FANO THREEFOLDS V_{22}

ALEXANDER KUZNETSOV

1. Introduction

Let W be a fixed 4-dimensional vector space. In the paper [8] S. Mukai has announced Theorem according to which every Fano threefold V_{22} can be embedded into the Ellingsrud–Piene–Strømme compactification 𝔚 of the space of twisted rational cubic curves in the projective space ℙ(W) as the zero locus of a certain global section of the vector bundle

K^{∗} ⊕ K^{∗} ⊕ K^{∗},  (1)

where K is the universal 3-dimensional vector bundle over 𝔚 (see [3], [6]).

The natural isomorphism H^{0}(𝔚, K^{∗}) ≅ S^{2}W shows that the global sections of (1) are in one-one correspondence with the nets of quadrics in the dual projective space ℙ(W^{∗}) which can be classified via their Hesse curve and a theta-characteristic on this curve (see [12], [7]). The Hesse curve is the curve of the degenerate quadrics in the net. Since dim W = 4 it is a plane quartic.

Now assume we are given a pair (C, θ), consisting of a plane quartic C and a theta-characteristic θ on C. Let A_{C} : ℂ^{3} → S^{2}W be the corresponding net of quadrics, s_{C} the corresponding global section of (1) and 𝔚_{C} the zero locus of s_{C}. We prove the following Theorem.

Main Theorem. The space 𝔚_{C} is a smooth Fano threefold V_{22}. Curve C is the curve, parameterizing lines in 𝔚_{C}. Intersection of lines in 𝔚_{C} is managed by the theta-characteristic θ.

The last statement means that lines l_{x}, l_{y} ⊂ 𝔚_{C} corresponding to points x, y ∈ C meet iff |x+θ−y| ≠ ∅. The embedding 𝔚_{C} ⊂ 𝔚 of our Fano threefold into the space of twisted cubics in ℙ(W) gives rise

to the following construction. The space 𝔚 is equipped with a natural “naive universal curve”. It is just the point-curve incidence subset in ℙ(W) × 𝔚. In the paper [6] a certain nice desingularization M̃ of the “naive curve” was constructed. It turns out that both natural projections π̃_{0} : M̃ → ℙ(W) and π̃_{1} : M̃ → 𝔚 are very simple. Both of them are consecutively projectivization of a vector bundle and a blow up (see [6, 4.7]).

We enlarge our picture by considering also the space M̃_{C} ⊂ M̃ that is the zero locus of the global section π̃_{1}^{∗}s_{C} of the vector bundle π̃_{1}^{∗}K^{∗} ⊕ π̃_{1}^{∗}K^{∗} ⊕ π̃_{1}^{∗}K^{∗} on M̃. The projections π̃_{i} can be, of course, restricted on the space M̃_{C}. We prove that the situation with the maps remains the same, namely both maps π̃_{0} : M̃_{C} → ℙ(W) and π̃_{1} : M̃_{C} → 𝔚_{C} are consecutively projectivization of a vector bundle and a blow up (see Theorem 1). However, both vector bundles now are 2-dimensional. The first of them E_{|𝔚_{C}} is just the restriction of the universal 2-dimensional bundle E over 𝔚 (see [3], [6]) to 𝔚_{C}. The second bundle R_{C}, which lives over ℙ(W) is a so-called mathematical instanton vector bundle with c_{2} = 3. Relation of such bundles to the nets of quadrics was investigated in [7]. Now it turns out that the space M̃_{C} and the projection π̃_{1} can be described purely in terms of the rational curves in ℙ(W) providing the most degenerate splitting of the bundle R_{C}. E.g. the exceptional lifts of the 2-jumping lines of R_{C} span the surface Σ^{−} ⊂ ℙ_{ℙ(W)}(R_{C}) which is to be blown up to get M̃_{C} (by the exceptional lift of the rational curve X ⊂ ℙ(W) we mean the exceptional section of the ruled surface ℙ_{X}(R_{C}_{|X}) ⊂ ℙ_{ℙ(W)}(R_{C})). The exceptional lifts of 5-jumping twisted cubics are but the fibers of the map π̃_{1}. Thus the geometry of the space 𝔚_{C} is found to be very strongly related to the geometry of the bundle R_{C}.

I was partially supported by CRDF grant RM1-265.

We apply the above results to the following problems. First we prove that

(O_{𝔚_{C}} , K_{|𝔚_{C}}^{∗} , E_{|𝔚_{C}}^{∗} , Λ^{2}K_{|𝔚_{C}}^{∗})

is a full and strong exceptional collection on 𝔚_{C}.

Second we give a precise description of rational curves of low degree (≤ 3) in 𝔚_{C}. In particular, we show that the linear system |K_{C}| is naturally a compactification of the space Rat_{2}(𝔚_{C}) of conics in 𝔚_{C} and that the space ℙ(W) is naturally a compactification of the space Rat_{3}(𝔚_{C}) of rational cubic curves in 𝔚_{C}. The latter seems to be the most surprising because the space 𝔚_{C} is by definition embedded into the compactification 𝔚 of the space of rational twisted cubics in ℙ(W)! It looks like the kind of duality the nature of which is quite mysterious.

Remark 1. In the paper [9] S. Mukai has introduced a construction of the Fano threefold V_{22} from a plane quartic F. This construction related to our as follows. The threefold VSP(6, F) constructed by Mukai coincides with the space 𝔚_{C}, when (C, θ) is the Scorza map (see [5]) of F.

The paper is organized as follows. In Section 2 we define the spaces 𝔚_{C} ⊂ 𝔚 and M̃_{C} ⊂ M̃ as the zero loci of the sections s_{C} and π̃_{1}^{∗}s_{C} respectively and give their description.

In Section 3 we deal with exceptional collections.

In Section 4 we give a description of rational curves in 𝔚_{C}.

This paper is a sequel to [6] and [7] so we will freely use notations and conventions thereof.

I am very indebted to A. Bondal for the attention and numerous advises. I also want to thank D. Orlov and A. Tyurin for helpful discussions. I am very grateful to the Max-Planck-Institut für Mathematik for hospitality.

2. The construction of the space 𝔚_{C}

2.1. Let W be a fixed 4-dimensional vector space, ℙ(W) its projectivization, 𝔚 the Ellingsrud-Piene-Strømme compactification of the space of rational twisted cubic curves in ℙ(W), M̃ the desingularization of the “naive universal curve” over 𝔚 (the “naive universal curve” is just the point-curve incidence subset in ℙ(W) × 𝔚) which was constructed in [6]. Finally let π̃_{0} : M̃ → ℙ(W) and π̃_{1} : M̃ → 𝔚 be the canonical projections.

