# Smoothing cones over K3 surfaces

Stephen Coughlan  Taro Sano

###### Abstract

We prove that the affine cone over a general primitively polarised K3 surface
of genus $g$ is smoothable if and only if $g\leq 10$ or $g=12$. We discuss
connections with the classification of Fano threefolds, and several examples
of singularities with special behaviour.

##  1 Introduction

### 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 $\leq 9$. Schlessinger’s 1973
criterion [26] can be used to show that the cone over an abelian variety of
dimension $\geq 2$ is never smoothable, and Mumford also used the same
criterion to show that the cone over a curve of genus $\geq 2$ 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 $g\leq 10$ 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

$R(S)=\bigoplus_{n=0}^{\infty}H^{0}(S,nL),$

so that the affine cone over $S,L$ is $X=C_{a}(S,L)=\operatorname{Spec}R(S)$.
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
$f\colon\widetilde{X}\to X$ at $P$, with exceptional divisor
$E\subset\widetilde{X}$ that is isomorphic to $S$ with normal bundle
$\mathcal{O}_{S}(-1)$. By the adjunction formula,
$K_{\widetilde{X}}=f^{*}K_{X}-E$, so $P$ is a log canonical 3-fold
singularity. Moreover, $P$ is an elliptic singularity; that is,
$R^{2}f_{*}\mathcal{O}_{\widetilde{X}}=\mathbb{C}_{P}$ and
$R^{1}f_{*}\mathcal{O}_{\widetilde{X}}=0$. 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 $\mathbb{C}^{*}$-action on $X$ induces a grading on
$T^{1}_{X}$, the space of isomorphism classes of infinitesimal deformations of
$X$:

$T^{1}_{X}=\bigoplus_{k\in\mathbb{Z}}T^{1}_{X}(k).$

######  Example 1.6.

Suppose $S$ is a quartic hypersurface in $\mathbb{P}^{3}$ defined by
$\sum_{i=0}^{3}x_{i}^{4}=0$, and let $X\subset\mathbb{A}^{4}$ be the affine
cone over $S$. Then

$T^{1}_{X}=\mathbb{C}[x_{0},x_{1},x_{2},x_{3}]/(x_{0}^{3},x_{1}^{3},x_{2}^{3},x_{3}^{3}),$

where the grading is shifted by $-4$. Thus $T^{1}_{X}(k)$ is nonzero in
degrees $-4\leq k\leq 4$ 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
$g\geq 13$, $T^{1}_{X}$ is concentrated in degree zero; that is,
$T^{1}_{X}(k)$ vanishes for $k\neq 0$. 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 $|k|\geq 2$ by Green’s conjecture for curves
on K3 surfaces. For $|k|\geq 1$, we interpret vanishing of $T_{X}^{1}(k)$ in
terms of ramification of the map of moduli stacks
$\varphi_{g,k}\colon\mathcal{P}_{g,k}\to\overline{\mathcal{M}}_{g_{k}}$, where
$\varphi_{g,k}$ maps a pair $S,C\in|kL|$ 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 $g\leq 10$ 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,
$C_{p}(V,L)=\operatorname{Proj}R(V)[x]$, 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 $\geq 13$ whose affine
cone is smoothable. These are hyperplane sections of the anticanonical model
of a Fano 3-fold with $b_{2}\geq 2$, 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 $\geq 2$ 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 $T^{1}_{X}(1)$ for a general K3
surface by extending work of Mori and Mukai on the uniruledness of the moduli
space of curves of genus 11. The last section contains the proof of Theorem
1.2 and the examples mentioned in 1.10. We also mention further questions
regarding hyperelliptic and trigonal K3 surfaces, and quasismooth K3 surfaces.

We work over the complex numbers.

#### Acknowledgements.

We thank Paul Hacking, Yoshinori Namikawa and Miles Reid for useful
discussions. We also thank Angelo Lopez and Shigeru Mukai for helpful comments
on the first version of this article. SC was supported by the DFG through
grant Hu 337-6/2, and ERC Advanced grant no. 340258, TADMICAMT. TS was
supported by Max Planck Institute for Mathematics, JSPS Research Fellowships
for Young Scientists and JST tenure track program.

##  2 Review and properties of graded $T_{X}^{1}$

###  2.1 Criterion for negative gradedness

Let $\operatorname{Art}_{\mathbb{C}}$ be the category of Artin local
$\mathbb{C}$-algebras with residue field $\mathbb{C}$ and
$(\operatorname{Sets})$ be the category of sets. For an algebraic scheme $X$,
let
${\operatorname{Def}_{X}}\colon{\operatorname{Art}_{\mathbb{C}}}\rightarrow(\operatorname{Sets})$
be the usual deformation functor (cf. [27, 2.4.1]). For a projective scheme
$X\hookrightarrow\mathbb{P}^{N}$, let
$\operatorname{Hilb}_{X}:=\operatorname{Hilb}_{X\hookrightarrow\mathbb{P}^{N}}\colon{\operatorname{Art}_{\mathbb{C}}}\rightarrow(\operatorname{Sets})$
be the Hilbert functor parametrizing embedded deformations of
$X\hookrightarrow\mathbb{P}^{N}$ (cf. [27, 3.2.1]).

Let $(X,L)$ be a polarized manifold, that is, $X$ is a smooth projective
variety such that $\dim X\geq 1$ and $L$ is an ample line bundle. Let
$C_{a}(X,L):=\operatorname{Spec}\bigoplus_{k\geq 0}H^{0}(X,kL)$ 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 $k\in\mathbb{Z}$. Then we have the following:

  1. (i)

The cone $C_{a}(X,L)$ is Cohen–Macaulay;

  2. (ii)

If $\omega_{X}\simeq L^{\otimes m}$ for some $m\in\mathbb{Z}$, 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
$k\in\mathbb{Z}_{\geq 0}$, let $A_{k}:=\mathbb{C}[t]/(t^{k+1})$ and
$T^{1}_{C_{a}}:=\operatorname{Def}_{C_{a}}(A_{1})$ be the tangent space to
$\operatorname{Def}_{C_{a}}$. Then by [22, Proposition 2.2] or [30, (3.2)],
the $\mathbb{C}^{*}$-action on $C_{a}$ induces a grading
$T^{1}_{C_{a}}=\bigoplus_{k\in\mathbb{Z}}T^{1}_{C_{a}}(k)$.

Let $\widetilde{C}_{a}\rightarrow C_{a}$ be the blowup at the vertex $P\in
C_{a}$. Then
$\widetilde{C}_{a}\simeq\operatorname{Tot}(L^{\vee}):=\operatorname{Spec}_{\mathcal{O}_{X}}\bigoplus_{k\geq
0}L^{k}$ is the total space of the dual line bundle $L^{\vee}$ of $L$. The
punctured cone $C_{a}^{\prime}:=C_{a}\smallsetminus P$ is isomorphic to
$\widetilde{C}_{a}\smallsetminus X$ if we view $X$ as the zero section in
$\widetilde{C}_{a}$. Note that
$C^{\prime}_{a}\simeq\operatorname{Spec}_{\mathcal{O}_{X}}\bigoplus_{k\in\mathbb{Z}}L^{k}$.
We have the inclusion map $\iota\colon C_{a}^{\prime}\to C_{a}$, the bundle
$\pi\colon\widetilde{C}_{a}\to X$, and the $\mathbb{C}^{*}$-bundle
$C_{a}^{\prime}\to X$, which we also denote by $\pi$.

Restricting the short exact sequence
$0\to\pi^{*}L\to\mathcal{T}_{\widetilde{C}_{a}}\to\pi^{*}\mathcal{T}_{X}\to 0$
of [26, §4] to $C_{a}^{\prime}$, we have

$0\to\mathcal{O}_{C_{a}^{\prime}}\to\mathcal{T}_{C_{a}^{\prime}}\to\pi^{*}\mathcal{T}_{X}\to
0,$ (1)

because $\pi^{*}L|_{C_{a}^{\prime}}$ is the trivial line bundle. More
precisely, by [30, Proposition 3.3], we have isomorphisms

$\mathcal{T}_{C_{a}^{\prime}}\simeq\pi^{*}\mathcal{E}_{L}\simeq\mathcal{T}_{\widetilde{C}_{a}}(-\log
X)\otimes\mathcal{O}_{C_{a}^{\prime}}$ (2)

where $\mathcal{E}_{L}$ is the extension

$0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{E}_{L}\rightarrow\mathcal{T}_{X}\rightarrow
0$ (3)

corresponding to $c_{1}(L)\in
H^{1}(X,\Omega^{1}_{X})\simeq\operatorname{Ext}^{1}(\mathcal{T}_{X},\mathcal{O}_{X})$.

