VECTOR FIELDS ON SMOOTH THREEFOLDS VANISHING ON COMPLETE INTERSECTIONS

THOMAS ECKL

1. Introduction

It is a well known fact that the existence of a vector field on a Kähler manifold with a special zero locus strongly influences the geometry of the manifold. For example, the plurigenera and some Hodge numbers vanish ([Kob72],[CL73]), and if the zero locus is projective-algebraic, the variety itself will be, too ([Hwa96]). In [Wah83] J. Wahl proved the following Theorem 1.1. Let X be a complex projective normal variety, L an ample line bundle, dim X > 1. If H^{0}(T_{X} ⊗ L^{−1}) ≠ 0 then:

(i) L ≅ O(E), where the effective divisor E is a normal variety.

(ii) X ≅ Proj A[t] with A = ⨁_{n=0}^{∞} H^{0}(E, O_{E}(nE)). X is the cone over E,

and t has weight 1 and E is the divisor at ∞ (t = 0). In particular every smooth complex projective variety with a vector field vanishing on an ample divisor is isomorphic to ℙ^{N}.

The aim of this paper is to look for similar statements in case of zero loci in higher codimensions, with ample normal bundle for example. The main result of this paper is in dimension 3: Theorem 1.2. Let X be a smooth complex projective threefold. Let D_{1}, D_{2} be two ample effective divisors on X such that the scheme theoretic intersection C = D_{1} ∩ D_{2} is an irreducible reduced curve. Let v ∈ H^{0}(X, T_{X}) be a vector field vanishing on C. Then X is isomorphic to ℙ^{3} or to the three-dimensional quadric Q_{3}.

Note that the irreducibility assumption on C is necessary: Let X ≅ ℙ^{2} × ℙ^{1} and p_{1} : X → ℙ^{2}, p_{2} : X → ℙ^{1} the projections, p ∈ L ⊂ ℙ^{2} a point in a line in ℙ^{2}. Let D_{1} ⊂ ℙ^{2} × ℙ^{1} be the blow up of ℙ^{2} in p, embedded in X. There is a point q ∈ ℙ^{1} such that the fibre p_{2}^{−1}(q) contains the strict transform of L. Let D_{2} ⊂ ℙ^{2} × ℙ^{1} be the union p_{2}^{−1}(q) ∪ p_{1}^{−1}(L). Then D_{1}, D_{2} are ample divisors of X and C = D_{1} ∩ D_{2} is the union of two lines. Choose homogeneous coordinates (x_{0} : x_{1} : x_{2}) on ℙ^{2} such that L = \{x_{0} = 0\}. Then the vector field v = x_{0} \frac{∂}{∂x_{1}} ∈ H^{0}(ℙ^{2}, T_{ℙ^{2}}) vanishes on L and the pullback p_{1}^{∗}v ∈ H^{0}(X, T_{X}) vanishes on C.

The proof of the theorem uses a modification of Wahl’s theorem and Lefschetz’ hyperplane theorem for an inductive argument to conclude that Pic(X) = ℤ. Since there exists a vector field with nonempty zero locus X is a smooth Fano threefold. The geometry of the ample divisors discovered by Wahl’s theorem rules out most of the possibilities in Iskhovskhi’s classification ([Isk77, Isk78]). The remaining cases are dealt with by looking at their vector fields.

Date: April 18, 2001.

1991 Mathematics Subject Classification. Primary 14M20, Secondary 14L99, 14F05.

Key words and phrases. Characterization of the threedimensional quadric, zero locus of vector fields.

The author gratefully acknowledges support by the DFG-Graduiertenkolleg “Komplexe Mannigfaltigkeiten” at the University of Bayreuth.

Notation. Throughout the present work, let X be a complex projective variety. The tangent sheaf T_{X} = Hom_{O_{X}} (Ω_{X}^{1}, O_{X}) is the dual of the sheaf of differentials Ω_{X}^{1} on X. If X is smooth, T_{X} is locally free. A vector field v ∈ H^{0}(X, T_{X}) is a global section of T_{X}. The groups ℂ^{∗}, ℂ^{+} denote the multiplicative and the additive group of complex numbers.