Recall that projections π̃_{0} and π̃_{1} factor as follows

[figure: M̃; ℙ(R); ℙ(E); ℙ(W); 𝔚; π̂_{0}; π̂_{1}; π_{0}; π_{1}; ↩^{i}; M̃ = M^{−}×_{M^{0}}M^{+}; M^{−} = ℙ_{M^{0}}(S^{−}); ℙ_{M^{0}}(S^{+}) = M^{+}; M^{0}; π̂_{0}; π̂_{1}; p^{−}; p^{+}]

Here R is a certain 11-dimensional exceptional vector bundle over ℙ(W) (see [6], [7]) and E is the universal 2-dimensional vector bundle over 𝔚 (see [3], [6]). The maps π_{0}, π_{1} are natural projections. The maps π̂_{0} and π̂_{1} are the blow ups of ℙ_{ℙ(W)}(R) (resp. ℙ_{𝔚}(E)) along the subvariety M^{−} = ℙ_{M^{0}}(S^{−}) ⊂ ℙ_{ℙ(W)}(R) (resp. M^{+} = ℙ_{M^{0}}(S^{+}) ⊂ ℙ_{𝔚}(E)) with common exceptional divisor M̃ = M^{−} ×_{M^{0}} M^{+} ⊂ M̃. Here M^{0} = ℙ(W^{∗}) × Gr(2, W) and S^{±} are certain 5-dimensional and 2-dimensional vector bundles over M^{0}.

Recall that over the space 𝔚 there is also a 3-dimensional universal vector bundle K (see [3], [6]) and we have the following exact sequence of coherent sheaves over M̃

0 ⟶ π̂_{0}^{∗}L_{R} ⊗ π̃_{0}^{∗}T_{ℙ(W)}(−3) ⟶ π̃_{1}^{∗}K ⟶ i_{∗}(p^{∗}Λ^{2}U ⊗ π̂_{1}^{∗}L^{+}).  (2)

Here L_{R} ⊂ π_{0}^{∗}R is the relative tautological linear subbundle on ℙ_{ℙ(W)}(R), L^{±} ⊂ p^{±}^{∗}S^{±} relative tautological linear subbundles on M^{±} = ℙ_{M^{0}}(S^{±}), p^{±} : M^{±} → M^{0} and p : M̃ = M^{−} ×_{M^{0}} M^{+} → M^{0} natural projections, U ⊂ W ⊗ O_{Gr(2,W)} tautological 2-dimensional subbundle on M^{0} = ℙ(W^{∗}) × Gr(2, W) (actually the pull back from the second factor) and i : M̃ → M̃ is the embedding.

2.1.1. Lemma. We have natural isomorphisms

H^{0}(𝔚, K^{∗}) ≅ H^{0}(ℙ_{𝔚}(E), π_{1}^{∗}K^{∗}) ≅ H^{0}(M̃, π̃_{1}^{∗}K^{∗}) ≅ H^{0}(M̃, π̂_{0}^{∗}L_{R}^{∗} ⊗ π̃_{0}^{∗}Ω_{ℙ(W)}(3)) ≅

≅ H^{0}(ℙ_{ℙ(W)}(R), L_{R}^{∗} ⊗ π_{0}^{∗}Ω_{ℙ(W)}(3)) ≅ Hom_{ℙ(W)}(R, Ω_{ℙ(W)}(3)) = S^{2}W.

Proof. The first, second, fourth and fifth isomorphisms follow from the projection formula applied to the maps π_{1}, π̂_{1}, π̂_{0}, π_{0} respectively. In order to prove the third isomorphism consider the dual sequence of (2).

0 → π̃_{1}^{∗}K^{∗} → π̂_{0}^{∗}L_{R}^{∗} ⊗ π̃_{0}^{∗}Ω_{ℙ(W)}(3) → i_{∗}(p^{∗}Λ^{2}U^{∗} ⊗ π̂_{0}^{∗}L^{−}) → 0.  (3)

We applied here the fact that N_{M̃/M̃} = π̂_{0}^{∗}L^{−} ⊗ π̂_{1}^{∗}L^{+} (see [6, 4.6]). One can easily see that

H^{•}(M̃, i_{∗}(p^{∗}Λ^{2}U^{∗} ⊗ π̂_{0}^{∗}L^{−})) = H^{•}(M̃, p^{∗}Λ^{2}U^{∗} ⊗ π̂_{0}^{∗}L^{−}) =

= H^{•}(M^{−}, p^{−}^{∗}Λ^{2}U^{∗} ⊗ L^{−}) = H^{•}(M^{0}, Λ^{2}U ⊗ p_{∗}^{−}L^{−}) = 0.

The third isomorphism follows. The last isomorphism can be proved by the direct calculation (alternatively, see [7, 3.2.4]).

2.1.2. The restriction of (3) on M̃ gives rise to the pair of exact sequences.

0 ⟶ p^{∗}Λ^{2}U^{∗} ⊗ π̂_{1}^{∗}L^{+}^{∗} ⟶ π̃_{1}^{∗}K_{|M̃}^{∗} ⟶ p^{∗}(η^{∗} ⊗ W/U) ⟶ 0;  (4)
0 ⟶ p^{∗}(η^{∗} ⊗ W/U) ⟶ (π̂_{0}^{∗}L_{R}^{∗} ⊗ π̃_{0}^{∗}Ω_{ℙ(W)}(3))_{|M̃} ⟶ p^{∗}Λ^{2}U^{∗} ⊗ π̂_{0}^{∗}L^{−} ⟶ 0

(see [6, 5.5]). Here η ⊂ W^{∗} ⊗ O_{ℙ(W^{∗})} stands for the tautological linear subbundle on M^{0} (actually, the pull back from the first factor.)

Lemma. We have natural isomorphisms

H^{0}(M̃, p^{∗}(η^{∗} ⊗ W/U)) ≅ H^{0}(M^{0}, η^{∗} ⊗ W/U) = W ⊗ W; H^{•}(M̃, p^{∗}Λ^{2}U^{∗} ⊗ π̂_{0}^{∗}L^{−}) = 0.

Proof. Evident.

2.1.3. Fix a 3-dimensional vector space N. Let (C, θ) ∈ Qu \ Qu^{L} be a non-Lüroth rigid plane quartic α : C ↪ ℙ(N) (see [7, 2.1.1]) and A_{C} : N ⟶ S^{2}W the corresponding net of quadrics in ℙ(W^{∗}), which accordingly to [7, 2.1.3] is a regular non-Lüroth net.

Taking into account the isomorphisms of Lemma 2.1.1 the net of quadrics A_{C} gives rise to the following data.

• s_{1} ∈ H^{0}(𝔚, N^{∗} ⊗ K^{∗});

• s_{2} ∈ H^{0}(ℙ_{𝔚}(E), N^{∗} ⊗ π_{1}^{∗}K^{∗});

• s_{3} ∈ H^{0}(M̃, N^{∗} ⊗ π̃_{1}^{∗}K^{∗});

• s_{4} ∈ H^{0}(M̃, N^{∗} ⊗ π̂_{0}^{∗}L_{R}^{∗} ⊗ π̃_{0}^{∗}Ω_{ℙ(W)}(3));

• s_{5} ∈ H^{0}(ℙ_{ℙ(W)}(R), N^{∗} ⊗ L_{R}^{∗} ⊗ π_{0}^{∗}Ω_{ℙ(W)}(3));

• μ_{C} ∈ Hom(R, N^{∗} ⊗ Ω_{ℙ(W)}(3)). Let Z_{k} be the zero locus of s_{k} (k = 1, …, 5) and R_{C} be the kernel of μ_{C}. Recall, that accordingly to the Theorem 1 of [7] R_{C} is an instanton 2-dimensional vector bundle over ℙ(W).

