Fano 3-folds
Shigeru MUKAI
Abstract: In the beginning of this century, G. Fano initiated the study of 3-dimensional projective varieties
X2g−2 ⊂ Pg+1 with canonical curve sections in connection with the Lüroth problem.1 After a quick review
of a modern treatment of Fano’s approach (§1), we discuss a new approach to Fano 3-folds via vector bundles,
which has revealed their relation to certain homogeneous spaces (§§2 and 3) and varieties of sums of powers(§§5 and 6). We also give a new proof of the gunus bound of prime Fano 3-folds (§4). In the maximum genus(g = 12) case, Fano 3-folds X22 ⊂ P13 yield a 4-dimensional family of compactiﬁcations of C3 (§8).

A compact complex manifold X is Fano if its ﬁrst Chern class c1 (X) is positive, or
equivalently, its anticanonical line bundle OX (−KX ) is ample. If OX (−KX ) is generated by
global sections and Φ|−KX | is birational, then its image is called the anticanonical model of
X. In the case dim X = 3, every smooth curve section C = X ∩ H1 ∩ H2 ⊂ Pg−1 of the
anticanonical model X ⊂ Pg+1 is canonical, that is, embedded by the canonical linear system|KC |. Conversely, every projective 3-fold X2g−2 ⊂ Pg+1 with a canonical curve section is
obtained in this way. The integer 12 (−KX )3 + 1 is called the genus of a Fano 3-fold X since
it is equal to the genus g of a curve section of the anticanonical model.
A projective 3-fold X2g−2 ⊂ Pg+1 with a canonical curve section is a complete intersectionof hypersurfaces if g ≤ 5. In particular, the Picard group of X is generated by OX (−KX ).
We call such a Fano 3-fold prime. If a Fano 3-fold X is not prime, then either −KX isdivisible by an integer ≥ 2 or the Picard number ρ of X is greater than one. See [15], [7]and [9] for the classiﬁcation in the former case and [24] and [25] in the latter case.
§1 Double projection The anticanonical line bundle OX (−KX ) is very ample if X is aprime Fano 3-fold of genus ≥ 5 (cf. [15] and [41]). To classify prime Fano 3-folds X2g−2 ⊂Pg+1 of genus g ≥ 6, Fano investigated the double projection from a line2 ℓ on X2g−2 , thatis, the rational map associated to the linear system |H − 2ℓ| of hyperplane sections singularalong ℓ.
Example 1 Let X16 ⊂ P10 be a prime Fano 3-fold of genus 9. Then the double projection
π2ℓ from a line ℓ ⊂ X16 is a birational map onto P3 . The union D of conics which intersectsℓ is a divisor of X and contracted to a space curve C ⊂ P3 of genus 3 and degree 7. The
inverse rational map P3 − → X16 ⊂ P10 is given by the linear system |7H − 2C| of surfacesof degree 7 which are singular along C.
The key for the analysis of π2ℓ is the notion of ﬂop. Let X − be the blow-up of X alongℓ. Since other lines intersect ℓ, X − is not Fano. But X − is almost Fano in the sense that| − KX − | is free and gives a birational morphism contracting no divisors. The anticanonicalmodel X̄ of X − is the image of the projection X− → P8 from ℓ. The strict transformD− ⊂ X − of D is relatively negative over X̄. By the theory of ﬂops ([33], [19]), there exists
another almost Fano 3-fold X + which has the same anticanonical model as X − and suchthat the strict transform D+ ⊂ X + of D− is relatively ample over X̄. X + is called theD− -ﬂop3 of X − .
1

A surface dominated by a rational variety is rational by Castelnuovo’s criterion. But this does not holdany more for 3-folds. See [5], [44] and [18].
2
The existence of a line is proved by Shokurov [42].
3
The smoothness of X + follows from [19, 2.4] or from the classiﬁcation [6, Theorem 15] of the singularityof X̄.