Acknowledgement. This paper is part of a PhD thesis written while the author was a member of the DFG-Graduiertenkolleg “Komplexe Mannigfaltigkeiten” at the University of Bayreuth. The author is grateful to his advisor Th. Peternell for pointing out the problem and continuous encouragement to work on it. He would like to thank the other members of the Graduiertenkolleg, especially S. Kebekus and H. Chr. von Bothmer, for creating a stimulating athmosphere and many helpful discussions on the topic.

2. A Theorem of Wahl

An easy corollary of Theorem 1.1 is the following result already proven in [MS78]: Theorem 2.1. Let X be a complex projective smooth variety, L an ample line bundle and T_{X} the tangent bundle. If there is a vector field v ∈ H^{0}(X, T_{X}) vanishing on an ample effective divisor D ∈ H^{0}(X, L) then

(X, L) = (ℙ^{n}, O(1)) or (ℙ^{1}, O(2)).

The proof of Theorem 1.1 uses the normality of X when applying the Kodaira vanishing theorem. A thorough analysis of this proof and the ideas in [MS78] shows that one can replace normality by the existence of an effective divisor in the ample line bundle to get a slightly weaker result: Theorem 2.2. Let X be a complex projective variety, dim X > 1, and L an ample line bundle with H^{0}(X, L) ≠ 0. If H^{0}(X, T_{X} ⊗ L^{−1}) ≠ 0 then there is a finitely generated graded ℂ-algebra A ⊂ ⨁_{n=0}^{∞} H^{0}(X, nL) and a homogeneous element T ∈ H^{0}(X, L) such that

A[T] = ⨁_{n=0}^{∞} H^{0}(X, nL).

In this case X is isomorphic to a cone over an ample Divisor E of Proj A.

Proof. The last part is an easy consequence of the first statements, cf. [BS95, 5.3].

The first part can be proven exactly as in [Wah83] provided that one is still able to construct a derivation of weight -1 on R = ⨁_{n=0}^{∞} H^{0}(X, nL): By assumption there is an effective divisor D ∈ H^{0}(X, L) and a vector field v ∈ H^{0}(X, T_{X}) such that v_{|D} = 0. Let G ⊂ Aut X be the subgroup of automorphisms fixing D. The existence of v implies that G is nontrivial. Furthermore the linear representation of G on the m-jets of a point x ∈ D is faithful if m ≫ 0. Therefore G is linear algebraic and contains a linear algebraic one parameter group H, i.e. H ≅ ℂ^{∗} or ℂ^{+}.

Since H stabilizes the divisor D there is a H-linearization of the line bundle L and a dual H-action τ on R = ⨁_{n=0}^{∞} H^{0}(X, L^{n}) (cf. [MF82, Prop. 1.5]). If H = ℂ^{∗} there are semi-invariant elements F_{0}, F_{1}, …, F_{m} of τ, i.e.

τ(t) · F_{i} = t^{χ(F_{i})} · F_{i}

which generate R as a ℂ-algebra ( χ(F) is the weight of the semi-invariant element F). Let F_{0} ∈ H^{0}(X, L) correspond to D.

Let R/F_{0} = ⨁_{m} H^{0}(D, mL_{|D}) be the homogeneous coordinate ring of D. Since ℂ^{∗} acts trivial on D via τ it acts semi-trivial on H^{0}(D, mL_{|D}) with weight χ_{m}, and the quotients \frac{χ_{m}}{m} are equal for all m ≥ 1. This implies

s := \frac{χ(F_{1})}{deg F_{1}} = … = \frac{χ(F_{m})}{deg F_{m}} ,

and s ≠ \frac{χ(F_{0})}{deg F_{0}} since otherwise ℂ^{∗} would act trivial on all of X.

Now one can twist the ℂ^{∗}-linearization: Let ℂ^{∗} act on R via

σ(t) · F = t^{−s deg F} · τ(t)F

for all homogeneous F ∈ R. Hence for an arbitrary polynomial P ∈ ℂ[X_{0}, …, X_{m}]:

σ(t)P (F_{0}, F_{1}, …, F_{m}) = P (t^{χ(F_{0})−s}F_{0}, F_{1}, …, F_{m}),

and χ(F_{0}) − s ≠ 0. It follows D_{σ}F_{1} = … = D_{σ}F_{m} = 0, D_{σ}F_{0} = (χ(F_{0}) − s)F_{0} where D_{σ} is the R-derivation corresponding to σ. One can divide D_{σ} by F_{0} to get a (−1)− derivation D_{−1}.