Lemma. We have 1). Z_{2} = π_{1}^{−1}(Z_{1}) = ℙ_{Z_{1}}(E_{|Z_{1}}); 2). Z_{3} = π̂_{1}^{−1}(Z_{2}); 3). Z_{4} = π̂_{0}^{−1}(Z_{5}); 4). Z_{2} \ M^{+} = Z_{3} \ M̃ = Z_{4} \ M̃ = Z_{5} \ M^{−}; 5). Z_{5} = ℙ_{ℙ(W)}(R_{C}).

Proof. Evident.

2.1.4. The Lemma 2.1.3, 4) shows that the spaces Z_{2} and Z_{5} coincide on the complements of M^{+} and M^{−}. We should also describe Z_{2} ∩ M^{+} and Z_{5} ∩ M^{−}. To this end consider the following additional data induced by A_{C}.

• s′_{0} ∈ H^{0}(M^{0}, N^{∗} ⊗ η^{∗} ⊗ W/U);

• s′_{2} ∈ H^{0}(M^{+}, N^{∗} ⊗ π_{1}^{∗}K_{|M^{+}}^{∗});

• s′_{3} ∈ H^{0}(M̃, N^{∗} ⊗ π̃_{1}^{∗}K_{|M̃}^{∗});

• s′_{4} ∈ H^{0} (M̃, N^{∗} ⊗ (π̂_{0}^{∗}L_{R}^{∗} ⊗ π̃_{0}^{∗}Ω_{ℙ(W)}(3))_{|M̃});

• s′_{5} ∈ H^{0} (M^{−}, N^{∗} ⊗ (L_{R}^{∗} ⊗ π_{0}^{∗}Ω_{ℙ(W)}(3))_{|M^{−}}).

Let Z′_{k} be the zero locus of s′_{k} (k = 0, 2, 3, 4, 5). The sequence (4) shows that on the space p^{+}^{−1}(Z′_{0}) the section s′_{2} can be considered as section

s′_{2}_{|p^{+}^{−1}(Z′_{0})} = s′_{1} ∈ H^{0}(p^{+}^{−1}(Z′_{0}), N^{∗} ⊗ p^{+}^{∗}Λ^{2}U^{∗} ⊗ L^{+}^{∗}).

Lemma. We have 1). Z′_{2} ⊂ p^{+}^{−1}(Z′_{0}) is the zero locus of s′_{1}; 2). Z′_{3} = π̂_{1}^{−1}(Z′_{2}); 3). Z′_{4} = π̂_{0}^{−1}(Z′_{5}); 4). Z′_{5} = p^{−}^{−1}(Z′_{0}); 5). Z_{2} ∩ M^{+} = Z′_{2}, Z_{3} ∩ M̃ = Z′_{3}, Z_{4} ∩ M̃ = Z′_{4}, Z_{5} ∩ M^{−} = Z′_{5}.

Proof. Evident.

2.1.5. Recall that in [7, 2.2.6] we have defined embeddings β : C → ℙ(W^{∗}) and γ : C → Gr(2, W) and proved that (β × γ)(C) ⊂ ℙ(W^{∗}) × Gr(2, W) is the zero locus of the section t_{C} ∈ H^{0}(ℙ(W^{∗}) × Gr(2, W), N^{∗} ⊗O_{ℙ(W^{∗})}(1)⊗(W/U)). This section evidently coincides with s′_{0} (recall that M^{0} = ℙ(W^{∗})× Gr(2, W) and η^{∗} = O_{ℙ(W^{∗})}(1)), hence Z′_{0} = (β × γ)(C) ⊂ ℙ(W^{∗}) × Gr(2, W). Since

H^{0}(p^{+}^{−1}(β × γ)(C), N^{∗} ⊗ p^{+}^{∗}Λ^{2}U^{∗} ⊗ L^{+}^{∗}) = H^{0}(C, (β × γ)^{∗}(N^{∗} ⊗ Λ^{2}U^{∗} ⊗ p_{∗}^{+}L^{+}^{∗})) =

= H^{0}(C, N^{∗} ⊗ (β × γ)^{∗}(Λ^{2}U^{∗} ⊗ S^{+}^{∗})) = Hom_{C}((β × γ)^{∗}S^{+}, N^{∗} ⊗ γ^{∗}Λ^{2}U^{∗}).

the section s′_{1} induces a homomorphism ν_{C} : (β × γ)^{∗}S^{+} ⟶ N^{∗} ⊗ γ^{∗}Λ^{2}U^{∗}. The proof of the following Lemma will be postponed till 2.1.12.

Lemma. The map ν_{C} is surjective.

2.1.6. Definition. Let S_{C}^{+} denote the kernel of the map ν_{C} and S_{C}^{−} denote (β × γ)^{∗}S^{−}. Both S_{C}^{+} and S_{C}^{−} are 2-dimensional vector bundles over C. The Lemmas 2.1.4 and 2.1.5 imply.

Corollary. Z′_{2} = ℙ_{C}(S_{C}^{+}), Z′_{5} = ℙ_{C}(S_{C}^{−}), Z′_{3} = ℙ_{C}(S_{C}^{−})×_{C} ℙ_{C}(S_{C}^{+}) ⊂ ℙ_{C}(S_{C}^{−})×_{C} ℙ_{C}((β ×γ)^{∗}S^{+}) = Z′_{4}. In particular, codim_{Z_{5}} Z′_{5} = 4 − 2 = 2 and codim_{Z_{2}} Z′_{2} = 4 − 2 = 2.

Note that the Lemma 2.1.4, 5) implies that we have canonical embeddings ℙ_{C}(S_{C}^{+}) ⊂ Z_{2}, ℙ_{C}(S_{C}^{−}) ⊂ Z_{5}.

2.1.7. Now we can give the final description. To this end we introduce the following notation. Let 𝔚_{C} = Z_{1} ⊂ 𝔚, M̃_{C} = Z_{3} ⊂ M̃, M̃_{C} = Z′_{3} ⊂ M̃. Let us denote the restriction of the maps π̃_{i}, π̂_{i} on M̃_{C} and of the maps π_{0} on ℙ_{ℙ(W)}(R_{C}) ⊂ ℙ_{ℙ(W)}(R), π_{1} on ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) ⊂ ℙ_{𝔚}(E) by the same letters.