We use the following criterion about the vanishing of the graded pieces
$T^{1}_{C_{a}}(k)$ when $L$ induces a projectively normal embedding in
$\mathbb{P}^{n}$.

######  Proposition 2.3.

([2, Theorem 12.1]) Let $(X,L)$ be a polarized manifold. Assume that $L$
induces a projectively normal embedding $\Phi_{|L|}\colon
X\hookrightarrow\mathbb{P}^{N}$.

  1. (i)

(Pinkham [22, Theorem 5.1]) Suppose that $T^{1}_{C_{a}}(k)=0$ for all $k>0$.
Write $C_{p}(X,L)\subset\mathbb{P}^{N+1}$ for the projective cone over
$\Phi_{|L|}\colon X\hookrightarrow\mathbb{P}^{N}$ as defined in 1.9. Then the
restriction map

$\operatorname{Hilb}_{C_{p}(X,L)}\rightarrow\operatorname{Def}_{C_{a}(X,L)}$

is formally smooth.

  2. (ii)

(Schlessinger [26, §4.3]) Suppose that $T^{1}_{C_{a}}(k)=0$ for all $k\neq 0$.
Then we have a canonical morphism of functors

$\operatorname{Hilb}_{X}\rightarrow\operatorname{Def}_{C_{a}(X,L)}$

and it is formally smooth. Every deformation of $C_{a}(X,L)$ is a cone.

We describe the eigencomponent $T^{1}_{C_{a}}(k)$ for an arithmetically
Gorenstein embedding $X\subset\mathbb{P}^{n}$.

######  Proposition 2.4.

Let $X\subset\mathbb{P}^{n}$ be a smooth, arithmetically Gorenstein variety of
dimension $\geq 2$, and let $C_{a}=C_{a}(X,\mathcal{O}_{X}(1))$ be the affine
cone over $X$. If $\dim X\geq 3$, then

$T^{1}_{C_{a}}(k)=H^{1}(X,\mathcal{T}_{X}(k)).$

If $X$ is a surface with $\omega_{X}=\mathcal{O}(c)$, then

$T^{1}_{C_{a}}(c-k)\simeq T^{1}_{C_{a}}(c+k)\simeq
H^{1}(X,\mathcal{T}_{X}(c+k))\text{ for }k\neq 0,$

and $T^{1}_{C_{a}}(c)\subseteq H^{1}(X,\mathcal{T}_{X}(c))$.

###### Proof.

We start from
$T^{1}_{C_{a}}=\operatorname{Ext}^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})$.
Since $\operatorname{codim}_{P}C_{a}\geq 3$ and $C_{a}$ is Cohen–Macaulay, we
have

$T^{1}_{C_{a}}=\operatorname{Ext}^{1}(\Omega^{1}_{C_{a}},\mathcal{O}_{C_{a}})=H^{1}(C_{a}^{\prime},\mathcal{T}_{C_{a}^{\prime}}).$

By the projection formula and the Leray spectral sequence, we know that
$H^{1}(\mathcal{O}_{C_{a}^{\prime}})=H^{1}(\bigoplus_{k\in\mathbb{Z}}\mathcal{O}_{X}(k))$
and
$H^{1}(\pi^{*}\mathcal{T}_{X})=H^{1}(\bigoplus_{k\in\mathbb{Z}}\mathcal{T}_{X}(k))$.

If $X$ is not a surface, then
$H^{1}(\mathcal{O}_{X}(k))=H^{2}(\mathcal{O}_{X}(k))=0$ for all $k$ because
$X$ is arithmetically Gorenstein. Thus the long exact sequence associated to
(1) above gives $T^{1}_{C_{a}}(k)=H^{1}(\mathcal{T}_{X}(k))$.

Now suppose that $X$ is an arithmetically Gorenstein surface with
$\omega_{X}=\mathcal{O}(c)$. Then $H^{1}(\mathcal{O}_{X}(c+k))$ vanishes for
all $k$, and this gives an inclusion of $T^{1}_{C_{a}}(c+k)$ in
$H^{1}(\mathcal{T}_{X}(c+k))$ for all $k$. For $k>0$, we have
$H^{2}(\mathcal{O}_{X}(c+k))=0$ by Kodaira vanishing, and thus
$T^{1}_{C_{a}}(c+k)=H^{1}(\mathcal{T}_{X}(c+k))$ for all $k>0$. Now
$T^{1}_{C_{a}}(c+k)\cong T^{1}_{C_{a}}(c-k)$ by a theorem of Wahl [31, §2.3],
and this completes the proof. ∎

###  2.5 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.

######  Definition 2.6.

(cf. Appendices to [16] and [9]) Let $(X,L)$ be a polarized manifold. Let

$R(X,L)[t]:=\bigoplus_{i\geq 0}\bigoplus_{k\geq 0}H^{0}(X,kL)\cdot t^{i}$

be a graded $\mathbb{C}$-algebra such that $\deg(\sigma_{k}\cdot t^{i})=k+i$,
where $\sigma_{k}\in H^{0}(X,kL)$. We define
$C_{p}(X,L):=\operatorname{Proj}R(X,L)[t]$ to be the projective cone over
$(X,L)$ and $H_{t}\subset C_{p}(X,L)$ to be the divisor defined by the
variable $t$. There is a natural $\mathbb{C}^{*}$-action on $R(X,L)[t]$ which
is trivial on $t$, and this induces a $\mathbb{C}^{*}$-action on $C_{p}(X,L)$.

######  Remark 2.7.

Let $\mathcal{O}_{C_{p}}(1)$ be the tautological sheaf corresponding to the
graded module $R(X,L)[t](1)$ shifted by degree $1$. Then we can check that
this is invertible and $H_{t}\in|\mathcal{O}_{C_{p}}(1)|$. Note that
$\mathcal{O}_{C_{p}}(1)$ is not necessarily very ample. We can also check that
$C_{p}\smallsetminus H_{t}\simeq C_{a}$. Let $\widetilde{C}_{p}\rightarrow
C_{p}$ be the blowup at the vertex $P$. Then
$\widetilde{C}_{p}\simeq\mathbb{P}(\mathcal{O}_{X}\oplus L^{\vee})$ is the
projectivization of $\mathcal{O}_{X}\oplus L^{\vee}$. We also have the
punctured cone $C_{p}^{\prime}:=C_{p}\smallsetminus P$ and
$C_{p}^{\prime}\simeq\widetilde{C}_{p}\smallsetminus
H_{0}\simeq\operatorname{Tot}(L)\simeq\operatorname{Spec}_{\mathcal{O}_{X}}\bigoplus_{k\geq
0}L^{-k}$, where $H_{0}\subset\widetilde{C}_{p}$ is the negative section.

Next we define two deformation functors associated to an algebraic scheme with
a closed subscheme and a line bundle.

######  Definition 2.8.

(cf. [27, 3.3.3, 3.4.4]) Let $X$ be an algebraic scheme, $D\subset X$ be its
closed subscheme, and $L$ be a line bundle on $X$.

  1. (i)

For $A\in\operatorname{Art}_{\mathbb{C}}$, a deformation of a pair $(X,D)$
over $A$ is a pair $(\xi,D_{A})$ which consists of $\xi=(X\hookrightarrow
X_{A}\rightarrow\operatorname{Spec}A)\in\operatorname{Def}_{X}(A)$, and a
closed subscheme $D_{A}\subset X_{A}$ which is flat over $A$ and has an
isomorphism $D_{A}\otimes_{A}k\simeq D$. Let $\operatorname{Def}_{(X,D)}(A)$
be the isomorphism classes of deformations of $(X,D)$ over
$A\in\operatorname{Art}_{\mathbb{C}}$. Then we obtain a deformation functor
$\operatorname{Def}_{(X,D)}\colon\operatorname{Art}_{\mathbb{C}}\rightarrow(\operatorname{Sets})$
of the pair $(X,D)$.

  2. (ii)

For $A\in\operatorname{Art}_{\mathbb{C}}$, a deformation of a pair $(X,L)$
over $A$ is a pair $(\xi,L_{A})$ which consists of $\xi=(X\hookrightarrow
X_{A}\rightarrow\operatorname{Spec}A)\in\operatorname{Def}_{X}(A)$, and
$L_{A}\in\operatorname{Pic}X_{A}$ with an isomorphism $L_{A}|_{X}\simeq L$.
Let $\operatorname{Def}_{(X,L)}(A)$ be the set of isomorphism classes of
deformations of $(X,L)$ over $A$. Then we obtain a deformation functor
$\operatorname{Def}_{(X,L)}\colon\operatorname{Art}_{\mathbb{C}}\rightarrow(\operatorname{Sets})$
of the pair $(X,L)$.