If H = ℂ^{+} let F_{0}, F_{1}, …, F_{m} still be homogeneous generators of R, the element F_{0} corresponding to D. Since the unipotent group ℂ^{+} fixes D = Proj(R/F_{0}) the group acts trivial on R/F_{0}. This implies for homogeneous F that

τ(t)F − F ∈ (F_{0}).

Once more one can divide the corresponding derivation D_{τ} by F_{0} and gets a (−1)− derivation D_{−1} on R.

Now one constructs an element t ∈ H^{0}(X, L) as in [Wah83, Lemma 2.7] with

D_{−1}t = 1.

Then R ≅ A[T] with A = {r ∈ R|D_{−1}r = 0} and deg T = 1 (cf. [Wah83, Prop. 2.4]), and the theorem follows. □

3. Reduction to Fano Manifolds with Picard number 1

As in Theorem 1.2 let D_{1}, D_{2} be two ample effective divisors on a smooth complex projective variety X of dimension 3 such that the scheme theoretic intersection C = D_{1} ∩ D_{2} is an irreducible reduced curve. This implies that D_{1}, D_{2} are irreducible and reduced.

Let furthermore v ∈ H^{0}(X, T_{X}) be a vector field vanishing on C. Lemma 3.1. If D_{2} is not stabilized by v then there will be an irreducible and reduced divisor D_{∞} stabilized by v, which is linearly equivalent to D_{1} and whose scheme theoretic intersection with D_{2} equals

D_{∞} ∩ D_{2} = D_{1} ∩ D_{2} = C.

Proof. Let G ⊂ Aut X be the connected and nontrivial algebraic subgroup of the automorphism group of X which fixes the zero locus Z(v) of v. Because the action of G on the vector space of m-jets at a fixed point will be faithful for m ≫ 0, the group G is linear algebraic.

Let H ⊂ G be the minimal algebraic subgroup whose Lie algebra contains v. A representation of G in GL(V) shows that H ≅ (ℂ^{∗})^{k} × (ℂ^{+})^{l} ([Bor91, II.7.3]). Because H is commutative, the fixed point locus X^{H} is contained in Z(v) and H stabilizes Z(v). By composing the various ℂ^{∗}- and ℂ^{+}-actions one can move the divisor D_{1} along orbits to a linearly equivalent divisor D_{∞} stabilized by H.

H(Z(v)) = Z(v) implies D_{1} ∩ D_{2} ⊂ D_{∞} ∩ D_{2}, linear equivalence means D_{1}.D_{2} = D_{∞}.D_{2}, consequently D_{1} ∩ D_{2} = D_{∞} ∩ D_{2}. And D_{∞} is reduced and irreducible because D_{∞} ∩ D_{2} is. □

Assume from now on that D_{1} is stabilized by the vector field v. If v_{|D_{1}} = 0, Wahl’s Theorem 1.1 will imply X ≅ ℙ^{3}. If v_{|D_{1}} ≠ 0, the variant Theorem 2.2 will imply that D_{1} is a cone C(C, L) for a (possibly singular) curve C and an ample line bundle L on C.

The cone D_{1} ≅ C(C, L) is the contraction of the section of the projective space bundle ℙ(O_{C} ⊕O_{C}(L)) belonging to the projection O_{C} ⊕O_{C}(L) → O_{C}. Therefore, H^{2}(D_{1}, ℚ) = ℚ, and the finitely generated abelian group H^{2}(D_{1}, ℤ) has rank 1. Now apply Lefschetz’ hyperplane theorem: the natural map

H^{2}(X, ℤ) ↪ H^{2}(D_{1}, ℤ)

is an injection. Since Num(X) is a torsion free quotient of NS(X) ⊂ H^{2}(X, ℤ), it follows that Num(X) = ℤ. There is an ample divisor H on X and r ∈ ℤ such that

K_{X} ≡ rH

On the other hand X can be covered by rational curves: Since there is a non trivial vector field with zeroes on X one of the groups ℂ^{∗} or ℂ^{+} is acting on X (s. proof of Lemma 3.1). The closures of the orbits are rational curves. Consequently, K_{X} is not nef ([Kol96, II.3.13.1]), not ample, and −K_{X} is ample. By [Isk77, Prop.1.15] (and [Sho80] for the proof of hypothesis 1.14 in [Isk77]), X is a smooth Fano threefold with

PicX ≅ H^{2}(X, ℤ) = ℤ.