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256

Theorem([23], [17]) Let X, ℓ and D be as in Example 1. Then the D− -ﬂop X + of theblow-up X − of X along ℓ is isomorphic to the blow-up of P3 along a space curve of genus 3and degree 7.
For the proof, the theory of extremal rays ([22]) is applied to the almost Fano 3-fold X + .
If X is a prime Fano 3-fold of genus 10, then X + is isomorphic to the blow-up of a smooth
3-dimensional hyperquadric Q3 ⊂ P4 along a curve of genus 2 and degree 7. In the casegenus 12, X + is the blow-up of a quintic del Pezzo 3-fold4 V5 ⊂ P6 along a quintic normal
rational curve.
§2 Bundle method A line on X2g−2 ⊂ Pg+1 can move in a 1-dimensional family. Hencethe double projection method does not give a canonical biregular description of X2g−2 ⊂Pg+1 . In the case g = 9, e.g., there are inﬁnitely many diﬀerent space curves5 C ⊂ P3 whichgive the same Fano 3-fold X16 ⊂ P10 . By the same reason, the double projection method
does not classify X2g−2 ⊂ Pg+1 over ﬁelds which are not algebraically closed. Even whena Fano 3-fold X is deﬁned over k ⊂ C, it may not have a line deﬁned over k. Our newclassiﬁcation makes up these defects. It is originated to solve the following:n
Problem6 : Classify all projective varieties X2g−2⊂ Pg+n−2 of dimension n ≥ 3 with a7canonical curve section .

We restrict ourselves to the case that every divisor on X is cut out by a hypersurface. In
contrast with the case g ≤ 5, the dimension n cannot be arbitrarily large in the case g ≥ 6.
In each case 7 ≤ g ≤ 10, the maximum dimension n(g) is attained by a homogeneous spaceΣ2g−2 .
Tableg n(g)
66

Σ2g−2 ⊂ Pg+n(g)−2r(E) χ(E) c1 (E)c2 (E)Hyperquadric section of the cone2
54of the Grassmann variety8
G(2, 5) ⊂ P9
7
10 10-dimensional spinor variety
5
10
48
SO(10, C)/P ⊂ P15
8
8
Grassmann variety G(2, 6) ⊂ P14
2
6
5
13
9
6
Sp(6, C)/P ⊂ P
3
6
8
10
5
G2 /P ⊂ P13
5
7
12
12
3
G(V, 3, N ) ⊂ P13 (See Theorem 3.)
3
7
10
We claim that every variety X ⊂ P with canonical curve section of genus g ≥ 6 is alinear section of the above Σ2g−2 ⊂ Pg+n(g)−2 . Since each Σ2g−2 has a natural morphism to a
Grassmann variety, vector bundles play a crucial role in our classiﬁcation. Instead of a line,
we show the existence of a good vector bundle E on X. Instead of the double projection,
we embed X into a Grassmann variety by the linear system |E| and describe its image.
The vector bundle is ﬁrst constructed over a general (K3) surface section S of X and thenextended to X applying a Lefschetz type theorem (cf. [8]).9 The numerical invariants of E
A smooth projective variety Vd ⊂ Pd+n−2 with a normal elliptic curve section is called del Pezzo. The
anticanonical class −KV is llinearly equivalent to (n − 1) times hyperplane section. All quintic del Pezzo
3-folds are isomorphic to each other (see [15] and [9]).
5The isomorphism classes of curves C are uniquely determined by the Torelli theorem since the intermediate Jacobian variety of X is isomorphic to the Jacobian variety of C.
6
Roth [36] [37] studied this problem by generalizing the double projection method.
7n2
The anticanonical class of X2g−2
is (n − 2)-times hyperplane section. In the case n = 2, X2g−2
is a(polarized) K3 surface. The integer g is called the genus of X.
8
G(s, n) denotes the Grassmann variety of s-dimensional subspaces of a ﬁxed n-dimensional vector space.
9
By our assumption on X and [21], there exists a surface section with Picard number one. Hence every
member of |OS (−KX )| is irreducible. We use this property to analyze Φ|E| .
4

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257

are as in the above table.10 All higher cohomology groups of E vanish and E is generated
by its global sections. The morphism11 Φ|E| : X −→ G(H 0 (E), r(E)) is an embedding if
g ≥ 7. The ﬁrst Chern class c1 (E) is equal to 2c1 (X) if g = 7 and equal to c1 (X) otherwise.
E is characterized by the following two properties:1) r(E), c1 (E) and c2 (E) are as above, and2) the restriction12 of E to a general surface section is stable.
In the case g = 9, |E| embeds X into the 9-dimensional Grassmann variety G(V, 3),
where V = H 0 (X, E). Consider the natural map
λ2 :

2
∧

H 0 (X, E) −→ H 0 (X,

2
∧

E).