######  Remark 2.9.

Let $(X,L)$ be a polarized manifold. Let
$T^{1}_{(X,L)}:=\operatorname{Def}_{(X,L)}(A_{1})$ be the tangent space for
$\operatorname{Def}_{(X,L)}$. It is known that $T^{1}_{(X,L)}\simeq
H^{1}(X,\mathcal{E}_{L})$ and we can take $H^{2}(X,\mathcal{E}_{L})$ as an
obstruction space, where $\mathcal{E}_{L}$ is the sheaf as in (3) (cf. [27,
Theorem 3.3.11]).

######  Definition 2.10.

Let
$\operatorname{Def}_{(C_{p}(X,L),H_{t})}\colon\operatorname{Art}_{\mathbb{C}}\rightarrow(\operatorname{Sets})$
be the deformation functor of a pair $(C_{p}(X,L),H_{t})$ (cf. [27, 3.4.4]).
Then we have a restriction morphism

$\Phi\colon\operatorname{Def}_{(C_{p}(X,L),H_{t})}\rightarrow\operatorname{Def}_{C_{a}(X,L)}$

which is induced by the open immersion $C_{a}(X,L)\hookrightarrow C_{p}(X,L)$.

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 $T^{1}_{C_{a}}(k)$ vanishes for all $k>0$. Then the restriction
morphism
$\Phi\colon\operatorname{Def}_{(C_{p},H_{t})}\rightarrow\operatorname{Def}_{C_{a}}$
is formally smooth.

###### Proof.

We follow the approach of [2, Theorem 12.1], that is, we shall prove the
following:

  1. (a)

The tangent map $d\Phi\colon T^{1}_{(C_{p},H_{t})}\rightarrow T^{1}_{C_{a}}$
is surjective;

  2. (b)

Given $\xi_{A}:=(C_{p,A},H_{t,A})\in\operatorname{Def}_{(C_{p},H_{t})}(A)$.
Let $\bar{\xi}_{A}:=\Phi(\xi_{A})\in\operatorname{Def}_{C_{a}}(A)$ be its
image and assume that $\bar{\xi}_{A}$ can be lifted over a small extension
$A^{\prime}\in\operatorname{Art}_{\mathbb{C}}$ of $A$. Then there exists a
lift $\xi_{A^{\prime}}\in\operatorname{Def}_{(C_{p},H_{t})}(A^{\prime})$ of
$\xi_{A}$ over $A^{\prime}$.

First we shall check (a). Let
$\Phi^{\prime}\colon\operatorname{Def}_{(C_{p}^{\prime},H_{t})}\rightarrow\operatorname{Def}_{C_{a}^{\prime}}$
be the restriction by an open immersion $\iota^{\prime}\colon
C_{a}^{\prime}\hookrightarrow C_{p}^{\prime}$. We have a commutative diagram

$\textstyle{T^{1}_{(C_{p},H_{t})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\Phi}$$\scriptstyle{\iota_{p}^{*}}$$\textstyle{T^{1}_{C_{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{a}^{*}}$$\textstyle{T^{1}_{(C_{p}^{\prime},H_{t})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\Phi^{\prime}}$$\textstyle{T^{1}_{C^{\prime}_{a}}.}$

The vertical homomorphisms are induced by open immersions $\iota_{p}\colon
C_{p}^{\prime}\hookrightarrow C_{p}$ and $\iota_{a}\colon
C_{a}^{\prime}\hookrightarrow C_{a}$ and they are injective since $C_{a}$ and
$C_{p}$ are normal. We will show that
$T^{1}_{(C_{p},H_{t})}=(d\Phi^{\prime})^{-1}(T^{1}_{C_{a}})$. This is
sufficient since we can construct a deformation of $(C_{p},H_{t})$ from
$\eta\in(d\Phi^{\prime})^{-1}(T^{1}_{C_{a}})\subset
T^{1}_{(C_{p}^{\prime},H_{t})}$ by taking the push-forward of the structure
sheaf by the open immersion $\iota_{p}\colon C_{p}^{\prime}\hookrightarrow
C_{p}$.

Let
$T^{1}_{(C^{\prime}_{p},H_{t})}=\bigoplus_{k\in\mathbb{Z}}T^{1}_{(C_{p}^{\prime},H_{t})}(k)$
and
$T^{1}_{C^{\prime}_{a}}=\bigoplus_{k\in\mathbb{Z}}T^{1}_{C^{\prime}_{a}}(k)$
be the eigendecompositions with respect to the $\mathbb{C}^{*}$-actions on
$C^{\prime}_{p}$ and $C^{\prime}_{a}$. Since all the homomorphisms in the
above diagram are $\mathbb{C}^{*}$-equivariant, we have a homomorphism
$d\Phi^{\prime}(k)\colon T^{1}_{(C_{p}^{\prime},H_{t})}(k)\rightarrow
T^{1}_{C^{\prime}_{a}}(k)$ for all $k\in\mathbb{Z}$. By the assumption on
$T^{1}_{C_{a}}(k)$, it is enough to check that $d\Phi^{\prime}(k)\colon
T^{1}_{(C_{p}^{\prime},H_{t})}(k)\rightarrow T^{1}_{C^{\prime}_{a}}(k)$ is
surjective for all $k\leq 0$.

Let $\mathcal{E}_{L^{\vee}}$ be the locally free sheaf on $X$ defined by the
extension associated to $c_{1}(L^{\vee})\in H^{1}(X,\Omega^{1}_{X})$ as in
short exact sequence (3). In fact, since $c_{1}(L)=-c_{1}(L^{\vee})$, we have
$\mathcal{E}_{L}\simeq\mathcal{E}_{L^{\vee}}$ by [27, 3.3.10]. Then by [30,
Proposition 3.3], we have $\mathcal{T}_{C_{p}^{\prime}}(-\log
H_{t})\simeq\pi_{p}^{*}\mathcal{E}_{L^{\vee}}$ where $\pi_{p}\colon
C_{p}^{\prime}\simeq\operatorname{Tot}(L)\rightarrow X$ is the projection.

Since $\pi_{p}$ is an affine morphism, we obtain an isomorphism

$T^{1}_{(C_{p}^{\prime},H_{t})}\simeq
H^{1}(C_{p}^{\prime},\mathcal{T}_{C_{p}^{\prime}}(-\log
H_{t}))\simeq\bigoplus_{k\leq 0}H^{1}(X,\mathcal{E}_{L^{\vee}}(kL))$

by the projection formula and
$(\pi_{p})_{*}\mathcal{O}_{C^{\prime}_{p}}\simeq\bigoplus_{k\geq
0}\mathcal{O}_{X}(-kL)$. Hence, for $k\leq 0$, we obtain

$T^{1}_{(C_{p}^{\prime},H_{t})}(k)\simeq H^{1}(X,\mathcal{E}_{L^{\vee}}(kL)).$

Let $\pi_{a}\colon C_{a}^{\prime}\rightarrow X$ be the $\mathbb{C}^{*}$-bundle
projection. Then we have
$\mathcal{T}_{C^{\prime}_{a}}\simeq\pi_{a}^{*}\mathcal{E}_{L^{\vee}}$ by (2)
and

$T^{1}_{C^{\prime}_{a}}\simeq
H^{1}(C_{a}^{\prime},\mathcal{T}_{C^{\prime}_{a}})\simeq\bigoplus_{k\in\mathbb{Z}}H^{1}(X,\mathcal{E}_{L^{\vee}}(kL))$
(4)

by the projection formula for the affine morphism $\pi_{a}$ and
$(\pi_{a})_{*}\mathcal{O}_{C^{\prime}_{a}}\simeq\bigoplus_{k\in\mathbb{Z}}\mathcal{O}_{X}(kL)$.
Hence, for $k\in\mathbb{Z}$, we obtain

$T^{1}_{C^{\prime}_{a}}(k)\simeq H^{1}(X,\mathcal{E}_{L^{\vee}}(kL)).$

We see that $d\Phi^{\prime}(k)$ is an isomorphism for all $k\leq 0$ by the
description of $T^{1}_{(C_{p}^{\prime},H_{t})}(k)$ and
$T^{1}_{C_{a}^{\prime}}(k)$, and the following commutative diagram for
$C_{p}^{\prime}\simeq\operatorname{Spec}_{\mathcal{O}_{X}}\bigoplus_{k\leq
0}L^{k}$ and
$C^{\prime}_{a}\simeq\operatorname{Spec}_{\mathcal{O}_{X}}\bigoplus_{k\in\mathbb{Z}}L^{k}$:

$\textstyle{(\pi_{p})_{*}\mathcal{O}_{C^{\prime}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\pi_{a})_{*}\mathcal{O}_{C^{\prime}_{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{k\leq
0}\mathcal{O}_{X}(kL)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{k\in\mathbb{Z}}\mathcal{O}_{X}(kL).}$

Next consider the situation in (b). We shall construct a lift
$(C_{p,A^{\prime}},H_{t,A^{\prime}})\in\operatorname{Def}_{(C_{p},H_{t})}(A^{\prime})$
of $(C_{p,A},H_{t,A})$. The transformation
$\Phi^{\prime}\colon\operatorname{Def}_{(C_{p}^{\prime},H_{t})}\rightarrow\operatorname{Def}_{C_{a}^{\prime}}$
induces a homomorphism

$o(\Phi^{\prime})\colon
H^{2}(C_{p}^{\prime},\mathcal{T}_{C_{p}^{\prime}}(-\log H_{t}))\rightarrow
H^{2}(C_{a}^{\prime},\mathcal{T}_{C_{a}^{\prime}})$

between the obstruction spaces of the functors. This is injective since we
have a commutative diagram

$\textstyle{H^{2}(C_{p}^{\prime},\mathcal{T}_{C_{p}^{\prime}}(-\log
H_{t}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{k\leq
0}H^{2}(X,\mathcal{E}_{L^{\vee}}(kL))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(C_{a}^{\prime},\mathcal{T}_{C^{\prime}_{a}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{k\in\mathbb{Z}}H^{2}(X,\mathcal{E}_{L^{\vee}}(kL)).}$

Thus we can lift $\iota_{p}^{*}(\xi_{A})$ in
$\operatorname{Def}_{(C_{p}^{\prime},H_{t})}(A)$ to $A^{\prime}$. Let
$(C^{\prime}_{p,A^{\prime}},H_{t,A^{\prime}})$ in
$\operatorname{Def}_{(C_{p}^{\prime},H_{t})}(A^{\prime})$ be the corresponding
deformation. We see that
$(\iota_{p})_{*}\mathcal{O}_{C^{\prime}_{p,A^{\prime}}}$ is flat over
$A^{\prime}$ since $\bar{\xi}_{A}$ can be lifted over $A^{\prime}$. Thus we
obtain a lift of $(C_{p,A},H_{t,A})$ over $A^{\prime}$ 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 $\phi\colon\mathcal{X}\rightarrow S$ of $X$ over some quasi-
projective curve $S$ whose fiber $\mathcal{X}_{s}:=\phi^{-1}(s)$ is smooth for
general $s\in S$.

As a corollary of Proposition 2.11, we obtain a quick proof of Pinkham’s
theorem on cones over elliptic curves.

######  Corollary 2.13.

([22, §9]) Let $E$ be an elliptic curve and $L$ be an ample line bundle on $E$
of degree $d$. Then $C_{a}(E,L)$ has a smoothing if and only if $d\leq 9$.

###### Proof.

Note that $T^{1}_{C^{\prime}_{a}}(k)\simeq
H^{1}(E,\mathcal{E}_{L^{\vee}}(kL))$ as in (4). This vanishes for all $k>0$ by
the exact sequence (3) and $H^{1}(E,kL)=0$ for all $k>0$. Since $C_{a}$ is
normal, we have a $\mathbb{C}^{*}$-equivariant injection
$T^{1}_{C_{a}}\hookrightarrow T^{1}_{C^{\prime}_{a}}$ and this induces an
injection $T^{1}_{C_{a}}(k)\hookrightarrow T^{1}_{C^{\prime}_{a}}(k)$. Thus we
obtain $T^{1}_{C_{a}}(k)=0$ for all $k>0$ and, by Proposition 2.11, the
restriction morphism
$\Phi\colon\operatorname{Def}_{(C_{p}(E,L),H_{t})}\rightarrow\operatorname{Def}_{C_{a}(E,L)}$
is formally smooth.

Assume that $C_{a}(E,L)$ is smoothable. By the formal smoothness of $\Phi$,
the projective cone $C_{p}(E,L)$ also has a smoothing
$\mathcal{C}_{p}\rightarrow S$ over some quasi-projective curve $S$. Since
$-K_{C_{p}(E,L)}$ is ample, the general fibre $\mathcal{C}_{p,s}$ is a smooth
del Pezzo surface of degree $d$. Hence we obtain $d\leq 9$ by the
classification of del Pezzo surfaces.

Conversely assume that $d\leq 9$. For a given $(E,L)$, we can explicitly
construct a smooth del Pezzo surface $S$ such that $E\in|{-}K_{S}|$ and
$L\simeq\mathcal{O}_{S}(-K_{S})|_{C}$. For example, first choose an embedding
$\iota_{E}\colon E\hookrightarrow\mathbb{P}^{2}$ as a cubic curve and take
$p_{1},\ldots,p_{9-d}\in E$ such that
$L\simeq\mathcal{O}_{E}(3)\otimes\mathcal{O}_{E}(-p_{1}-\cdots-p_{9-d})$. Let
$S\rightarrow\mathbb{P}^{2}$ be the blow-up at $p_{1},\ldots,p_{9-d}$. Note
that we can choose the embedding $\iota_{E}$ such that $p_{1},\ldots,p_{9-d}$
are in general position and $S$ is del Pezzo. Then we see that
$L\simeq\mathcal{O}_{S}(-K_{S})|_{C}$ by the construction. For this $S$, we
see that $C_{a}(E,L)$ is a hypersurface in $C_{a}(S,-K_{S})$ since
$H^{0}(S,-mK_{S})\rightarrow H^{0}(E,mL)$ is surjective by
$H^{1}(S,-(m-1)K_{S})=0$ for all $m\geq 1$. Thus we can construct a smoothing
by sweeping out the cone. ∎

Using the functor $\operatorname{Def}_{(X,L)}$ 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 $\dim X\geq 1$ and $H^{1}(X,kL)=0$ for all $k>0$.

  1. (i)

We can define a canonical morphism of functors

$\Gamma\colon{\operatorname{Def}_{(X,L)}}\rightarrow\operatorname{Def}_{C_{a}(X,L)}.$

  2. (ii)

Suppose that $T^{1}_{C_{a}}(k)=0$ for $k\neq 0$. Then the morphism $\Gamma$ is
formally smooth. Thus $C_{a}(X,L)$ has only conical deformations.

######  Remark 2.15.

The morphism $\Gamma$ of Proposition 2.14 can also be formulated as a morphism
of functors

$\Gamma_{\\!w}\colon{{\operatorname{Hilb}^{w}_{X}}}\rightarrow\operatorname{Def}_{C_{a}(X,L)},$

where $\operatorname{Hilb}^{w}_{X}$ denotes the Hilbert functor of embedded
deformations of $X$ in weighted projective space $w\mathbb{P}^{n}$ via the
$\operatorname{Proj}$-construction $X=\operatorname{Proj}\bigoplus_{k\geq
0}H^{0}(X,kL)$.

######  Remark 2.16.

Christophersen–Kleppe treated a similar result (cf. [4, Corollary 5.2]).
However they assume $H^{2}(X,\mathcal{O}_{X})=0$ and the deformations preserve
polarizations (cf. [14, Example 21.2.5]). A general projective variety $X$ has
non-algebraic deformations. By considering deformations of $(X,L)$, we can
avoid this problem.

###### Proof.

(i) For $A\in\operatorname{Art}_{\mathbb{C}}$ and
$(X_{A},L_{A})\in\operatorname{Def}_{(X,L)}(A)$, let

$C_{a}(X_{A},L_{A}):=\operatorname{Spec}\bigoplus_{k\geq
0}H^{0}(X_{A},kL_{A}).$

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 $\Gamma$ by letting
$\Gamma_{A}((X_{A},L_{A})):=C_{a}(X_{A},L_{A})$.

(ii) We follow the approach of [2, Theorem 12.1] as in Proposition 2.11. That
is, we shall prove the following:

  1. (a)

The tangent map $d\Gamma\colon T^{1}_{(X,L)}\rightarrow T^{1}_{C_{a}}$ is
surjective;

  2. (b)

Given $\xi_{A}:=(X_{A},L_{A})\in\operatorname{Def}_{(X,L)}(A)$. Let
$\bar{\xi}_{A}:=\Gamma(\xi_{A})\in\operatorname{Def}_{C_{a}}(A)$ be its image
and assume that $\bar{\xi}_{A}$ can be lifted over a small extension
$A^{\prime}\in\operatorname{Art}_{\mathbb{C}}$ of $A$. Then there exists a
lift $\xi_{A^{\prime}}\in\operatorname{Def}_{(X,L)}(A^{\prime})$ of $\xi_{A}$
over $A^{\prime}$.