A classification of these Fano threefolds is given by the following table (cf. [Isk77],[Isk78]):

r (H)^{3} b_{3}/2 g X
4  1  0 33 ℙ^{3}
3  2  0 28 Q ⊂ ℙ^{4}, the quadric
2  1 21 5 V_{1}, a covering of the cone over the Veronese surface
2  2 21 9 V_{2}, a double covering of ℙ^{3}
2  3  5 13 V_{3} ⊂ ℙ^{4} , a cubic
2  4  2 17 V_{4} ⊂ ℙ^{5} , an intersection of two quadrics
2  5  0 21 V_{5}, the intersection Gr(1, 4) ⊂ ℙ^{9} with ℙ^{6}
1  2 52 2 V′_{2} , a double covering of ℙ^{3}
1  4 30 3 V′_{4} ⊂ ℙ^{4} , a quartic
1  4 30 3 V′′_{4}, a double covering of a quadric

r (H)^{3} b_{3}/2 g X
1  6 20 4 V_{6} ⊂ ℙ^{5} , an intersection of a quadric with a cubic
1  8 14 5 V_{8} ⊂ ℙ^{6}, an intersection of three quadrics
1  10 10 6 V_{10} ⊂ ℙ^{7}
1  12  5 7 V_{12} ⊂ ℙ^{8}
1  14  5 8 V_{14} ⊂ ℙ^{9}
1  16  3 9 V_{16} ⊂ ℙ^{10}
1  18  2 10 V_{18} ⊂ ℙ^{11}
1  22  0 12 V_{22} ⊂ ℙ^{13}

In this table, H is the ample generator of Pic(X), r is the index of V, i.e. −K_{X} = r · H, and g = −(K_{X})^{3}/2 + 1 is the genus of X.

The ample divisor H is very ample except when X is of type V_{1}, V_{2}, V′_{2}, V′′_{4}. These cases are dealt with in the last section, while in the next section one assumes that H is very ample.

4. An estimate for the degree

By assumption D = d · H and D_{2} = d_{2} · H are very ample divisors on X. Theorem 2.2 implies that D is a cone C(C, O(D_{2|C})) over a (possibly singular) curve C with vertex P, defined by the divisor D_{2} restricted to C.

Claim 4.1. d_{2} = 1.

Proof. By construction C is linearly equivalent to D_{2|D}, hence very ample. The

corresponding embedding D ↪ X →^{|D_{2}|} ℙ maps C into a hyperplane, and the cone D consists of the lines through the cone vertex P (not in the hyperplane) and points Q ∈ C. Such a line L cuts C transversally in one point, consequently (in X):

1 = C.L = d_{2}H.L.

H.L > 0 implies the hypothesis. □

Let π : X̃ → X be the blow-up of X in the cone vertex P and E the exceptional divisor. Because of the universal property of the blow-up the strict transform S of D also is the blow-up of D in P. By [Ful84, 6.7.1] the effective Cartier divisor S is linearly equivalent to π^{∗}(D) − μ_{D}E for a μ_{D} ∈ ℤ, the multiplicity of the point P in the variety D.

Claim 4.2. μ_{D} = dH^{3}.

Proof. Since H is very ample, by Bertini there exist two smooth hyperplane sections H_{1}, H_{2} ∈ |H| with H_{1} ∩ H_{2} ∩ D = {P }. Then H_{1} ∩ H_{2} does not contain any line L from P to a point Q ∈ C.