The kernel is generated by a nondegenerate bivector σ on V . Hence the image of X is∧contained in the zero locus G(V, 3, σ) of the global section of 2 E corresponding to σ, where
E is the universal quotient bundle on G(V, 3). G(V, 3, σ) is a 6-dimensional homogeneous
space of Sp(V, σ) and a projective variety Σ16 ⊂ P13 with a canonical curve section of genus
9. In the case dim X = 3, we have
Theorem 2 A prime Fano 3-fold X16 ⊂ P10 of genus 9 is isomophic to the intersection of
Σ16 and a linear subspace P10 in P13 .
By the above characterization, E is deﬁned over k ⊂ C if X is so. Hence the theorem holds
true for every Fano 3-fold X16 ⊂ P10
k over k ⊂ C such that X ⊗ C is prime.
The results are similar for g = 7, 8 and 10. In the case g = 7 and 10, the natural∧∧
mappings σ2 : S 2 H 0 (X, E) −→ H 0 (X, S 2 E) and λ4 : 4 H 0 (X, E) −→ H 0 (X, 4 E) areconsidered instead of λ2 . In the case g = 6, X is a double cover of a linear section ofG(2, 5) ⊂ P9 if the linear subspace P passes through the vertex of the Grassmann cone.
Otherwise, X is isomorphic to the complete intersection of a 6-dimensional hyperquadricQ ⊂ P and G(2, 5) ⊂ P9 .
§3 Fano 3-fold of genus 12 A prime Fano 3-fold13 X of genus 12 cannot be an ample
divisor of a 4-fold. But the vector bundle E gives a canonical description of X in the
12-dimensional Grassmann variety G(V, 3), V = H 0 (X, E). Consider the natural map λ2 :∧∧2 0
H (X, E) −→ H 0 (X, 2 E) as in the case g = 9. Its kernel N is of dimension 3. Let{σ1 , σ2 , σ3 } be a basis of N .
Theorem 3 A prime Fano 3-fold X22 ⊂ P12 of genus 12 is isomorphic to the common zero∧
locus G(V, 3, N ) of the three global sections of 2 E corresponding to σ1 , σ2 and σ3 , where E
is the universal quotient bundle on G(V, 3).
The third Chern number deg c3 (E) is equal to 2. Hence every general global section of E
vanishes at two points. Conversely, since V is of dimension 7, there exists a nonzero global
section sx,y vanishing at x and y for every pair of distinct points x and y. If x and y are
general, then sx,y is unique up to constant multiplications. The correspondence (x, y) 7→ [sx,y ]gives the birational mappings Π : S 2 X− → P∗ (V ) ≅ P6 and Πx : X− → P∗ (Vx ) ≅ P3 forgeneral x, where Vx ⊂ V is the space of global sections of E which vanish at x. In particular,
X is rational. The birational mapping Πx is the same as the triple projection of X22 ⊂ P13
from x.
The bundle method gives another canonical description of prime Fano 3-folds of genus
12 in the variety of twisted cubics ([29, §3]). This description is useful to analyze the double
projection of X22 ⊂ P13 from a line.
10

The bundle method works for other values of g, e.g., 18 and 20 and gives a description of polarized K3
surfaces (see [30]).
11
For a vector space V , G(V, r) denotes the Grassmann variety of r-dimensional quotient space of V .
12
The restriction of E is rigid and characterized by its numerical invariants and stability ([27, §3]).
13
Prime Fano 3-folds of genus 12 were omitted in [38, Chap. V, §7] and ﬁrst constructed by Iskovskih [16].