Theorem 1. 1) The projections π̃_{0} : M̃_{C} → ℙ(W) and π̃_{1} : M̃_{C} → 𝔚_{C} factor as follows:

[figure: M̃_{C}; ℙ_{ℙ(W)}(R_{C}); ℙ_{𝔚_{C}} (E_{|𝔚_{C}}); ℙ(W); 𝔚_{C}; π̂_{0}; π̂_{1}; π_{0}; π_{1}; ↩^{i}; M̃_{C} = ℙ_{C}(S_{C}^{−})× _{C}ℙ_{C}(S_{C}^{+}); ℙ_{C}(S_{C}^{−}); ℙ_{C}(S_{C}^{+}); C; π̂_{0}; π̂_{1}; p^{−}; p^{+}]

Here R_{C} is the mathematical instanton with c_{2} = 3 corresponding to the curve C in the sense of [7, 3.1.2], E is the universal 2-dimensional vector bundle over 𝔚 (see [3], [6]), S_{C}^{±} are certain 2-dimensional vector bundles over C, defined in 2.1.6. The maps π_{0} and π_{1} are natural projections. The maps π̂_{0} and π̂_{1} are the blow ups along ℙ_{C}(S_{C}^{−}) and ℙ_{C}(S_{C}^{+}) respectively with common exceptional divisor M̃_{C} = ℙ_{C}(S_{C}^{−}) ×_{C} ℙ_{C}(S_{C}^{+}) ⊂ M̃_{C}.

2) The birational transform q = π̂_{1} · π̂_{0}^{−1} : ℙ_{ℙ(W)}(R_{C}) −−→ ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) is a flop at ℙ_{C}(S_{C}^{−}).

Proof. Since π̂_{0} : M̃ → ℙ_{ℙ(W)}(R) is the blow up along M^{−} the irreducibility of π̂_{0}(M̃_{C}) = ℙ_{ℙ(W)}(R_{C}) and of M̃_{C} = M̃_{C} ∩ M̃ = M̃_{C} ∩ π̂_{0}^{−1}(M^{−}) implies that M̃_{C} irreducible. Now the Lemma 2.1.3 implies that M̃_{C} is a strict transform of ℙ_{ℙ(W)}(R_{C}) with respect to π̂_{0} and of ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) with respect to π̂_{1}. Hence M̃_{C} is the blow up of ℙ_{ℙ(W)}(R_{C}) along ℙ_{ℙ(W)}(R_{C}) ∩ M^{−} = Z′_{5} = ℙ_{C}(S_{C}^{−}) and the blow up of ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) along ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) ∩ M^{+} = Z′_{2} = ℙ_{C}(S_{C}^{+}). Now the rest of the first part of the Theorem follows from the same properties of M̃ (see 2.1).

The proof of the second part of the Theorem will be given below.

2.1.8. Theorem 1 implies that Pic M̃_{C} = ℤ^{3}. We have a basis (h, r, d) in Pic M̃_{C} formed by the following elements: h = c_{1}(O_{ℙ(W)}(1)) the positive generator of Pic ℙ(W) = ℤ, r = c_{1}(L_{R}^{∗}_{|ℙ_{ℙ(W)}(R_{C})}) ∈

Pic ℙ_{ℙ(W)}(R_{C}), d = [M̃_{C}] ∈ Pic M̃_{C} (we mean that all above Picard groups are embedded into Pic M̃_{C} via π̃_{i}^{∗}, π̂_{i}^{∗}). We also have in Pic M̃_{C} elements H = c_{1}(K_{|𝔚_{C}}^{∗}) = c_{1}(E_{|𝔚_{C}}^{∗}) ∈ Pic 𝔚_{C} and

e = c_{1}(L_{E}^{∗}_{|ℙ_{𝔚_{C}} (E_{|𝔚_{C}})}) ∈ Pic_{𝔚_{C}} (E_{|𝔚_{C}}). All of them are restrictions of the analogous elements in Pic M̃, hence satisfy the same relations

e = r + 2h, H = 3r + 5h − d,  r = 2H − 5e + 2d, h = 3e − H − d  (5)

(compare with [6, (21)]). This implies that (H, e, d) is also a basis in Pic M̃_{C}, hence H is a generator of Pic 𝔚_{C} = ℤH.

The canonical classes of our spaces are expressed as follows:

K_{ℙ(W)} = −4h,  K_{𝔚_{C}} = −H,
K_{ℙ_{ℙ(W)}(R_{C})} = −4h − 2r,  K_{ℙ_{𝔚_{C}} (E_{|𝔚_{C}})} = −2e,
KM̃_{C} = −4h − 2r − d = −2e − d.

2.1.9.

Proof of the Theorem 1, 2). Recall that spaces ℙ_{ℙ(W)}(R) and ℙ_{𝔚}(E) are equipped by the maps p^{−} (resp. p^{+}) into a certain variety M_{2/3}, such that p^{−} (resp p^{+}) is a contraction of M^{−} = ℙ_{M^{0}}(S^{−}) (resp. M^{+} = ℙ_{M^{0}}(S^{+})) onto M^{0} (see [6, Remark 12]). Therefore the composition of the natural embedding ℙ_{ℙ(W)}(R_{C}) ⊂ ℙ_{ℙ(W)}(R) (resp. ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) ⊂ ℙ_{𝔚}(E)) and of the map p^{−} (resp. p^{+}) is a contraction of ℙ_{C}(S_{C}^{−}) (resp. ℙ_{C}(S_{C}^{+})) onto C. Hence we have the following commutative diagram

ℙ_{ℙ(W)}(R_{C}) ⟶^{q} ℙ_{𝔚_{C}} (E_{|𝔚_{C}})
p^{−}↓  ^{p^{+}}↓
Y  Y

where

M_{2/3} ⊃ Y = p^{−}(ℙ_{ℙ(W)}(R_{C})) = \overline{p^{−}(ℙ_{ℙ(W)}(R_{C}) \ ℙ_{C}(S_{C}^{−}))} =

= \overline{p^{+}(ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) \ ℙ_{C}(S_{C}^{+}))} = p^{+}(ℙ_{𝔚_{C}} (E_{|𝔚_{C}})).

Applying [6, 5.10] and 2.1.8 we get

(K_{ℙ_{ℙ(W)}(R_{C})})_{|ℙ_{C}(S_{C}^{−})} ∈ p^{−}^{∗} Pic C, (K_{ℙ_{𝔚_{C}} (E_{|𝔚_{C}})})_{|ℙ_{C}(S_{C}^{+})} ∈ p^{+}^{∗} Pic C

and we see that q satisfies the definition of a flop.

Remark 2. It is easy to see that M̃_{C} = ℙ_{ℙ(W)}(R_{C}) ×_{Y} ℙ_{𝔚_{C}} (E_{|𝔚_{C}}). Indeed, we have

\begin{matrix}M̃_{C} = M̃_{C} ⊔ M̃_{C} \ M̃_{C}, & ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) = ℙ_{C}(S_{C}^{+}) ⊔ ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) \ ℙ_{C}(S_{C}^{+}), \\ ℙ_{ℙ(W)}(R_{C}) = ℙ_{C}(S_{C}^{−}) ⊔ ℙ_{ℙ(W)}(R_{C}) \ ℙ_{C}(S_{C}^{−}), & Y = C ⊔ Y \ C \end{matrix}

and evidently

M̃_{C} = ℙ_{C}(S_{C}^{−}) ×_{C} ℙ_{C}(S_{C}^{+}),  M̃_{C} \ M̃_{C} = Y \ C = ℙ_{ℙ(W)}(R_{C}) \ ℙ_{C}(S_{C}^{−}) = ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) \ ℙ_{C}(S_{C}^{+}).