Let $\iota\colon C^{\prime}_{a}\hookrightarrow C_{a}$ be the open immersion of
the punctured neighborhood. Then
$\iota^{*}\colon{\operatorname{Def}_{C_{a}}}\rightarrow\operatorname{Def}_{C_{a}^{\prime}}$
is the restriction by $\iota$, and we define
$\Gamma^{\prime}:=\iota^{*}\circ\Gamma\colon{\operatorname{Def}_{(X,L)}}\rightarrow\operatorname{Def}_{C_{a}^{\prime}}$
to be the composition. Then the tangent map $d\Gamma^{\prime}$ is decomposed
as

$d\Gamma^{\prime}\colon{T^{1}_{(X,L)}}\xrightarrow{d\Gamma}T^{1}_{C_{a}}\xrightarrow{\iota^{*}}T^{1}_{C_{a}^{\prime}}$

and $\iota^{*}$ is injective since the cone $C_{a}$ is normal.

We shall prove $d\Gamma$ is surjective. We can describe $d\Gamma^{\prime}$ as
the natural homomorphism

$d\Gamma^{\prime}\colon{H^{1}(X,\mathcal{E}_{L})}\rightarrow
H^{1}(C_{a}^{\prime},\mathcal{T}_{C_{a}^{\prime}})\simeq
H^{1}(C_{a}^{\prime},\pi_{a}^{*}\mathcal{E}_{L}),$

where $\pi_{a}\colon C_{a}^{\prime}\rightarrow X$ is the
$\mathbb{C}^{*}$-bundle projection. Note that we have
$H^{1}(C_{a}^{\prime},\pi_{a}^{*}\mathcal{E}_{L})\simeq\bigoplus_{k\in\mathbb{Z}}H^{1}(X,\mathcal{E}_{L}(kL))$
and $d\Gamma^{\prime}$ is induced by the adjunction morphism
$\mathcal{E}_{L}\rightarrow(\pi_{a})_{*}\pi_{a}^{*}\mathcal{E}_{L}$. Thus we
see that $d\Gamma^{\prime}$ is an isomorphism onto the degree $0$ part of the
target space. Hence $d\Gamma$ is an isomorphism onto $T^{1}_{C_{a}}(0)$. By
this and the assumption $T^{1}_{C_{a}}(k)=0$ for $k\neq 0$, we see that
$d\Gamma$ is surjective. This proves (a).

Next we shall prove (b). The obstruction class to lifting $\xi_{A}$ to
$A^{\prime}$ is $o(\xi_{A})\in H^{2}(X,\mathcal{E}_{L})$. We show that
$o(\xi_{A})=0$. The morphism
$\Gamma^{\prime}\colon\operatorname{Def}_{(X,L)}\rightarrow\operatorname{Def}_{C_{a}^{\prime}}$
induces a linear map

$o_{\Gamma^{\prime}}\colon H^{2}(X,\mathcal{E}_{L})\rightarrow
H^{2}(C_{a}^{\prime},\mathcal{T}_{C_{a}^{\prime}})$

between the obstruction spaces. By the construction of the obstruction map and
$\pi_{a}^{*}\mathcal{E}_{L}\simeq\mathcal{T}_{C_{a}^{\prime}}$, we have a
commutative diagram

$\textstyle{H^{2}(X,\mathcal{E}_{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{o_{\Gamma^{\prime}}}$$\scriptstyle{o^{\prime}}$$\scriptstyle{o^{\prime\prime}}$$\textstyle{H^{2}(C_{a}^{\prime},\mathcal{T}_{C_{a}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{2}(C_{a}^{\prime},\pi_{a}^{*}\mathcal{E}_{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{2}(X,(\pi_{a})_{*}\pi_{a}^{*}\mathcal{E}_{L}).}$

Note that $o^{\prime\prime}$ is induced by the adjunction homomorphism
$\mathcal{E}_{L}\rightarrow(\pi_{a})_{*}\pi_{a}^{*}\mathcal{E}_{L}$ which is a
split injection. Hence $o^{\prime\prime}$ is injective and
$o_{\Gamma^{\prime}}$ is also injective.

We see that $o_{\Gamma^{\prime}}(o(\xi_{A}))=0$ since
$\bar{\xi}_{A}\in\operatorname{Def}_{C_{a}}(A)$ and its image
$\bar{\xi}^{\prime}_{A}\in\operatorname{Def}_{C_{a}^{\prime}}(A)$ can be
extended over $A^{\prime}$. Hence we have $o(\xi_{A})=0$ 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 $n\geq 2$ and $L$ an ample line
bundle on $X$. Then the affine cone $C_{a}=C_{a}(X,L)$ has only conical
deformations.

###### Proof.

We see that $\iota^{*}\colon T^{1}_{C_{a}}\rightarrow T^{1}_{C_{a}^{\prime}}$
is injective since $C_{a}$ is $S_{2}$ and $\operatorname{codim}_{C_{a}}P\geq
2$. We have an isomorphism
$T^{1}_{C_{a}^{\prime}}\simeq\bigoplus_{k\in\mathbb{Z}}H^{1}(X,\mathcal{E}_{L}(kL))$.
Note that $\mathcal{T}_{X}\simeq\mathcal{O}_{X}^{\oplus n}$. Thus
$H^{1}(X,\mathcal{T}_{X}(kL))=0$ and $H^{1}(X,kL)=0$ for any $k\neq 0$ by
Serre duality, Kodaira vanishing and $n\geq 2$. Hence we obtain
$H^{1}(X,\mathcal{E}_{L}(kL))=0$ and thus $T^{1}_{C_{a}}(k)=0$ for $k\neq 0$.
To conclude the proof, we apply Proposition 2.14 (ii). ∎

###  2.18 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.

######  Example 2.19.

Let $X\subset\mathbb{A}^{6}$ be the codimension 3 variety defined by the
$4\times 4$ Pfaffians of the skew matrix $M$ with homogeneous entries of
degrees $\left(\begin{smallmatrix}-1&1&1&1\\\ &1&1&1\\\ &&3&3\\\
&&&3\end{smallmatrix}\right).$ The general such $M$ is

$\begin{pmatrix}0&x_{1}&x_{2}&x_{3}\\\ &x_{4}&x_{5}&x_{6}\\\ &&f_{1}&f_{2}\\\
&&&f_{3}\end{pmatrix},$

and $X=C_{a}(S,K_{S})$ is the cone over a divisor $S$ of bidegree $(3,4)$ in
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ under the Segre embedding in
$\mathbb{P}^{5}$ ($S$ is the canonical model of a surface of general type with
$p_{g}=6$ and $K^{2}=11$). Indeed, the $4\times 4$ Pfaffians of $M$ are

$x_{1}x_{5}-x_{2}x_{4},\ x_{1}x_{6}-x_{3}x_{4},\ x_{2}x_{6}-x_{3}x_{5},\
x_{1}f_{3}-x_{2}f_{2}+x_{3}f_{1},\ x_{4}f_{3}-x_{5}f_{2}+x_{6}f_{1},$

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 $X^{\prime}$ of any
deformation of $X$ is defined by the Pfaffians of

$M^{\prime}=\begin{pmatrix}\lambda&x_{1}&x_{2}&x_{3}\\\ &x_{4}&x_{5}&x_{6}\\\
&&f^{\prime}_{1}&f^{\prime}_{2}\\\ &&&f^{\prime}_{3}\end{pmatrix},$

where $f^{\prime}_{i}=f_{i}+h_{i}$ for some polynomials $h_{i}$.

We first show that $X$ is smoothable. Let $\lambda$ be a nonzero constant, and
choose $h_{i}$ sufficiently general with some terms of degree $\leq 1$. Since
$\lambda$ is constant, Pfaffians 1 and 2 are redundant, and $X^{\prime}$ is a
nonsingular complete intersection for suitably chosen $h_{i}$.

Now suppose that we restrict ourselves to deformations $X^{\prime}$ that are
induced by a deformation of the projective cone $C_{p}(S)$ over $S$. Then
$\lambda\equiv 0$ for degree reasons, and $h_{i}$ must have degree $\leq 3$ —
in particular, we see that the above smoothing is not induced by $C_{p}(S)$.
Since $\lambda=0$, $X^{\prime}$ passes through the origin, and a computation
of the partial derivatives of Pfaffians 3, 4 and 5 shows that the Jacobian
matrix of $X^{\prime}$ must have rank $\leq 2$ there. Thus $X^{\prime}$ 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
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ of bidegree $(k,k+1)$ for any $k\geq 3$.