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Remark 4 The third Betti number of a prime Fano 3-fold of of genus g ≥ 7 is equal 2(n(g)−3). In particular, prime Fano 3-folds of genus 12 have the same homology group as P3 .
§4 Genus bound The descriptions given in §§2 and 3 complete the classiﬁcation of prime
Fano 3-folds by virtue of Iskovskih’s genus bound:
Theorem 5 The genus g of a prime Fano 3-fold satisﬁes g ≤ 10 or g = 12.
This is proved in the course of the classiﬁcation by the double projection method. Here
we sketch a simple proof using a correspondence between the moduli spaces of K3 surfaces
and curves. Let Fg be the moduli space of polarized K3 surfaces (S, h) of degree 2g − 2. Asmooth member of |h| is a curve of genus g. Hence we obtain the rational map φg from the∐Pg -bundle Pg := (S,h)∈Fg |h| over Fg to the moduli space Mg of stable curves of genus g.
The key observation is this.
Proposition 6 If a prime Fano 3-fold of genus g exists, then the rational map φg : Pg − →Mg is not generically ﬁnite.
By a simple deformation argument, we have that the generic hyperplane section (S, h) of
the generic prime Fano 3-fold is generic in Fg . Take a generic pencil P of hyperplane sectionsof X2g−2 ⊂ Pg+1 . The isomorphism classes of the members of P vary since the pencil P
contains a singular member. But every member of P contains the base locus of P , which isa curve of genus g. This shows the proposition.
Since dim Pg = g + 19 and dim Mg = 3g − 3, dim Pg ≤ dim Mg holds if and only if
g ≥ 11. We recall the proof of the generic ﬁniteness of φ11 in [25]. Let C ⊂ P5 be a sextic
normal elliptic curve and S a smooth complete intersection of three hyperquadrics containing
C. Let H be a general hyperplane section of S and put Γ = H ∪ C. The S is a K3 surface
and Γ is a stable curve of genus 11.
Theorem([25, (1.2)]) For every embedding i : Γ → S ′ of Γ in to a K3 surface S ′ , there exists
an isomorphism I : S → S ′ whose restriction to Γ coincides with i.
This implies that the point ξ ∈ P11 corresponding to (S, Γ) is isolated in φ−1
11 (φ11 (ξ)).
Hence φ11 is generically ﬁnite and a prime Fano 3-fold of genus 11 does not exist. The
non-existence of prime Fano 3-folds of genus ≥ 13 is proved in a similar way. Note thatthe elliptic curve C induces an elliptic ﬁbration of S, which we denote by π : S → P1 . We
consider the case in which π has two singular ﬁbers of the following types:i) E1 ∪ E2 ∪ E3 with (E2 .E3 ) = (E3 .E1 ) = (E1 .E2 ) = 1, andii) E2′ ∪ E4 with (E2′ .E4 ) = 2,
where Eν is isomorphis to P1 and satisﬁes (Eν .H) = ν for evey 1 ≤ ν ≤ 4. It is easy toconstruct a stable curve Γg of genus ≥ 13 on S from Γ by adding ﬁbres of π. For example,Γ ∪ E3 , Γ ∪ E4 and Γ ∪ E2 ∪ E3 are of genus 13, 14 and 15, respectively. Note that to add
one general ﬁbre of π increases the genus by 6. Byt the above theorem, it is easy to showthat every embedding of Γg into a K3 surface S ′ is extended to an isomorphism from S onto
S ′ . Hence we have
Theorem 7 The rational map φg : Pg − → Mg is generically ﬁnite if and only if g = 11 org ≥ 13.
This completes the proof of Theorem 5.
Remark 8 The map φg is generically of maximal rank except for g = 10, 12. In the case of
g = 10, the image of φ10 is is a divisor of M10 (See [28]).

MUKAI: Fano 3-folds

259§5 Theory of polars Prime Fano 3-folds of genus 12 are related to the classical problem
on sums of powers, which is a polynomial version of the Warring problem. Let Fd be ahomogeneous polynomial of degree d in n variables.
∑d1) Are there N linear forms f1 , · · · , fN such that Fd = N
1 fi ?
2) If so, then how many?
In the following cases, every general Fd is a sum of d-th powers of N linear forms and theexpression is unique:(1) n = 2 and d = 2N (Sylvester[43]),(2) n = 4, d = 3 and N = 5(Sylvester’s pentahedral theorem [34] [39]), and(3) n = 3, d = 5 and N = 7(Hilbert [14, p. 153], Richmond [34] and Palatini [32]).
∏
We consider the case n = 3. Let C and Γ be the plane curves deﬁned by Fd and N
1 fi ,∑N drespectively. Γ is called a polar N -side of C if Fd = 1 fi . The name comes from thefollowing:
Example 9 Let C be a smooth conic and ℓ1 , ℓ2 and ℓ3 three distinct lines. Then the following
are equivalent:(1) △ = ℓ1 + ℓ2 + ℓ3 is a polar 3-side of C in the above sense, and(2) the triangle △ is self polar with respect to C, that is, each side is the polar of its
opposite vertex.
§6 Variety of sums of powers We regard the set of polar N -sides of C : Fd (x, y, z) = 0 as
a subvariety of the projective space of plane curves of degree N . We denote its closure14 by
V SP (C, N ) or V SP (Fd , N ). The homogeneous forms of degree N form a vector space ofdimension 21 (d + 1)(d + 2). The N -ples of linear forms form a vector space of dimension 3N .
Hence the dimension of V SP (C, N ) is expected to be 3N − 21 (d + 1)(d + 2) for general C.
In the case (d, N ) = (2, 3), this is true.
Proposition 10 If C is a smooth conic, then V SP (C, 3) is a smooth quintic del Pezzo
3-fold.
Let V2 be the vector space of quadratic forms. If △ : f1 f2 f3 = 0 is a polar 3-side of C, thenthe deﬁning equation F2 of C is contained in the subspace < f12 , f22 , f32 > of V2 . Therefore,△ determines a 2-dimensional subspace W of V ∗ := V2 /CF2 . Hence we have the morphismfrom V SP (C, 3) to the 6-dimensional Grassmann variety G(2, V ∗ ) ⊂ P9 . Let q : V2 −→ C
be the linear map associated to the dual conic of C. For a pair of quadratic forms f and g,
consider the three minors Ji (f, g), i = 1, 2, 3, of the Jacobian matrix(