Remark 3. One can easily prove that 1). the space Y ⊂ ℙ^{7} is a complete intersection of three quadrics; 2). the embedding C ⊂ Y ⊂ ℙ^{7} is given by the complete linear system |2K_{C} + θ|; 3). the curve C is the singular locus of Y; 4). the space M̃_{C} is the blow-up of Y along C.

2.1.10. Corollary. The space 𝔚_{C} is a smooth threefold.

Proof. First, note that ℙ_{𝔚_{C}} (E_{C}) is a smooth 4-fold, because q is a flop and ℙ_{ℙ(W)}(R_{C}) is a smooth 4-fold. It is remained to recall that the projection π_{1} : ℙ_{𝔚_{C}} (E_{C}) ⟶ 𝔚_{C} is a smooth ℙ^{1}-fibration.

2.1.11. Now we can prove the first part of the Main Theorem which has been formulated in the Introduction.

Proposition (S. Mukai, [8]). The space 𝔚_{C} is a smooth Fano threefold V_{22}.

Proof. The Betti numbers of 𝔚_{C} one can easily compute using the Theorem 1. The answer is

b_{2i}(𝔚_{C}) = 1, (i = 0, 1, 2, 3) b_{2i+1}(𝔚_{C}) = 0, (i = 0, 1, 2).

Hence, from Iskovskih’s classification Theorem [4] it follows that in order to prove the Proposition it suffices to check that the canonical class K_{𝔚_{C}} of 𝔚_{C} generates the Picard group. But this has been done in 2.1.8.

Remark 4. If C is a double conic (the non-reduced case, see [7, 2.1.2]) then the space 𝔚_{C} is the socalled “Japanese” threefold V_{22} which had been overlooked by Iskovskih and was found by S. Mukai and H. Umemura in [10].

Remark 5. Recalling that the space 𝔚 is the space of rational cubic curves in ℙ(W) defined by a determinantal net of quadrics and the fact that the fiber of K over the point P ∈ 𝔚 is just the net N_{P} ⊂ S^{2}W^{∗} corresponding to P imply that 𝔚_{C} is nothing else then the subspace in 𝔚 of cubics P with the net of quadrics N_{P} orthogonal to the net N ⟶^{A_{C}} S^{2}W (with respect to natural pairing S^{2}W ⊗ S^{2}W^{∗} → ℂ) (compare with [8]).

2.1.12.

Proof of Lemma 2.1.5. Recall that

S^{+} = \frac{W^{∗}/η ⊗ W/U}{O} ≅ \frac{W^{∗}/η ⊗ U^{⊥}}{Λ^{2}U} ⊗ Λ^{2}U^{∗}

and let

ν̂_{C} : (β × γ)^{∗}(W^{∗}/η ⊗ U^{⊥}) ⟶ N^{∗} ⊗ O_{C}, ν̂_{C}^{∗} : N ⊗ O_{C} ⟶ (β × γ)^{∗}(η^{⊥} ⊗ W/U)

be the maps induced by ν_{C}. It suffices to prove that ν̂_{C}^{∗} is an embedding of vector bundles, that is for any x ∈ C the induced map

N ⟶ β(x)^{⊥} ⊗ W/γ(x) = Hom(W^{∗}/β(x), W/γ(x))

is an embedding. But this map evidently coincides with the map

g_{α(x)} : W^{∗}/β(x) ⟶ W/U, W^{∗} ∋ ψ ↦ A_{C}(α(x), ψ) mod U.

But if for any ψ ∈ W^{∗} we have g_{α(x)}(ψ) = 0 then it means that dim Im A_{C}(α(x)) ≤ 2, that is A_{C}(α(x)) is a quadric of corank 2 which contradicts the regularity of the net A_{C}.

3. The exceptional collection.

3.1. Recall that in [6, 4.1] on the space ℙ_{ℙ(W)}(R) there was defined a 2-dimensional vector bundle E^{−} as a unique non-trivial extension of O(−r) by O(h)

0 ⟶ O(h) ⟶ E^{−} ⟶ O(−r) ⟶ 0.

We will denote by the same letter its restriction on ℙ_{ℙ(W)}(R_{C}). Again, it is an unique extension of O(−r) by O(h).

Proposition. We have

q_{∗}O_{ℙ_{ℙ(W)}(R_{C})} = O_{ℙ_{𝔚_{C}} (E_{|𝔚_{C}})},
q_{∗}E^{−}(2r + 2h) = π_{1}^{∗}E_{|𝔚_{C}}^{∗} ,
q_{∗}π_{0}^{∗}Ω_{ℙ(W)}(r + 3h) = K_{|𝔚_{C}}^{∗} ,
q_{∗}π_{0}^{∗}Ω_{ℙ(W)}^{2}(2r + 6h) = Λ^{2}K_{|𝔚_{C}}^{∗}

For any coherent sheaf G on ℙ_{ℙ(W)}(R_{C}) we have q_{∗}(G(−r − 2h)) = (q_{∗}G)(−e).

Proof. The first is trivial. Restricting the second sequence of [6, 5.6] on M̃_{C} we get an exact sequence

0 ⟶ π̂_{0}^{∗}E^{−}(−r − 3h) ⟶ π̃_{1}^{∗}E_{|𝔚_{C}} ⟶ i_{∗}(L^{+} ⊗ …) ⟶ 0,

where … stans for some sheaf which is the pull back from C. Twisting on OM̃_{C} (3r +5h) = OM̃_{C} (H +d) (see (5)) and pushing forward along π̂_{1} we get

0 ⟶ q_{∗}E^{−}(2r + 2h) ⟶ π̂_{1}_{∗}π̃_{1}^{∗}E_{|𝔚_{C}} (H + d) ⟶ π̂_{1}_{∗}(i_{∗}(L^{+} ⊗ …) ⊗ OM̃_{C} (3r + 5h)) ⟶ 0.

But by [6, 5.10] we have OM̃_{C} (3r + 5h)_{|M̃_{C}} = L^{−} ⊗ …, hence the fourth sheaf in the above sequence is i_{∗}π̂_{1}_{∗}(L^{−} ⊗ L^{+} ⊗ …) = 0. On the other hand, pushing forward exact sequence

0 ⟶ π̃_{1}^{∗}E_{|𝔚_{C}} (H) ⟶ π̃_{1}^{∗}E_{|𝔚_{C}} (H + d) ⟶ i_{∗}i^{∗}π̃_{1}^{∗}E_{|𝔚_{C}} (H + d) ⟶ 0,

and applying [6, 5.10] and evident formula

E_{|𝔚_{C}} (H) = E_{|𝔚_{C}} ⊗ OM̃_{C} (c_{1}(E_{|𝔚_{C}}^{∗})) = E_{|𝔚_{C}}^{∗}

we get the second formula of the Proposition. The third and fourth formulas similarly follow from (3) and from the third sequence of [6, 5.6]. The fifth formula follows from (5) and from the projection formula.

3.2. The following Theorem provides 𝔚_{C} with an exceptional collection.

Theorem 2. The following is full and strong exceptional collection on 𝔚_{C}.

