (S, Γ g ) ∈ P g,1 , we see by Theorem 4.8 that ϕ −1 g,1 (Γ g ) = {(S, Γ g )}. This is the construction due to Mukai.
Next we consider the case k > 1. The linear system |Γ g | is free. Indeed, if there is a fixed curve C ⊂ Bs |Γ g |, then we see that C = E i for some i and (Γ g − C) 2 = 0 by Saint-Donat's classification. This does not happen since Γ g − C is ample by construction. Take a smooth member C g,k−1 ∈ |(k − 1)Γ g | and define Γ g,k := Γ g ∪ C g,k−1 . Thus (S, Γ g,k ) is in P g,k and ϕ −1 g,k (Γ g,k ) = {(S, Γ g,k )}. Note that for g ≤ 10, ϕ g is not generically finite for dimension reasons, and ϕ 12 is also not generically finite (essentially due to the existence of Fano 3-folds of genus 12, cf. [3]). Thus combining Theorem 4.1 and Proposition 2.4 we get See the next section for various comments about the generality assumption.

Smoothings and Fano threefolds
We prove the main theorem, after which we present several remarks and examples.
Theorem 5.1. Let S be a general K3 surface of genus g. The affine cone over S is smoothable if and only if g ≤ 10 or g = 12.
Proof. Let X = C a (S, O(1)) denote the affine cone over S. If g = 11 or g ≥ 13, then X has only conical deformations by Corollary 4.10 and Proposition 2.3. If g ≤ 10 or g = 12 then by [3, Corollary 4.1], S is an anticanonical member of a smooth Fano 3-fold W , that is, S ∈ |−K W |. Let C a (W, −K W ) be the affine cone over the pair (W, −K W ) and σ ∈ H 0 (W, −K W ) be the defining section of S. Then we can regard C a (S, L) as a divisor in C a (W, −K W ) defined by the section σ since H 0 (W, −mK W ) → H 0 (S, kL) is surjective for all m ≥ 1 by H 1 (W, −(m−1)K W ) = 0. Let X ⊂ C a (W, −K W ) × A 1 be a divisor defined by a function σ + λ, where λ is the parameter of the affine line A 1 . Then we have a deformation X → A 1 of X. The fibre over 0 is X, and the general fibre X t is nonsingular, because the general fibers avoid the vertex. This is a smoothing of X by sweeping out the anticanonical member of W .

5.2
The cone over a K3 surface with g = 11 or g ≥ 13 can nevertheless be smoothable If a K3 surface S is an anticanonical section of a Fano 3-fold with b 2 ≥ 2 from the Mori-Mukai classification [17], then C a (S, O(1)) is smoothable. Thus there are K3 surfaces of genus 11 and ≥ 13 whose affine cone is smoothable. For such K3 surfaces, Theorem 4.1 does not apply, and H 1 (Ω 1 S (L)) does not vanish. Example 5.3. Let S be a hypersurface of bidegree (2,3) in P 1 ×P 2 . Since S is a section of |−K P 1 ×P 2 |, we see that (S, −K P 1 ×P 2 | S ) is a K3 surface of genus 28. Nevertheless, we obtain a smoothing of the affine cone C a (S, O(1)), simply by sweeping out the cone inside C a (P 1 × P 2 , −K P 1 ×P 2 ).
Example 5.4. Suppose S is a K3 of genus 13. By [17], there are five distinct deformation families of Fano 3-folds with g = 13. Two each with b 2 = 2 and b 2 = 3, and one with b 2 = 4. The cone C a (S, O(1)) over a general S is not smoothable, but if we specialise C a (S, O(1)) to C a (S ′ , O(1)) where S ′ is the hyperplane section of one of the above Fano 3-folds, then C a (S ′ , O(1)) is smoothable. Thus we see that there are at least five strata in the moduli space of genus 13 K3 surfaces, for which the cone over a K3 surface in such a stratum is smoothable.
Example 5.5. Similarly, if S is a K3 of genus 11, then by [17], there are four families of Fano 3-folds with g = 11. Three with b 2 = 2 and one with b 2 = 3. The general K3 of genus 11 is not a hyperplane section of any Fano 3-fold. 5.6 K3 surfaces of genus > 32 Suppose S is a K3 surface of genus ≥ 13. The only smoothings of C a (S, O(1)) that we know of, are induced by Fano 3-folds appearing in the classification of Mori-Mukai [17], in the same way as the above examples. If S has genus > 32, exceeding the maximum appearing in [17], then any smoothing of C a (S, O(1)) does not lift to the projective cone C p (S, O(1)). In spite of Example 2.19, we expect that C a (S, O(1)) is not smoothable for any S of sufficiently large genus. An equivalent question (cf. [3, §5.4]) is the following: Is ϕ g,k actually finite and unramified for g > 32?
5.7 K3 surfaces whose affine cone has at least two distinct smoothings We give an example of a K3 surface S of genus 7 which is a hyperplane section of two topologically distinct anticanonical Fano 3-folds. It follows that the affine cone C a (S, O(1)) has two topologically distinct smoothings, obtained by sweeping out the cone over the two different Fano 3-folds. First recall the following famous example: Example 5.8. The degree 6 del Pezzo surface Y is a hyperplane section of V 1 = V : (1, 1) ⊂ P 2 × P 2 and V 2 = P 1 × P 1 × P 1 . Thus C a (Y, −K Y ) has two distinct smoothings.
Inspired by this, we found the following example: Example 5.9. Let Y be the degree 6 del Pezzo surface, and take π : S → Y a double cover branched in B ∈ |−2K Y |. Then S is a K3 surface of degree 12 in P 7 . By Example 5.8, The W i are Fano 3-folds with distinct topology. Indeed, W 1 (respectively W 2 ) is number 2.6b (resp. 3.1) of the classification [17]. Moreover, both W 1 and W 2 contain S as a section of |−K W i |, because W i ∩ π * i H i = S. Thus the affine cone C a (S, O S (1)) is a hyperplane section of C a (W i , −K W i ) ⊂ A 9 for each i, and so C a (S, O S (1)) has two topologically distinct smoothings.

5.10
Hyperelliptic and trigonal K3 surfaces In view of Theorem 3.1, it would be interesting to systematically study cones over hyperelliptic and trigonal K3 surfaces, and other K3 surfaces with Clifford index ≤ 2. These K3 surfaces are not general in the sense of Theorem 1.2. For example, we would expect that the genus bound on smoothable cones over hyperelliptic K3 surfaces is given by the genus bound on hyperelliptic Fano 3-folds.

5.11
Quasismooth K3 surfaces It would be very interesting to generalise Theorem 1.2 to the case of affine cones over quasismooth K3 surfaces embedded in weighted projective space. Some applications of this are worked out in [7]. This motivates future work.