As in the proof of the previous claim the H_{i} intersect every line L from P to a point Q ∈ C − H_{i} exactly in P with intersection multiplicity 1, i.e. transversal. Therefore the strikt transforms D̃, H̃_{1}, H̃_{2} do not intersect at all. Since the H_{i} are smooth, the intersection multiplicities of the H_{i} in P are μ_{H_{i}} = 1. Then, by [Ful84, 12.4.8]

μ_{D} = μ_{D}μ_{H_{1}}μ_{H_{2}} = D.H_{1}.H_{2} = dH^{3}.
□

The strict transform S of the cone D is isomorphic to the ℙ^{1}− bundle

S ≅ ℙ(O_{C} ⊕ O_{C}(C))

over the curve C. Let f : Ĉ → C the normalization of the possibly singular curve C and let the ℙ^{1}− bundle

Ŝ ≅ ℙ(OĈ ⊕ f^{∗}O_{C}(C))

over Ĉ be the normalization of S.

Thus one has the following diagram:

\xymatrix{ & X̃ \ar[r]^{π} & X \\ Ŝ \ar[r]^{f} \ar[d]_{p̂} & S \ar@{^{(}->}[u] \ar[r]^{π} \ar[d]^{p} & D \ar@{^{(}->}[u] \\ Ĉ \ar[r]^{f} & C}

Claim 4.3. H^{3} ≤ 4.

Proof. By the adjunction formulas one gets

−K_{X|D} = −K_{D} + D_{|D}

and

K_{S} = K_{X̃|S} + S_{|S}
= π^{∗}(K_{X})_{|S} + 2E_{|S} + π^{∗}(D)_{|S} − μE_{|S} =
= π^{∗}(K_{X|D} + D_{|D}) + (2 − μ)E_{|S} = π^{∗}(K_{D}) + (2 − μ)E_{|S}.

The nonnormal locus N on S is given by the conductor ideal of the normalization f : Ŝ → S, and its support consists of whole fibers of the ℙ^{1}−bundle S → C. From the subadjunction formula for normalization (and the formula for the canonical bundle on smooth ruled surfaces), it follows

f^{∗}K_{S} = KŜ − N = −2C_{0} + kF, k ∈ ℤ,

where C_{0} ≅ Ĉ is the section with negative self intersection on the smooth ruled surface Ŝ and F is a fiber.

Let FŜ be a fiber of p̂ : Ŝ → Ĉ and F_{S} a general fiber of p : S → C. Then f_{∗}FŜ = F_{S}. Furthermore, π_{∗}F_{S} = L, where L is a line from the cone vertex P to a point Q ∈ C. Since E_{|S} is a section of the projective line bundle S over C, the intersection multiplicity F_{S}.E_{|S} = 1. Therefore:

−2 = (−2C_{0} + kF).F = f^{∗}K_{S}.FŜ = K_{S}.F_{S} =
= π^{∗}(K_{D}).F_{S} + (2 − μ)E_{|S}.F_{S} = K_{D}.π_{∗}F_{S} + 2 − H^{3}d =
= K_{D}.L + 2 − H^{3}d.

Let r_{D} be defined by −K_{D} = −r_{D}H_{|D}. As in the proof of Claim 4.1, H_{|D}.L = 1. Consequently:

−2 = −r_{D}H_{|D}.L + 2 − H^{3}d = −r_{D} + 2 − H^{3}d,

and this implies r_{D} = 4 − H^{3}d. But the index r_{X} of the Fano threefold X is ≥ 1, hence

r_{X} = r_{D} + d = 4 − (H^{3} − 1)d ≥ 1,

and H^{3} ≤ 4. □

Now, the following cases must be considered:

• H^{3} = 4: Then d = 1 and r_{X} = 1. Under the assumption that H is very ample, X must be a quartic.

• H^{3} = 3: Then still d = 1, but r_{X} = 2. So X must be a cubic.

• H^{3} = 2: Then d = 1, 2 or 3, corresponding to r_{X} = 3, 2, 1, and X must be a quadric (if H is very ample).

• H^{3} = 1: This implies r_{X} = 4 and X = ℙ^{3}.

5. Special Fanos

In this section, vector fields vanishing on the reduced and irreducble intersection of two (very) ample divisors on ℙ^{3} and the quadric Q_{3} will be constructed, and it will be shown that such vector fields do not exist on a cubic, on a quartic and on varieties of type V_{1}, V_{2}, V′_{2}, V′′_{4}, where H is not very ample.