fx fy fzgx gy gz)and put σi (f, g) = q(Ji (f, g)). Then σi are skew-symmetric forms on V2 and F2 is theircommon radical. Therefore, each σi , i = 1, 2, 3, determine three hyperplanes Hi of P9 =∧P∗ ( 2 V ∗ ). V SP (C, 3) is isomorphic to the quintic del Pezzo 3-fold G(2, V ∗ ) ∩ H1 ∩ H2 ∩ H3 .
Now we consider plane quartic curves C : F4 (x, y, z) = 0. The dimension count?

3N − 15 = dim V SP (C, N )does not hold for N = 5:
Let {∂1 = ∂ 2 /∂x2 , · · · , ∂6 = ∂ 2 /∂z 2 } be a basis of the space of homogeneous second orderpartial diﬀerential operators.
14

The closure is taken in the symmetric product SymN P2 . But this is a temporary deﬁnition. In practice,
we choose a suitable model of SymN P2 to deﬁne V SP (C, N ).

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Theorem (Clebsch [4]) If a plane quartic curve C : F4 (x, y, z) = 0 has a polar 5-side, thenΩ(F ) := det(∂i ∂j F )1≤i,j≤6 = 0.
In particular, general plane quartic curves have no polar 5-sides.
In other words, polar 5-sides are not equally distributed to quartic curves. Once a quartic
curve has a polar 5-side, it has a 1-dimensional family of polar 5-sides. (The same happens
for polar 2-sides of conics.)
Polar 6-sides of plane quartics was studied by Rosanes [35] and Scorza [40]. The dimension
count is correct for N = 6 and we obtain 3-folds.
Theorem 11 (1) If a quartic curve C has no polar 5-sides or no complete quadrangles as
its polar 6-sides, then the variety V SP (C, 6) of polar 6-sides of C is a prime Fano 3-fold of
genus 12.
(2) Conversely every prime Fano 3-fold X of genus 12 is obtained in this way. The
isomorphism class of C is uniquely determined by that of X.
By virtue of Theorem 3, it suﬃce to show that G(V, 3, N ) is isomorphic to V SP (C, 6).
Let V3 be the vector space of cubic forms. If Γ : f1 f2 · · · f6 = 0 is a polar 6-side of C,
then the partial derivatives Fx , Fy and Fz of the deﬁning equation F4 are contained in <f13 , f23 , · · · , f63 >. Hence Γ determines a 3-dimensional subspace of V ∗ := V3 / < Fx , Fy , Fz >and we obtain a morphism φ from V SP (C, 6) to G(3, V ∗ ). Three skew-symmetric forms
σ1 , σ2 and σ3 on V ∗ are deﬁned as in the case of V SP (F2 , 3) and the image of φ is containedin G(3, V ∗ , σ1 , σ2 , σ3 ).
∧Conversely, let V and N ⊂ 2 V be as in Theorem 3. The multiplication in the exterior∧∧•
V induces the map σ3 : S 3 N −→ 6 V . This is surjective and its kernel is ofalgebradimension 3.
Lemma 12 There exists a quartic polynomial F (x, y, z) ∈ S 4 N whose partial derivatives
FX , FY and FZ form a basis of the kernel of σ3 , where {x, y, z} is a basis of N and {X, Y, Z}
is its dual.
The conics on (the anticanonical model of) G(V, 3, N ) is parametrized by the projective
plane15 P∗ (N ) For every point x of G(V, 3, N ), there exist exactly six conics {Zλi }1≤i≤6 ,λi ∈ P∗ (N ), passing through x, counted with their multiplicities. . Let Λi , 1 ≤ i ≤ 6, be the∑lines on P(N ) with coordinates λi . Then Γ = 61 Λi is a polar 6-side of the plane curve C
on P(N ) deﬁned by the quartic form F (x, y, z) in the lemma. This correspondence x 7→ Γ
gives the inverse of the above morphism φ.
Remark 13 (1) Assume that a plane quartic C ′ has a polar 5-side and that the 5lines arein general position.
When C in Theorem 11 deforms to C ′ , the variety V SP (C, 6) deforms to a Fano 3-fold
X ′ with an ordinary double point. X ′ is isomorphic to the anti canonical model of P(E),where E is a stable vector bundle on P2 with c1 = 0 and c2 = 4 (cf. [2]).
(2) If C is a general plane septic curve, then the variety V SP (C, 10) is a polarized K3
surface of genus 20.