(O_{𝔚_{C}} , K_{|𝔚_{C}}^{∗} , E_{|𝔚_{C}}^{∗} , Λ^{2}K_{|𝔚_{C}}^{∗})  (6)

Proof. Consider the following collection on ℙ_{𝔚_{C}} (E_{|𝔚_{C}})

(O_{𝔚_{C}} (−e), K_{|𝔚_{C}}^{∗} (−e), E_{|𝔚_{C}}^{∗} (−e), Λ^{2}K_{|𝔚_{C}}^{∗} (−e), O_{𝔚_{C}} , K_{|𝔚_{C}}^{∗} , E_{|𝔚_{C}}^{∗} , Λ^{2}K_{|𝔚_{C}}^{∗})  (7)

(we omit π_{1}^{∗} for short). According to [11] the collection (6) is full and exceptional iff the collection (7) is, so we will prove that (7) is full exceptional collection.

To this end we will represent it as the push forward along q of a certain exceptional collection on ℙ_{ℙ(W)}(R_{C}). Since the construction is long we will divide it on steps.

3.2.1. First we chose the following full exceptional collection on ℙ(W)

(O_{ℙ(W)}(−h), Ω_{ℙ(W)}^{2}(2h), Ω_{ℙ(W)}(h), O_{ℙ(W)})

3.2.2. Then we construct an exceptional collection on ℙ_{ℙ(W)}(R_{C}) as in [11]

(O(−h), Ω_{ℙ(W)}^{2}(2h), Ω_{ℙ(W)}(h), O, O(r + h), Ω_{ℙ(W)}^{2}(r + 4h), Ω_{ℙ(W)}(r + 3h), O(r + 2h))

(we omit π_{0}^{∗} for short). Its second part is the first twisted by O(r + 2h) which can be considered as relative O(1). But this collection is not the required one. We yet have to make some mutations.

3.2.3. We translate O(−h) and Ω_{ℙ(W)}^{2}(2h) from left to right. Since K_{ℙ(R_{C})} = 2r + 4h (see 2.1.8) it follows from [1] that we’ll get the following full exceptional collection

(Ω_{ℙ(W)}(h), O, O(r + h), Ω_{ℙ(W)}^{2}(r + 4h), Ω_{ℙ(W)}(r + 3h), O(r + 2h), O(2r + 3h), Ω_{ℙ(W)}^{2}(2r + 6h))

3.2.4. We make the left mutation of O(f + h) through O and of O(2r + 3h) through O(r + 2h). Since

Hom^{i}(O, O(r + h)) = H^{i}(O(r + h)) = H^{i}(ℙ(W), R_{C}^{∗}(h)) = ℂ, for i = 1 and 0, for others

(see [7, 3.2.2]) the result of mutation is but the unique non-trivial extension of O by O(r + h) (resp. of O(r + 2h) by O(2r + 3h)) that is E^{−}(r) (resp. E^{−}(2r + 2h)). Therefore the following collection is full and exceptional

(Ω_{ℙ(W)}(h), E^{−}(r), O, Ω_{ℙ(W)}^{2}(r + 4h), Ω_{ℙ(W)}(r + 3h), E^{−}(2r + 2h), O(r + 2h), Ω_{ℙ(W)}^{2}(2r + 6h))

3.2.5. We make the left mutation of Ω_{ℙ(W)}^{2}(r +4h) through O and of Ω_{ℙ(W)}^{2}(2r +6h) through O(r +2h). Since

Hom^{•}(O(r + 2h), Ω_{ℙ(W)}^{2}(2r + 6h)) = Hom^{•}(O, Ω_{ℙ(W)}^{2}(r + 4h)) = H^{•}(Ω_{ℙ(W)}^{2}(r + 4h)) =

= H^{•}(ℙ(W), R_{C}^{∗} ⊗ Ω_{ℙ(W)}^{2}(4h)) = H^{•}(ℙ(W), R_{C}^{∗} ⊗ T_{ℙ(W)}) = Hom^{•}(R_{C}, T_{ℙ(W)}) = 0.

(see [7, 3.1.6]) the mutation degenerates into the trivial transposition and the following collection is full and exceptional

(Ω_{ℙ(W)}(h), E^{−}(r), Ω_{ℙ(W)}^{2}(r + 4h), O, Ω_{ℙ(W)}(r + 3h), E^{−}(2r + 2h), Ω_{ℙ(W)}^{2}(2r + 6h), O(r + 2h))

3.2.6. We translate O(r + 2h) from right to left. Similarly to 3.2.3 we get the following full exceptional collection

(O(−r − 2h), Ω_{ℙ(W)}(h), E^{−}(r), Ω_{ℙ(W)}^{2}(r + 4h), O, Ω_{ℙ(W)}(r + 3h), E^{−}(2r + 2h), Ω_{ℙ(W)}^{2}(2r + 6h))

3.2.7. Note, that the first part of the above collection coincides with the second part twisted by O(−r − 2h). Therefore by the Proposition 3.1 its push forward along q is the collection (7). According to [2] the push forward along flop is an equivalence of categories hence the collection (7) (and therefore the collection (6)) is full and exceptional.

3.2.8. It is remained to prove that collection (6) is strong. But this evidently follows from [1].

4. Rational curves on 𝔚_{C}

4.1. First we will describe a special class of rational curves in ℙ_{ℙ(W)}(R_{C}).

4.1.1. Let M := H^{0}(ℙ(W), R_{C}(2)).

Lemma. We have H^{>0}(ℙ(W), R_{C}(2)) = 0, dim M = 8. For any line L ⊂ ℙ(W) the restriction map H^{0}(ℙ(W), R_{C}(2)) ⟶ H^{0}(L, R_{C}(2)_{|L}) is surjective.

Proof. Consider the following resolvent of the sheaf O_{L} on ℙ(W)

0 ⟶ L ⊗ O_{ℙ(W)}(−1) ⟶ T_{ℙ(W)}(−2) ⟶ O_{ℙ(W)} ⟶ O_{L} ⟶ 0.

The spectral cohomology sequence of this resolvent tensored by R_{C}(2) with application of [7] ??? gives

0 ⟶ L ⟶ H^{0}(ℙ(W), R_{C}(2)) ⟶ H^{0}(L, R_{C}(2)_{|L}) ⟶ 0, H^{>0}(ℙ(W), R_{C}(2)) = H^{>0}(L, R_{C}(2)_{|L})

and Lemma follows because according to [7, 3.3.2] R_{C} has no d-jumping lines with d > 2.

4.1.2. Definition. For any rational curve X ⊂ ℙ(W) (resp. X ⊂ 𝔚_{C}) let X̃ denote the exceptional section of the Hirzebruch surface π_{0}^{−1}(X) = ℙ_{X}(R_{C}_{|X}) (resp. π_{1}^{−1}(X) = ℙ_{X}(E_{|X})).

Definition. Let Σ^{−} ⊂ ℙ_{ℙ(W)}(R_{C}) be the surface spanned by L̃ while L runs through the curve γ(C) of 2-jumping lines of R_{C}.

Note that H^{0}(ℙ_{ℙ(W)}(R_{C}), O(r + 2h)) = H^{0}(ℙ(W), R_{C}(2)) = M.