5.1. V_{1}, V_{2}, V′_{2} und V′′_{4}. These Fano varieties are described in [Isk78]: (a) The morphism φ_{K_{V_{1}}^{−1}} : V_{1} → W_{4} ⊂ ℙ^{6} induced by the complete linear

system |K_{V_{1}}^{−1}| is a 2:1-covering of the cone W_{4} over the Veronese surface F_{4} ⊂ ℙ^{5}. It branches over the smooth divisor S ⊂ W_{4} cut out by a cubic hypersurface not containing the cone vertex.

(b) The morphism φ_{H} : V_{2} → ℙ^{3} is a 2:1-covering with smooth branching divisor S ⊂ ℙ^{3} of degree 4.

(c) The morphism φ_{H} : V′_{2} → ℙ^{3} is a 2:1-covering with smooth branching divisor S ⊂ ℙ^{3} of degree 6.

(d) The morphism φ_{H} : V′′_{4} → Q_{3} ⊂ ℙ^{4} is a 2:1-covering of the threedimensional quadric with smooth branching divisor S ⊂ Q_{3} cut out by a hypersurface of degree 4.

Assume that on one of these Fano threefolds there exists a vector field v ∈ H^{0}(X, T_{X}) vanishing on a curve. Then there is a linear algebraic group G acting on X and inducing v. Since the coverings are induced by complete linear systems, G acts on the bases W of the coverings, too. Furthermore G stabilizes the branching divisors S ∈ W since these divisors describe the locus of the points where the rank of the differentials df_{x} : T_{X,x} → T_{W,f(x)} drops.

Therefore, v restricted to S is a (non trivial) vector field on the branching divisor. By assumption, v should vanish on a curve C. Since the branching divisor S is ample, S intersects C, and v_{|S} has a zero. As explained in section 3 this implies the existence of a ℂ^{+}- or ℂ^{∗}-action on S. Consequently, S is covered by rational curves, i.e. the closures of the orbits. S is uniruled, and K_{S} is not nef.

On the other hand it is easy to compute K_{S} since the branching divisors are complete intersections:

(a) On varieties of type V_{1} is K_{S} = O_{S}(2), because the Veronese surface and the cone over it are generated by three quadrics.

(b) If X is of type V_{2}, then K_{S} = O_{S}.

(c) X of type V′_{2}: then K_{S} = O_{S}(2).
(d) X of type V′′_{4}: then K_{S} = O_{S}(1).

In all cases K_{S} is nef, contradiction.

5.2. Quadrics, cubics and quartics. The starting point is the following Lemma 5.1. All vector fields on a hypersurface H ∈ ℙ^{n}, n > 2, are induced by equivariant vector fields on ℙ^{n}.

Proof. Set d := deg H. First consider the structure sheaf sequence of H tensorized by the tangent bundle T_{ℙ^{n}},

0 → T_{ℙ^{n}}(−d) → T_{ℙ^{n}} → T_{ℙ^{n}} ⊗ O_{H} → 0,

and the beginning of the corresponding long exact sequence,

0 → H^{0}(ℙ^{n}, T_{ℙ^{n}}(−d)) → H^{0}(ℙ^{n}, T_{ℙ^{n}}) →^{ν} H^{0}(H, T_{ℙ^{n}} ⊗ O_{H}) → H^{1}(ℙ^{n}, T_{ℙ^{n}}(−d)).

The Euler sequence

0 → O_{ℙ^{n}}(−d) → O_{ℙ^{n}}(1 − d)^{n+1} → T_{ℙ^{n}}(−d) → 0

and H^{i}(ℙ^{n}, O_{ℙ^{n}}(k)) = 0 for 0 < i < n, k ∈ ℤ imply H^{1}(ℙ^{n}, T_{ℙ^{n}}(−d)) = 0. Therefore, ν is surjective. By the normal sequence

0 → T_{H} → T_{ℙ^{n}} ⊗ O_{H} → N_{H/ℙ^{n}} → 0

one finally has H^{0}(H, T_{H}) ⊂ H^{0}(H, T_{ℙ^{n}} ⊗ O_{H}). □

From now on, let H ⊂ ℙ^{4} be a smooth hypersurface of degree d = 2, 3, 4 with a vector field v ∈ H^{0}(H, T_{H}) vanishing on the irreducible and reduced intersection C = H_{1} ∩ H_{2} of two very ample divisors. The lemma above shows that v is induced by a vector field on ℙ^{4} also called v.