For a vector space V , P∗ (V ) is the projective space of 1-dimensional subspaces of V . P(V ), or P∗ (V ),is its dual.
15

MUKAI: Fano 3-folds

261§7 Almost homogeneous Fano 3-fold Varieties of sums of powers give two examples ofalmost homogeneous spaces of SO(3, C) and their compactiﬁcations. Apply Theorem 11 toa double conic, say 2C0 : (XZ + Y 2 )2 = 0.
The variety V SP (2C0 , 6) is a Fano 3-fold and has an action of SO(3, C). It is easy tocheck
30(XZ + Y 2 )2 = 25Y 4 +4∑(ζ i X + Y + ζ −i Z)4 ,i=0

where ζ is a ﬁfth root of unity. The polar 6-side
Γ:Y

4
∏

(ζ i X + Y + ζ −i Z) = 0

i=0intersects the 2-sphere C0 at the 12 vertices of a regular icosahedron. The stabilizer groupat Γ of SO(3, C) is the icosahedral group ≅ A5 . Hence we have
Theorem 14 The variety V SP (2C0 , 6) is a smooth equivariant compactiﬁcation ofSO(3, C)/Icosa.
Similarly the quintic del Pezzo 3-fold V SP (C0 , 3) is a smooth equivariant compactiﬁcationof the quotient of SO(3, C) by an octahedral group ≅ S4 by Proposition 10. These twocompactiﬁcations are described in [31, §§3 and 6] by another method. We remark thatQ3 and P3 are also almost homogeneous spaces of SO(3, C). The stabilizer groups are
tetrahedral group ≅ A4 and a dihedral group of order 6 ≅ S3 , respectively.
§8 Compactiﬁcation of C3 There are four types of Fano 3-folds with the same homology
growps as P3 : P3 itself, Q3 ⊂ P4 , V5 ⊂ P6 and the 6-dimensional family of prime Fano 3folds X22 ⊂ P13 of genus 12 (see Remark 4). These Fano 3-folds are related to not only
SO(3, C) but also C3 , the aﬃne 3-space. It is well-known that P3 and Q3 are smoothcompactiﬁcations of C3 with irreducible boundary divisors. The quintic del Pezzo 3-fold
V5 ⊂ P6 is a compactiﬁcations of C3 in two ways (see [10] and [13]).
Furushima has found that the almost homogeneous Fano 3-fold U22 := V SP (2C0 , 6) also
is a Compactiﬁcation of C3 . This fact is proved in three ways usingi) the deﬁning equation ([31] p.506) of U22 ⊂ P12 (see [11]),ii) the double projection of U22 ⊂ P13 from a line (see [12]), andiii) the action of a torus C∗ ⊂ SO(3, C) on U22 (see [1] and [20]).
In the last case, U22 is decomposed into a disjoint union of aﬃne spaces by virtue of [3]. The
four compactiﬁcations by P3 , Q3 and V5 are rigid but that by U22 is not. In fact, by a carefulanalysis of the double projection of V SP (C, 6) ⊂ P13 from a line, we have
Theorem 15 The variety V SP (C, 6) in Theorem 11 is a compactiﬁcation of C3 if C has a
non-ordinary singular point.
The variety V SP (C, 6) has a line ℓ (on its anticanonical model) with normal bundle O(1) ⊕O(−2) corresponding to a non-ordinary singular point of C. Let D be the union of conics
which intersect ℓ as in Example 1. Then the complement of D is isomorphic to C3 .

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Department of Mathematics
School of Sciences
Nagoya University
464 Furō-chō, Chikusa-ku
Nagoya, Japan