##  3 Vanishing of $T_{X}^{1}(k)$ for $|k|\geq 2$

######  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
$|k|\geq 2$.

### 3.2

The Clifford index of a smooth curve $C$ is

$\operatorname{Cliff}C=\min\\{d-2r\mid r\geq 1,\ d\leq g-1\\},$

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
$\left[\frac{g-1}{2}\right]$, and using this terminology, Clifford’s theorem
states that $\operatorname{Cliff}C\geq 0$ 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 $\operatorname{Cliff}C$ for any $C$ in $|L|$.

The generic polarized K3 surface of genus $g$ has maximal Clifford index
$\left[\frac{g-1}{2}\right]$, and so the hypothesis of Theorem 3.1 holds for
general K3 surfaces of genus $g\geq 7$. 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 $H_{1}\cap H_{2}\cap
H_{3}\cap Q$ inside the Plücker embedding of $\mathrm{Gr}(2,5)$ in
$\mathbb{P}^{9}$, 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.

###  3.4 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 $k\leq-2$.

######  Theorem 3.5.

(Wahl [31, Corollary 2.8]) Suppose the free resolution of $\mathcal{O}_{S}$
begins with

$\mathcal{O}_{S}\leftarrow\mathcal{O}_{\mathbb{P}}\leftarrow\mathcal{O}_{\mathbb{P}}(-2)^{a}\leftarrow\mathcal{O}_{\mathbb{P}}(-3)^{b}\leftarrow\dots.$
(5)

Then $T^{1}_{X}(k)=0$ for $k\leq-2$.

Let $(S,L)$ be a polarized K3 surface. By [25], we can choose $C\in|L|$ a
nonsingular irreducible curve. Since $C$ is a hyperplane section of
$S\subset\mathbb{P}^{g}$ and the coordinate ring of $S$ is Gorenstein, the
Betti numbers of $\mathcal{O}_{C}$ are the same as those of $\mathcal{O}_{S}$.
Moreover, by adjunction $C\subset\mathbb{P}^{g-1}$ 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 $C\subset\mathbb{P}^{g-1}$ of genus $g$, the
free resolution of $\mathcal{O}_{C}$ as an
$\mathcal{O}_{\mathbb{P}^{g-1}}$-module is

$\mathcal{O}_{C}\leftarrow\mathcal{F}_{0}\leftarrow\mathcal{F}_{1}\leftarrow\dots\leftarrow\mathcal{F}_{g-2}\leftarrow
0$

where $\mathcal{F}_{0}=\mathcal{O}_{\mathbb{P}}^{\beta_{0,0}}$,
$\mathcal{F}_{i}=\bigoplus_{j=1,2}\mathcal{O}_{\mathbb{P}}(-i-j)^{\beta_{i,j}}$
for $i=1,\dots,g-3$ and
$\mathcal{F}_{g-2}=\mathcal{O}_{\mathbb{P}}(-g-2)^{\beta_{g-2,3}}$. This data
is represented in a Betti table as follows:

$\begin{array}[]{cccccccccc}0&1&2&\dots&g-4&g-3&g-2\\\ \hline\cr\beta_{0,0}\\\
&\beta_{1,1}&\beta_{2,1}&\dots&\beta_{g-4,1}&\beta_{g-3,1}\\\
&\beta_{1,2}&\beta_{2,2}&\dots&\beta_{g-4,2}&\beta_{g-3,2}\\\
&&&&&&\beta_{g-2,3}\end{array}$

Thus for Wahl’s criterion (5) to be verified, we need
$\beta_{1,2}=\beta_{2,2}=0$. This is equivalent to
$\beta_{g-3,1}=\beta_{g-4,1}=0$ by Koszul duality. Now, Green’s conjecture
relates non-vanishing of certain Betti numbers with existence of special
linear systems on $C$:

######  Conjecture 3.7 (Green [11]).

Let $C$ be a canonical curve in $\mathbb{P}^{g-1}$. Then

$\beta_{p,1}(C,K_{C})\neq 0\iff C\text{ has a }g^{r}_{d}\text{ with }d\leq
g-1,r\geq 1\text{ and }d-2r\leq g-2-p.$

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 $d-2r\leq g-2-(g-4)=2$, which
means that the Clifford index of $S$ must be $>2$. ∎

###  3.8 Higher index Fano 3-folds, imprimitive embeddings and smoothings

Let $S$ be a general K3 surface of genus $\leq 6$, and write
$X=C_{a}(S,\mathcal{O}(1))$. Now $S$ has Clifford index $\leq 2$, and in fact,
$T^{1}_{X}(k)$ does not vanish for some $|k|\geq 2$. 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 $Y=C_{a}(S,\mathcal{O}_{S}(I))$ over the
nonprimitive embedding of $S$ by $\mathcal{O}(I)$. By Proposition 2.4, we have

$T^{1}_{Y}(k)\cong T^{1}_{X}(kI).$

In the table below, we list all nonsingular Fano 3-folds $W,A$ of Fano index
$I>1$ with $\operatorname{Pic}W\simeq\mathbb{Z}$ 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 $\leq 6$. Firstly, as a
special subspace $\bigoplus_{k\leq 0}T^{1}_{X}(kI)$ of $T^{1}_{X}$
corresponding to a deformation of $C_{a}(S,\mathcal{O}(1))$ with total space
$C_{a}(W,A)$. Secondly, as a deformation of $C_{a}(S,\mathcal{O}_{S}(I))$ with
total space $C_{a}(W,-K_{W})$. We work this out in a series of examples below.

$\begin{array}[]{ccccc}g&(W,A)&\text{Index}&\text{Fano description}\\\
\hline\cr 2&W_{6}\subset\mathbb{P}(1,1,1,2,3)&2&\text{del Pezzo 3-fold of
degree 1}\\\ 3&W_{4}\subset\mathbb{P}(1^{4},4)&4&\mathbb{P}^{3}\\\
&W_{4}\subset\mathbb{P}(1^{4},2)&2&\text{del Pezzo 3-fold of degree 2}\\\
4&W_{2,3}\subset\mathbb{P}(1^{5},2)&2&\text{cubic 3-fold}\\\
&W_{2,3}\subset\mathbb{P}(1^{5},3)&3&\text{quadric 3-fold}\\\
5&W_{2,2,2}\subset\mathbb{P}(1^{6},2)&2&\text{intersection of two quadrics}\\\
6&W=H_{1}\cap H_{2}\cap
H_{3}\cap\mathrm{Gr}(2,5)\subset\mathbb{P}^{6}&2&\text{del Pezzo 3-fold of
degree 5}\end{array}$

######  Remark 3.9.

The only higher index Fano 3-folds that are missing from the table are
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathbb{P}^{2}\times\mathbb{P}^{2}$. These have Picard rank $\rho>1$, and in
these cases, the K3 surfaces in question have genus $7$ but do not have
maximal Clifford index. These 3-folds make an appearance in Section 5.7.

######  Example 3.10.

Consider the cone $X=C_{a}(S,\mathcal{O}(1))$ over the quartic K3 surface from
Example 1.6. Now, $T^{1}_{X}$ contains an 11-dimensional subspace
$T^{1}_{X}(-2)\oplus T^{1}_{X}(-4)$, corresponding to the deformation
$\mathcal{X}\to\Delta$ defined by

$\mathcal{X}\colon(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+t_{00}x_{0}^{2}+t_{01}x_{0}x_{1}+\dots+t_{33}x_{3}^{2}+u=0)\subset\mathbb{A}^{4}\times\Delta,$

where $\Delta=\mathbb{C}^{11}(t_{ij},u)$. Let
$\lambda,\mu\colon\mathbb{C}\to\Delta$ be maps

$\lambda\colon x\mapsto(x^{2},\dots,x^{2},x^{4})\text{ and }\mu\colon
y\mapsto(y,\dots,y,y^{2}),$

where for simplicity, we assume all coefficients are 1. Performing base change
with respect to $\lambda$ or $\mu$ induces one parameter smoothings of $X$,
which we denote by $\mathcal{X}_{\lambda}$ and $\mathcal{X}_{\mu}$. The total
space of $\mathcal{X}_{\lambda}$ is the affine cone $C_{a}(V,\mathcal{O}(1))$
over a quartic Fano 3-fold $V_{4}\subset\mathbb{P}^{4}$, and $\lambda$ sweeps
out the hyperplane section in $C_{a}(V)$. On the other hand, the total space
of $\mathcal{X}_{\mu}$ is the affine cone $C_{a}(W,\mathcal{O}(1))$ over
$W_{4}\subset\mathbb{P}(1,1,1,1,2)$, and $\mu$ sweeps out the weighted
hyperplane section of weight 2 inside $C_{a}(W)$.