Claim 4.1 and the cases at the end of section 4 show that v stabilizes a cone D = H ∩ H_{1} cut out by a hyperplane, and the zero locus of v is the intersection C = H ∩H_{1} ∩H_{2} with another hyperplane. The cone vertex P ∈ D is not contained in H_{2} since otherwise, D ∩ H_{2} contains more than one line because of deg H > 1.

Now choose homogeneous coordinates (x_{0} : x_{1} : x_{2} : x_{3} : x_{4}) on ℙ^{4} such that the cone D lies on the hyperplane H_{1} = {x_{4} = 0}, and the cone basis lies in the plane H_{1} ∪ H_{2} = {x_{3} = x_{4} = 0}. Furthermore, let P = (0 : 0 : 0 : 1 : 0). Lemma 5.2. The hypersurface H is given by a polynomial

h = f(x_{0}, x_{1}, x_{2}) + x_{4}g(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}),

where deg f = d, deg g = d − 1. The coefficients of the monomial x_{3}^{d−1} in g and of the monomial x_{i}x_{4}^{d−1} in h do not vanish (at least for one 0 ≤ i ≤ 4).

Proof. The polynomial h may be written as

h = f(x_{0}, x_{1}, x_{2}) + x_{3}k(x_{0}, x_{1}, x_{2}, x_{3}) + x_{4}g(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}).

Since the intersection with H_{1} = {x_{4} = 0} is supposed to be a cone over a basis C ⊂ {x_{3} = x_{4} = 0} with vertex (0 : 0 : 0 : 1 : 0), this cone D is given by the equation f(x_{0}, x_{1}, x_{2}) = 0, and k ≡ 0.

Since the hypersurface is smooth, the rest follows from computing the gradient of h: the coefficient of x_{3}^{d−1} must not be 0 because otherwise the vertex (0 : 0 : 0 : 1 : 0) will be a singularity in H, too. Similarly, the coefficient of at least one of the monomials x_{i}x_{4}^{d−1}, i ≤ 4, must not vanish, because otherwise the point (0 : 0 : 0 : 0 : 1) lies in H and will be not smooth. □

Now, by the Euler sequence vector fields on ℙ^{4} correspond to homogeneous derivations D = ∑_{i,j} a_{ij}x_{i} \frac{∂}{∂x_{j}} of weight 0 on ℂ[x_{0}, …, x_{4}], modulo the Euler vector fields ∑_{i} ax_{i} \frac{∂}{∂x_{i}}.

Lemma 5.3. The vector field v corresponds to the derivation D_{v} = ∑_{i,j} a_{ij}x_{i} \frac{∂}{∂x_{j}} given by (a non trivial scalar multiple of) the matrix

(a_{ij}) = ( \begin{matrix}0 & ⋯ & ⋯ & 0 & 0 \\ ⋮ & 0 & & ⋮ & ⋮ \\ ⋮ & & 0 & 0 & ⋮ \\ 0 & ⋯ & 0 & 1 & 0 \\ ∗ & ∗ & ∗ & ∗ & a \end{matrix} ) .

Proof. Since deg C > 1, the curve C is no line. Since vector fields on projective spaces always vanish on complete linear subspaces, v vanishes on the plane x_{3} = x_{4} = 0. On the other hand, v does not vanish on the hyperplane x_{4} = 0, because it contains the cone D, and v should be non trivial on D. Since every vector field on a cone vanishes in the vertex, v vanishes in P = (0 : 0 : 0 : 1 : 0).

This implies the hypothesis, because the zero locus of vector fields on ℙ^{4} consists of the eigenspaces of the transposed corresponding matrix. □

Now, v stabilizes the hypersurface H exactly when the derivation D_{v} maps the principal ideal (h) ⊂ ℂ[x_{0}, …, x_{4}] describing H to itself, i.e. D_{v}h = λh for a λ ∈ ℂ. Since a_{ij} = 0 for i, j = 0, 1, 2,

D_{v}h = D_{v}f + D_{v}x_{4} · g + x_{4}D_{v}g = 0 + x_{4}(ag + D_{v}g),

and consequently λ = 0.