Lemma. The complete linear system |r + 2h| on ℙ_{ℙ(W)}(R_{C}) is base-point free. The map

ξ^{−} : ℙ_{ℙ(W)}(R_{C}) ⟶ ℙ(M^{∗})

given by this linear system is a contraction of Σ^{−} ⊂ ℙ_{ℙ(W)}(R_{C}) on C.

Proof. Since R_{C} has no d-jumping lines with d > 2 the restriction R_{C}(1)_{|L} is generated by its global sections for any non-2-jumping line L. Hense the restriction of |r + 2h| on π_{0}^{−1}(L) is ample if L is not 2-jumping.

On the other hand, if L is 2-jumping, then π_{0}^{−1}(L) = ℙ_{L}(O_{L}(−2) ⊕ O_{L}(2)) hence the restriction ξ_{|π_{0}^{−1}(L)}^{−} is a contraction of the exceptional section L̃.

4.1.3. Note that from the above Lemma it follows that for any 2-jumping lines L_{1} ≠ L_{2} we have L̃_{1} ∩ L̃_{2} = ∅, hence Σ^{−} is a smooth ruled surface over the curve C.

For any x ∈ C let l_{x}^{−} = p^{−}^{−1}(x) ⊂ ℙ_{C}(S_{C}^{−}) ⊂ ℙ_{ℙ(W)}(R_{C}).

Lemma. We have Σ^{−} = ℙ_{C}(S_{C}^{−}) ⊂ ℙ_{ℙ(W)}(R_{C}) as schemes over C.

Proof. From [6, 5.10] it follows that the map ξ^{−} contracts ℙ_{C}(S_{C}^{−}) ⊂ ℙ_{ℙ(W)}(R_{C}) on C, hence it is remained to note that by [6, 5.4] the map

γ′ : C → Gr(2, W) x ↦ π_{0}(l_{x}^{−})

coincides with γ.

Remark 6. Since r + 2h = e the Remark 2 implies that these divisor classes are but pull-backs from Y along p^{−} and p^{+} respectively. Therefore the map ξ^{−} factors as ξ^{−} = ξ · p^{−}, where ξ : Y → ℙ(M^{∗}) = ℙ^{7} is a closed embedding (compare with Remark 3).

4.1.4. Corollary. a) The complete linear system |e| on ℙ_{𝔚_{C}} (E_{|𝔚_{C}}) is base-point free. The map ξ^{+} given by |e| factors as ξ · p^{+}.

b) The bundle E_{|𝔚_{C}}^{∗} on 𝔚_{C} is generated by its global sections H^{0}(𝔚_{C}, E_{|𝔚_{C}}^{∗}) ≅ M.

4.2. Lines in 𝔚_{C}. For any x ∈ C let l_{x}^{+} = p^{+}^{−1}(x) ⊂ ℙ_{C}(S_{C}^{+}) ⊂ ℙ_{𝔚_{C}} (E_{|𝔚_{C}}).

4.2.1. Recall that a curve l ⊂ 𝔚_{C} is called line (resp. conic, cubic) if K_{𝔚_{C}} · l = H · l = 1 (resp. 2, 3). Now we will prove the second part of the Main Theorem.

Proposition. For all x ∈ C the curve π_{1}(l_{x}^{+}) is a line in 𝔚_{C}. If l ⊂ 𝔚_{C} is a line then l̃= l_{X}^{+} for some x ∈ C.

Proof. The first part immidiately follows from [6, 5.10].

Given a line l ⊂ 𝔚_{C} we have E_{|l}^{∗} = O_{l} ⊕O_{l}(1) (because E_{|𝔚_{C}}^{∗} is generated by its global sections), hence e · l̃= 0, that is l̃ is contracted by p^{+}, hence l̃= l_{x}^{+} for some x ∈ C.

4.3. Intersection of lines.

4.3.1. Definition. A rational cubic X ⊂ ℙ(W) (possibly reducible) is called superjumping for R_{C} if we have H^{1}(X, R_{C}(1)_{|X}) ≠ 0. The usual resolvent

0 ⟶ O_{ℙ(W)}(−3)^{⊕2} ⟶ O_{ℙ(W)}(−2)^{⊕3} ⟶ O_{ℙ(W)} ⟶ O_{X} ⟶ 0

tensored by R_{C}(1) and [7, 3.2.2] show that cubic X is superjumping iff the net of quadrics in ℙ(W) intersecting on X is ortogonal (with respect to natural pairing S^{2}W ⊗ S^{2}W^{∗} → ℂ) to the net A_{C} : N ⟶ S^{2}W defining instanton R_{C}.

4.3.2. Lemma. For any point P ∈ 𝔚_{C} the curve X = π̃_{0}(π̃_{1}^{−1}(P)) is a superjumping cubic (possibly reducible).

Proof. Follows from the Remark 5.

4.3.3. Assume that C is smooth.

Lemma. If a superjumpig cubic X ⊂ ℙ(W) is a union X = L_{1} ∪L∪L_{2} of three lines such that L_{1} ∩L ≠ ∅ and L_{2} ∩ L ≠ ∅ then L_{1} = γ(x_{1}), L_{2} = γ(x_{2}) for some x_{1}, x_{2} ∈ C and (x_{1}, x_{2}) ∈ T_{θ}.

Proof. Let φ_{1} ∈ W^{∗} (resp. φ_{2}) be an equation of the hyperplane H_{1} (resp. H_{2}) containing L_{1} ∪ L (resp. L_{2} ∪L). Let ψ_{1} ∈ W^{∗} (resp. ψ_{2}) be an equation of a hyperplane H′_{1} ≠ H_{1} (resp. H′_{2} ≠ H_{2}) such that L_{1} = H_{1} ∩H′_{1} (resp. L_{2} = H_{2} ∩H′_{2}). Then the net of quadrics intersecting on X is ⟨φ_{1}ψ_{2}, φ_{1}φ_{2}, ψ_{1}φ_{2}⟩ ⊂ S^{2}W^{∗}. The orthogonality with A_{C} implies that φ_{1} = β(x_{1}), φ_{2} = β(x_{2}) for some x_{1}, x_{2} ∈ C and γ(x_{1}) = L_{2}, γ(x_{2}) = L_{1}. Since L_{1} ⊂ H_{1} it follows from [7, 3.3.3] that (x_{1}, x_{2}) ∈ T_{θ}.

Remark 7. If C is a double conic (the non-reduced case, see [7, 2.1.2 and Remark 8]) then the same arguments show that no superjumping cubics for R_{C} are union of three lines.

4.3.4. Now we will prove the last part of the Main Theorem. Assume that C is smooth.

Proposition. Let x_{1}, x_{2} ∈ C. Lines l_{1} = π_{1}(l_{x_{1}}^{+}) and l_{2} = π_{1}(l_{x_{2}}^{+}) in 𝔚_{C} meet iff (x_{1}, x_{2}) ∈ T_{θ}.

Proof. Assume that l_{1} ∩ l_{2} = P ∈ 𝔚_{C}. Then π̃_{0}(π̃_{1}^{−1}(P)) = π_{0}(l_{x_{1}}^{−} ∪ q^{∗}π_{1}^{−1}(P) ∪ l_{x_{2}}^{−}) is a union of three lines and the Proposition follows from.