The subspace $T^{1}_{X}(-2)\oplus T^{1}_{X}(-4)$ was chosen so that
$\mathcal{X}_{\mu}$ admits a weighted $\mathbb{C}^{*}$-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
$C_{a}(W,\mathcal{O}(1))$, where $W$ is the 3-fold
$W_{4}\subset\mathbb{P}^{4}(1,1,1,1,k)$ for $k=3,4$. When $k=3$, $W$ has a
$\frac{1}{3}(1,1,1)$ quotient singularity, while $k=4$ gives
$W\simeq\mathbb{P}^{3}$.

######  Example 3.11.

Continuing with the quartic K3 surface $S$, we now take $I=4$ and consider the
affine cone $Y=C_{a}(S,\mathcal{O}(4))$. We see that $Y$ is smoothable,
because it is a hyperplane section of
$C_{a}(\mathbb{P}^{3},-K_{\mathbb{P}^{3}})$. The smoothing given by sweeping
out this hyperplane corresponds to the 1-dimensional vector space
$T^{1}_{X}(-4)\cong T^{1}_{Y}(-1)$.

######  Example 3.12.

Consider the affine cone $X=C_{a}(S,\mathcal{O}(1))$ 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 $C_{a}(W,\mathcal{O}(1))$ over the del Pezzo 3-fold of index 2:
$W=H_{1}\cap H_{2}\cap H_{3}\cap\mathrm{Gr}(2,5)$ in $\mathbb{P}^{6}$. 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.

##  4 Vanishing of $T^{1}(k)$ for $|k|=1$

The following is a key result of this section.

######  Theorem 4.1 (cf. Beauville [3, §5.2], Mukai [19, §4]).

Let $S$ be a general K3 surface with primitive polarisation $L$ of genus
$g=11$ or $g\geq 13$. Then

$H^{1}(S,\Omega^{1}_{S}(kL))=0$

for any $k\geq 1$.

###  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 $\mathcal{F}_{g}$ be the moduli stack of polarized K3 surfaces $(S,L)$
such that $L$ is primitive and $L^{2}=2g-2$. We define $\mathcal{P}_{g,k}$ to
be the moduli stack of pairs $(S,C)$, where $C$ is a stable curve in $|kL|$
for some line bundle $L$ such that $(S,L)\in\mathcal{F}_{g}$. Finally, we
write $\overline{\mathcal{M}}_{g}$ for the moduli stack of stable curves of
genus $g$. Then we have a forgetful map

$\varphi_{g,k}\colon\mathcal{P}_{g,k}\rightarrow\overline{\mathcal{M}}_{g_{k}}$
(6)

sending $(S,C)$ to $C$, where $g_{k}$ is the genus of $C\in|kL|$. Following
the construction of [3, §3], we can show that $\mathcal{P}_{g,k}$ 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
$\varphi_{g}\colon\mathcal{P}_{g}\to\overline{\mathcal{M}}_{g}$, and Mukai
[19, Theorem 7] proves that $\varphi_{g}$ is generically finite when $g=11$
and $g\geq 13$ (see below for definitions and notation). Beauville’s proof
actually shows that for each $k>0$, the vanishing of
$H^{1}(S,\Omega^{1}_{S}(kL))$ is equivalent to generic finiteness of the
forgetful map $\varphi_{g,k}$ defined in (6) above.

######  Remark 4.3.

After the first version of this article appeared, Angelo Lopez and Shigeru
Mukai informed us that $\varphi_{g}$ is actually birational onto its image for
$g=11$ and $g\geq 13$, cf. [5], [20].

### 4.4

When $k\geq 2$, 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.

######  Lemma 4.5.

Let $(S,L)$ be a polarized K3 surface as in Theorem 4.1. Suppose $k\geq 2$ and
$H^{1}(\Omega^{1}_{S}(k))=0$. Then $H^{1}(\Omega^{1}_{S}(k+1))=0$.

###### Proof.

Let $C\in|L|$ be a smooth member. We compute $H^{1}(T_{S}(k))$. First use the
long exact sequence associated to

$0\to\mathcal{O}_{C}(k)\to\Omega^{1}_{S}|_{C}(k+1)\to\Omega^{1}_{C}(k+1)\to 0$

to show that $h^{1}(\Omega^{1}_{S}|_{C}(k+1))=0$ for $k\geq 2$. Then for
$k\geq 2$, the lemma follows from the long exact sequence associated to

$0\to\Omega^{1}_{S}(k)\to\Omega^{1}_{S}(k+1)\to\Omega^{1}_{S}|_{C}(k+1)\to 0.$

∎

######  Remark 4.6.

By the Riemann–Roch formula, we can estimate the dimension of $T_{X}^{1}(1)$
using $h^{1}(T_{S}(1))\approx-\chi(T_{S}(1))=20-(2g-2)$. This shows that Fano
3-folds of genus $g>10$ are superabundant: they are “not expected” to exist.

###### Proof of Theorem 4.1.

Let $(S,L)$ be a general K3 surfaces polarised by $L$ with $L^{2}=2g-2$, and
take a smooth curve $C\in|kL|$. By duality, it is enough to show that
$H^{1}(S,\Omega^{1}_{S}(-C))=0$.

Twisting the standard short exact sequence
$0\to\Omega^{1}_{S}\to\Omega^{1}_{S}(\log C)\to\mathcal{O}_{C}\to 0$ by
$\mathcal{O}(-C)$, we get the following exact sequence of cohomology

$0\rightarrow
H^{1}(S,\Omega^{1}_{S}(-C))\xrightarrow{\kappa}H^{1}(S,\Omega^{1}_{S}(\log
C)(-C))\xrightarrow{\tau}H^{1}(C,\mathcal{O}_{C}(-C))$ (7)

where $\kappa$ is injective because
$h^{0}(\mathcal{O}_{C}(-C))=h^{1}(2K_{C})=0$ by Serre duality. Now by [3,
(5.1)], $\tau$ is the tangent map at $(S,C)$ to the forgetful map
$\varphi_{g,k}$ from (6). Hence $\varphi_{g,k}$ is unramified at $(S,C)$ if
and only if $\tau$ is injective. Thus by Proposition 4.9 below,
$H^{1}(S,\Omega^{1}_{S}(-C))=0$ for general $(S,C)$. ∎

###  4.7 Mukai’s construction

In order to prove that $\varphi_{g,k}$ is generically finite, we use the
following theorem of Mori–Mukai [18]:

#### Assumption $(*)$

Let $S\subset\mathbb{P}^{m}$ be a smooth K3 surface with $m\geq 5$ such that
$S$ is set-theoretically an intersection of quadrics, and the map
$H^{0}(\mathcal{O}_{\mathbb{P}^{m}}(1))\to H^{0}(\mathcal{O}_{S}(1))$ is an
isomorphism. Suppose that $S$ contains an irreducible smooth curve $C$ such
that $H^{0}(\mathcal{O}_{S}(1))\to H^{0}(\mathcal{O}_{C}(1))$ is an
isomorphism and $\deg C\geq m+1$, so that $p_{a}(C)>0$. Let $H=S\cap L$ be a
smooth transverse hyperplane section of $S$ such that $C\cap H$ is in general
position in $L$.

######  Theorem 4.8 (Mori–Mukai [18, Theorem 1.2]).

Let $S$, $\Gamma=C+H$ be a pair satisfying assumption $(*)$. Then for every
embedding $i\colon\Gamma\to S^{\prime}$ into a K3 surface $S^{\prime}$, there
exists an isomorphism $I\colon S\to S^{\prime}$ whose restriction to $\Gamma$
coincides with $i$.

######  Proposition 4.9.

The map $\varphi_{g,k}$ is generically finite for $g=11$ or $g\geq 13$.

###### Proof.

It is enough to construct a pair $(S,\Gamma_{g,k})\in\mathcal{P}_{g,k}$ such
that

$\varphi_{g,k}^{-1}(\Gamma_{g,k})=\\{(S,\Gamma_{g,k})\\}.$

In [19, §4], Mukai constructed such pairs for $g=11$, $g\geq 13$ when $k=1$.
We summarise the construction here.