This gives immediately a quadric with a vector field vanishing as in Theorem 1.2: Let h = x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}x_{4} be the equation of the matrix and let v correspond to the derivation D_{v} = x_{3} \frac{∂}{∂x_{3}} − x_{4} \frac{∂}{∂x_{4}} . Then, D_{v}h = 0, and v vanishes on the smooth quadric x_{0}^{2} + x_{1}^{2} + x_{2}^{2} = 0 contained in the plane x_{3} = x_{4} = 0 and in the cone vertex (0 : 0 : 0 : 1 : 0). Furthermore, v stabilizes the cone x_{0}^{2} + x_{1}^{2} + x_{2}^{2} = 0 contained in the hyperplane x_{4} = 0.

Why are there no such vector fields on cubics and quartics ?

(a) By Lemma 5.2 the monomial x_{3}^{d−1}x_{4} has a coefficient c ≠ 0 in h, but there is no monomial of the form x_{3}^{d−1}x_{i}, i ≤ 3, in h.

(b) The coefficient of x_{3}^{d−1}x_{4} in D_{v}h = 0 is 0. Decomposing h in monomials and D_{v} in “monomial” derivations of the form a_{ij}x_{i} \frac{∂}{∂x_{j}}, one sees that only the derivations

a_{3j}x_{3} \frac{∂}{∂x_{j}} x_{j}x_{3}^{d−2}x_{4}, a_{4j}x_{4} \frac{∂}{∂x_{j}} x_{j}x_{3}^{d−1}x_{j}

contribute to the coefficient of x_{3}^{d−1}x_{4}. Since a_{3j} = 0 for i = 0, 1, 2, 4 and the coefficients of x_{3}^{d−1}x_{i}, i ≤ 3, vanish, the coefficient of x_{3}^{d−1}x_{4} in D_{v}h is the coefficient of

c(a_{33}x_{3} \frac{∂}{∂x_{3}} x_{3}^{d−1}x_{4} + a_{44}x_{4} \frac{∂}{∂x_{4}} x_{3}^{d−1}x_{4}),

i.e. c(d − 1 + a).

(c) Consequently, a = 1 − d. This number is an eigenvalue of the matrix corresponding to the derivation, different from the other Eigen values. Therefore, in appropriate coordinates the matrix is diagonal, i.e.

D_{v} = x_{3} \frac{∂}{∂x_{3}} + (1 − d)x_{4} \frac{∂}{∂x_{4}} .

(d) The difference between quadrics and cubics resp. quartics comes from the fact that the monomials x_{3}x_{4}^{d−1} and x_{3}^{d−1}x_{4} are different for d > 2. The coefficient c of x_{3}x_{4}^{d−1} in h must be 0 for d > 2, because otherwise the coefficient of x_{3}x_{4}^{d−1} in Dh equals c(1 − (d − 1)^{2}) ≠ 0. The same is true for the monomials x_{i}x_{4}^{d−1} with i = 0, 1, 2, 4. But this a contradiction to Lemma 5.2.

Remark 5.4. There is another argumentation for cubics: The Fano variety F (V_{3}) of the lines on the cubic V_{3} is a smooth variety with a very ample canonical divisor ([CG72, 7.8,10.13]). Furthermore, through a general point there are exactly 6 lines ([Tju72]).

But as already shown in the beginning of the section, the existence of a vector field with zeroes implies an effective linear algebraic group operation with fixed points on V_{3}. Every group operation on V_{3} induces a group operation on F (V_{3}), and this operation must be trivial because of the very ample canonical divisor. Therefore, lines on V_{3} are stabilized by the operation, and the intersection point of 6 lines is fixed. The operation on V_{3} is trivial, too.

5.3. The projective space ℙ^{3}. A vector field vanishing on a line is given by D_{v} = x_{2} \frac{∂}{∂x_{2}} + x_{3} \frac{∂}{∂x_{3}} .

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Thomas Eckl, Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany

E-mail address: thomas.eckl@uni-bayreuth.de