Remark 8. If C is a double conic (i.e. 𝔚_{C} is the Japanese threefold) then the Remark 7 implies that distinct lines in 𝔚_{C} do not meet.

4.4. Rational cubics in 𝔚_{C}.

Proposition. Let P ∈ ℙ(W) be a point not lying on a 2-jumping line of R_{C} (i.e. P ∈ ℙ(W) \ π_{0}(Σ^{−})). Then π̃_{1}(π̃_{0}^{−1}(P)) is an irreducible rational cubic curve in 𝔚_{C}. Any irreducible rational cubic curve in 𝔚_{C} can be obtained thus.

Proof. The first follows from (5) and 4.1.3.

Let X be an irreducible rational cubic in 𝔚_{C}. Then either E_{|X}^{∗} = O_{X} ⊕O_{X}(3) or E_{|X}^{∗} = O_{X}(1)⊕O_{X}(2). The first is impossible because then X̃ were contracted by p^{+} which is impossible by the Proposition 4.2.1. Therefore e · X̃ = 1. Since X̃ ⊄ ℙ_{C}(S_{C}^{+}) we have d · X̃ ≥ 0, hence by (5) h · X̃ ≤ 0. This means that h · X̃ = 0 and d · X̃ = 0, hence X̃ = π̃_{0}^{−1}(P) for some point P ∈ ℙ(W) not lying on a 2-jumping line of R_{C}.

The Proposition shows that ℙ(W) is a natural compactification of the space Rat_{3}(𝔚_{C}) of irreducible rational twisted cubics in 𝔚.

4.5. Conics in 𝔚_{C}. Assume that C is smooth.

4.5.1. Lemma. Let X ⊂ 𝔚_{C} be an irreducible conic and D_{X} = x_{1} + ⋯ + x_{r} is the divisor of points in C, corresponding to the lines in 𝔚_{C} meeting X. Then D_{X} ∈ |K_{C}|.

Proof. To prove the Lemma we degenerate X into the union l_{1} ∪ l_{2} of intersecting lines. By the Proposition 4.3.4 we have l_{i} = π_{1}(l_{y_{i}}^{+}) for some y_{1}, y_{2} ∈ C such that (y_{1}, y_{2}) ∈ T_{θ}. This means that T_{θ}(y_{1}) = y_{2} + x_{1} + x_{2}, T_{θ}(y_{2}) = y_{1} + x_{3} + x_{4}. Therefore

D_{X} = x_{1} + x_{2} + x_{3} + x_{4} ∼ T_{θ}(y_{1}) − y_{2} + T_{θ}(y_{2}) − y_{1} ∼ (y_{1} + θ − y_{2}) + (y_{2} + θ − y_{1}) ∼ 2θ ∼ K_{C}.

4.5.2. Proposition. Let P ∈ ℙ(W) be a point lying on exactly one 2-jumping line L = γ(x) of R_{C} (i.e. P is a smooth point of the surface π_{0}(Σ^{−})). Then π̃_{1}(π̃_{0}^{−1}(P)) = X ∪ π_{1}(l_{x}^{+}) is the union of an irreducible conic X ⊂ 𝔚_{C} and of a line π_{1}(l_{x}^{+}) meeting X. This defines a 4-fold covering ζ = π̃_{1} · π̃_{0}^{−1} : π_{0}(Σ^{−}) \ sing(π_{0}(Σ^{−})) → Rat_{2}(𝔚_{C}) of the space Rat_{2}(𝔚_{C}) of irreducible conics on 𝔚_{C} by π_{0}(Σ^{−}) \ sing(π_{0}(Σ^{−})).

Proof. The first is easy consequence of (5) and 4.1.3.

To prove the second we consider an arbitrary conic X ⊂ 𝔚_{C}. The same arguments as in proof of the Proposition 4.4 show that E_{|X}^{∗} = O_{X}(1)⊕O_{X}(1), hence π_{1}^{−1}(X) ≅ X ×ℙ^{1}. On the other hand it is known that X meets exactly four lines (see 4.5.1) on 𝔚_{C} hence the surface Σ^{+} intersects π_{1}^{−1}(X) in four points. Let X̃_{1}, X̃_{2}, X̃_{3}, X̃_{4} be the horizontal sections of π_{1}^{−1}(X) passing through one of these four points. Then the relations (5) show that h · X̃_{i} = 0 and d · X̃_{i} = 1, hence X̃_{i} = π̃_{0}^{−1}(P_{i}) for some P_{i} ∈ L_{i} ⊂ ℙ(W), where L_{i} = l_{x_{i}}^{−} is a certain 2-jumping line of R_{C}.

4.5.3. Let Σ̃ ⊂ Gr(2, N) × C = {(ν, x) | ν ∋ α(x)} be the incidence subspace. We have a natural map

σ : Σ̃ ⟶ π_{0}(Σ^{−}) (ν, x) ↦ A_{C}(ν, β(x)) ∈ γ(x) ⊂ ℙ(W).

It is easy to see that this map is a desingularization of π_{0}(Σ^{−}).

Proposition. Let Rat_{2}(𝔚_{C}) be the space of conics on 𝔚_{C}. There is an embedding κ : Rat_{2}(𝔚_{C}) ⊂ Gr(2, N) such that the composition

Σ̃ ⟶^{σ} π_{0}(Σ^{−}) ⟶^{ζ} Rat_{2}(𝔚_{C}) ⟶^{κ} Gr(2, N)  (8)

is induced by the projection of p_{1} : Gr(2, N) × C → Gr(2, N) on the first factor.

Proof. The Lemma 4.5.1 implies that for any X ∈ Rat_{2}(𝔚_{C}) we have p_{2}(ζ^{−1}(X)) ∈ |K_{C}|, where p_{2} : Gr(2, N) × C → C is the projection. Therefore there is defined a map κ : Rat_{2}(𝔚_{C}) → ℙ(N^{∗}) = Gr(2, N) such that α^{−1}(κ(X) ∩ α(C)) = p_{2}(ζ^{−1}(X)), hence the composition (8) is equal to p_{1} (see definition of Σ̃). It is remained to show that κ is embedding. But this is evident because both ζ and p_{1} are 4-fold coverings.

The Propositions 4.5.2 and 4.5.3 show that Gr(2, N) = ℙ(N^{∗}) = |K_{C}| is the natural compactification of the space Rat_{2}(𝔚_{C}) of irreducible conics in 𝔚_{C}.

Remark 9. If C is a double conic (i.e. 𝔚_{C} is the Japanese threefold) then slightly modified arguments of 4.5.2 and 4.5.3 show that in this case ℙ(N^{∗}) is also can be considered as the natural compactification of the space Rat_{2}(𝔚_{C}).

Remark 10. One can apply the same method to description of the rational curves of higher degree. E.g. one can show that the space Gr(2, W) is the natural compactification of the space Rat_{4}(𝔚_{C}) of irredusible rational curves of degree 4 in 𝔚_{C}.

References

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