Let $E\subset\mathbb{P}^{5}$ be a sextic normal elliptic curve. Let
$S:=Q_{1}\cap Q_{2}\cap Q_{3}$ be a smooth complete intersection of three
quadrics $Q_{i}$ which contain $E$. Since $S$ contains an elliptic curve,
there is an elliptic fibration $S\rightarrow\mathbb{P}^{1}$. Let
$H\in|\mathcal{O}_{S}(1)|$ be a general hyperplane section. We can assume that
$S$ contains the following singular fibres:

  1. $(I_{3})$

$E_{1}\cup E_{2}\cup E_{3}$ with $E_{i}\cdot E_{j}=1$ for all $i\neq j$

  2. $(I_{2})$

$E_{2}^{\prime}\cup E_{4}$ with $E_{2}^{\prime}\cdot E_{4}=2$,

where $E_{i}\simeq\mathbb{P}^{1}$ and $E_{i}\cdot H=i$ for all $i=1,\ldots,4$.
Let $\Gamma:=E\cup H$. Then $\Gamma$ is a stable curve of genus $11$. We can
check that $(S,\Gamma)$ satisfies Assumption $(*)$.

Let

$\begin{array}[]{ll}\Gamma_{13}:=\Gamma\cup E_{3},&\Gamma_{16}:=\Gamma\cup
E_{3}\cup E_{4},\\\ \Gamma_{14}:=\Gamma\cup E_{4},&\Gamma_{17}:=\Gamma\cup
E_{1}\cup E_{3}\cup E_{4},\\\ \Gamma_{15}:=\Gamma\cup E_{2}\cup
E_{3},&\Gamma_{18}:=\Gamma\cup E_{2}\cup E_{3}\cup E_{4}.\end{array}$

We construct $\Gamma_{g}$ for $g\geq 19$ by adding smooth fibres to
$\Gamma_{i}$ for $13\leq i\leq 18$. Then since
$(S,\Gamma_{g})\in\mathcal{P}_{g,1}$, we see by Theorem 4.8 that
$\varphi_{g,1}^{-1}(\Gamma_{g})=\\{(S,\Gamma_{g})\\}$. This is the
construction due to Mukai.

Next we consider the case $k>1$. The linear system $|\Gamma_{g}|$ is free.
Indeed, if there is a fixed curve $C\subset\operatorname{Bs}|\Gamma_{g}|$,
then we see that $C=E_{i}$ for some $i$ and $(\Gamma_{g}-C)^{2}=0$ by Saint-
Donat’s classification. This does not happen since $\Gamma_{g}-C$ is ample by
construction. Take a smooth member $C_{g,k-1}\in|(k-1)\Gamma_{g}|$ and define
$\Gamma_{g,k}:=\Gamma_{g}\cup C_{g,k-1}$. Thus $(S,\Gamma_{g,k})$ is in
$\mathcal{P}_{g,k}$ and
$\varphi_{g,k}^{-1}(\Gamma_{g,k})=\\{(S,\Gamma_{g,k})\\}$. ∎

Note that for $g\leq 10$, $\varphi_{g}$ is not generically finite for
dimension reasons, and $\varphi_{12}$ 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

######  Corollary 4.10.

Let $S$ be a general K3 surface with primitive polarisation $L$ of genus $g$
and write $X=C_{a}(S,L)$. Then $T^{1}_{X}$ is concentrated in degree 0 if and
only if $g=11$ or $g\geq 13$.

See the next section for various comments about the generality assumption.

##  5 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 $g\leq 10$ or $g=12$.

###### Proof.

Let $X=C_{a}(S,\mathcal{O}(1))$ denote the affine cone over $S$. If $g=11$ or
$g\geq 13$, then $X$ has only conical deformations by Corollary 4.10 and
Proposition 2.3. If $g\leq 10$ or $g=12$ then by [3, Corollary 4.1], $S$ is an
anticanonical member of a smooth Fano 3-fold $W$, that is, $S\in|{-}K_{W}|$.
Let $C_{a}(W,-K_{W})$ be the affine cone over the pair $(W,-K_{W})$ and
$\sigma\in H^{0}(W,-K_{W})$ 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 $\sigma$
since $H^{0}(W,-mK_{W})\rightarrow H^{0}(S,kL)$ is surjective for all $m\geq
1$ by $H^{1}(W,-(m-1)K_{W})=0$. Let $\mathcal{X}\subset
C_{a}(W,-K_{W})\times\mathbb{A}^{1}$ be a divisor defined by a function
$\sigma+\lambda$, where $\lambda$ is the parameter of the affine line
$\mathbb{A}^{1}$. Then we have a deformation
$\mathcal{X}\rightarrow\mathbb{A}^{1}$ of $X$. The fibre over $0$ is $X$, and
the general fibre $\mathcal{X}_{t}$ 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 $g\geq 13$ can nevertheless
be smoothable

If a K3 surface $S$ is an anticanonical section of a Fano 3-fold with
$b_{2}\geq 2$ from the Mori–Mukai classification [17], then
$C_{a}(S,\mathcal{O}(1))$ is smoothable. Thus there are K3 surfaces of genus
$11$ and $\geq 13$ whose affine cone is smoothable. For such K3 surfaces,
Theorem 4.1 does not apply, and $H^{1}(\Omega^{1}_{S}(L))$ does not vanish.

######  Example 5.3.

Let $S$ be a hypersurface of bidegree $(2,3)$ in
$\mathbb{P}^{1}\times\mathbb{P}^{2}$. Since $S$ is a section of
$|{-K_{\mathbb{P}^{1}\times\mathbb{P}^{2}}}|$, we see that
$(S,-K_{\mathbb{P}^{1}\times\mathbb{P}^{2}}|_{S})$ is a K3 surface of genus
$28$. Nevertheless, we obtain a smoothing of the affine cone
$C_{a}(S,\mathcal{O}(1))$, simply by sweeping out the cone inside
$C_{a}(\mathbb{P}^{1}\times\mathbb{P}^{2},{-K_{\mathbb{P}^{1}\times\mathbb{P}^{2}}})$.

######  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 $C_{a}(S,\mathcal{O}(1))$ over a
general $S$ is not smoothable, but if we specialise $C_{a}(S,\mathcal{O}(1))$
to $C_{a}(S^{\prime},\mathcal{O}(1))$ where $S^{\prime}$ is the hyperplane
section of one of the above Fano 3-folds, then
$C_{a}(S^{\prime},\mathcal{O}(1))$ 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 $\geq 13$. The only smoothings of
$C_{a}(S,\mathcal{O}(1))$ 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 $C_{a}(S,\mathcal{O}(1))$ does not lift to the
projective cone $C_{p}(S,\mathcal{O}(1))$. In spite of Example 2.19, we expect
that $C_{a}(S,\mathcal{O}(1))$ is not smoothable for _any_ $S$ of sufficiently
large genus. An equivalent question (cf. [3, §5.4]) is the following:

> Is $\varphi_{g,k}$ 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 $C_{a}(S,\mathcal{O}(1))$ 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
$V_{1}=V\colon(1,1)\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ and
$V_{2}=\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. 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 $\pi\colon S\to Y$ a
double cover branched in $B\in|{-2K_{Y}}|$. Then $S$ is a K3 surface of degree
12 in $\mathbb{P}^{7}$. By Example 5.8, $Y=V_{i}\cap H_{i}$ for some
$H_{i}\in|{-\frac{1}{2}K_{V_{i}}}|$. Take $\pi_{i}\colon W_{i}\to V_{i}$ a
double cover branched in $X_{i}\in|{-K_{V_{i}}}|$, where $X_{i}$ are chosen so
that $X_{i}\cap H_{i}=B$ since $H^{0}(V_{i},-K_{V_{i}})\rightarrow
H^{0}(Y,-2K_{Y})$ is surjective. 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 $W_{i}\cap\pi_{i}^{*}H_{i}=S$. Thus the
affine cone $C_{a}(S,\mathcal{O}_{S}(1))$ is a hyperplane section of
$C_{a}(W_{i},-K_{W_{i}})\subset\mathbb{A}^{9}$ for each $i$, and so
$C_{a}(S,\mathcal{O}_{S}(1))$ 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 $\leq 2$. 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.

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Stephen Coughlan
Institut für Algebraische Geometrie,
Leibniz Universität Hannover,
30167 Hannover, Germany

Current address: Mathematisches Institut,
Lehrstuhl Mathematik VIII,
Universitätsstrasse 30,
95447 Bayreuth, Germany
<EMAIL_ADDRESS>
Taro Sano
Department of Mathematics
Faculty of Science, Kobe University,
1-1, Rokkodai-cho, Nada-ku, Kobe, 657-0029, Japan
<EMAIL_ADDRESS>