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SPECIAL BIRATIONAL TRANSFORMATIONS OF TYPE (2,1) 5 1 BAOHUA FU AND JUN-MUK HWANG 0 2 n a J Abstract. A birational transformation Φ : Pn 99K Z ⊂ PN, 9 where Z ⊂ PN is a nonsingular variety of Picard number 1, is 1 calledaspecial birational transformation of type (a,b)ifΦ isgiven ] by a linear system of degree a, its inverse Φ−1 is given by a linear G system of degree b and the base locus S ⊂ Pn of Φ is irreducible A andnonsingular. Inthispaper,weclassifyspecialbirationaltrans- . formations of type (2,1). In addition to previous works [AS] and h [R2] on this topic, our proofemploys natural C∗-actionson Z in a t a crucialway. TheseC∗-actionsalsorelateourresulttotheproblem m studied in [FH]. [ 1 1. Introduction v 0 Recall (e.g. Section 2 in [AS] or Definition 4.1 in [R2]) that a bira- 1 tional transformation 4 4 Φ : Pn 99K Z ⊂ PN 0 1. where Z ⊂ PN is a nonsingular projective variety of Picard number 1 0 is called a special birational transformation of type (a,b) if 5 1 (1) the base locus S ⊂ Pn of Φ is irreducible and nonsingular; : (2) the rational map Φ is given by a linear system belonging to v Xi OPn(a); and (3) the inverse rational map Φ−1 is given by a linear system belong- r a ing to O (b). Z When Z = Pn, this is a special Cremona transformation, a classical topic in projective algebraic geometry. It is a challenging problem to classify special birational transformations. Even for special Cremona transformations, a complete classification is still missing. Special Cre- mona transformations of type (2,2) have been classified by Ein and Shepherd-Barron in [ES] by relating them to Severi varieties classified by Zak ([Z]). In [R2], special Cremona transformations of types (2,3) and (2,5) have been classified. Recently Alzati and Sierra ([AS]) have Baohua Fu is supported by National Scientific Foundation of China (11225106 and 11321101). 1 2 BAOHUAFU AND JUN-MUKHWANG extended [ES] to a classification of special birational transformations of type (2,2) for a wider class of Z. In this paper, we will give a complete classification of special birational transformations of type (2,1). This classification can be de- scribed in terms of the classification of the base locus S ⊂ Pn, which is contained in a hyperplane Pn−1 ⊂ Pn for the type (2,1). Our main result is the following. Theorem 1.1. The base locus Sd ⊂ Pn−1 of a special birational transformations of type (2,1) is projectively equivalent to one of the following: (a) Qd ⊂ Pd+1 for d ≥ 1; (b) P1 ×Pd−1 ⊂ P2d−1 for d ≥ 3; (c) the 6-dimensional Grassmannian Gr(2,5) ⊂ P9; (d) the 10-dimensional Spinor variety S ⊂ P15; 5 (e) a nonsingular codimension ≤ 2 linear section of P1 ×P2 ⊂ P5; (f) a nonsingular codimension ≤ 3 linear section of Gr(2,5) ⊂ P9. The description of the varieties in Theorem 1.1 as well as a more precise formulation of the result will be given in Section 2. Our proof of Theorem 1.1, to be given in Section 6, uses previous works on this topic in [AS] and [R2]. The main strategy is an inductive argument on VMRT (see Proposition 4.6) developed by Russo in [R2]. There are two new ingredients in our approach: the use of natural C∗-actions on Z, which reveals topological relations between the base loci of Φ and Φ−1, and a study of the intersection of entry loci on the base locus of Φ, which exhibits a delicate structure in the projective geometry of the base locus. The part on C∗-action is presented in Section 3 and the part on the intersection of entry loci is presented in Section 5. The use of C∗-actions on Z is motivated by our previous work [FH] on projective manifolds with nonzero prolongations. As a matter of fact, we will see in Section 7 that a prime Fano manifold Z is the target of a special birational transformation of type (2,1) if and only if it has nonzero prolongation. By this correspondence, we can use Theorem 1.1 to give a new proof (see Theorem 7.13) of the main classification result of [FH]. This new proof corrects an error in the classification in [FH], as explained in Remark 7.14. We will work over complex numbers. For simplicity, a nonsingular irreducible variety will be called a manifold. Acknowledgments: We are grateful to José Carlos Sierra for explain- ing to us the examples in Proposition 2.11 and Proposition 2.12 and for SPECIAL BIRATIONAL TRANSFORMATIONS OF TYPE (2,1) 3 a careful reading of the draft. We would like to thank Francesco Russo for pointing out a mistake in a previous version and for numerous sug- gestions. We are grateful to Qifeng Li for discussions on Section 5 and to Fyodor Zak for suggesting the argument in Remark 2.13 (3). 2. Statement of the classification of special quadratic manifolds Definition 2.1. Let U be a vector space and let σ : Sym2U → W be a surjective linear map to a vector space W. Let us denote by W∗ ⊂ Sym2U∗ the annihilator of Ker(σ), which is naturally dual to W. (1) Let ψo : PU 99K PW be the rational map defined by the linear system W∗ ⊂ Sym2U∗ ≃ H0(PU,O(2)). The scheme-theoretic base locus of ψo will be denoted by B(σ) ⊂ PU and the proper image of PU under ψo will be denoted by Y(σ) ⊂ PW. (2) Fix a 1-dimensional vector space T with a fixed identification T = C. Define a rational map φo : P(T ⊕U) 99K P(T ⊕U ⊕W) by [t : u] 7→ [t2 : tu : σ(u,u)] for t ∈ T,u ∈ U. The proper image of P(T ⊕ U) under φo will be denoted by Z(σ) ⊂ P(T ⊕U ⊕W). The scheme-theoretic base locus of φo coincides with B(σ) ⊂ PU = P(0⊕U). We will say that σ is a special system of quadrics if (a) the base locus subscheme B(σ) ⊂ PU isirreducible andnonsingular and(b)theimage Z(σ) of φo is nonsingular. A projective manifold S ⊂ PU is called a special quadratic manifold if S = B(σ) for a special system of quadrics σ : Sym2U → W. In this case, the rational map ψo comes from a morphism ψ : Bl (PU) → Y(σ) ⊂ PW S where Bl (PU) is the blow-up of PU along S. We will denote by F ⊂ S Bl (PU) its exceptional divisor. The rational map φo comes from a S morphism φ : Bl (P(T ⊕U)) → Z(σ) ⊂ P(T ⊕U ⊕W) S where Bl (P(T ⊕ U)) is the blow-up of P(T ⊕ U) along S ⊂ PU = S P(0⊕U). We will denote by E ⊂ Bl (P(T⊕U)) its exceptional divisor. S 4 BAOHUAFU AND JUN-MUKHWANG We have the commuting diagram ψ F ⊂ Bl (PU) −→ Y(σ) ⊂ PW S ∩ ∩ ∩ ∩ φ E ⊂ Bl (P(T ⊕U)) −→ Z(σ) ⊂ P(T ⊕U ⊕W). S ∼ NotethatφsendsU = P(T⊕U)\PU isomorphicallytoZ(σ)\P(U⊕W). Thus Z(σ) is a rational variety and φo : P(T ⊕ U) 99K Z(σ) is a birational map. Itfollowsimmediately fromthedefinitionthatforaspecial quadratic manifold S ⊂ PU, the birational map φo : P(T ⊕ U) 99K Z(σ) is a special birational transformation of type (2,1) in the sense of Section 1. Conversely we have Proposition 2.2. Let Φ : Pn 99K Z ⊂ PN be a special birational transformation of type (2,1) as defined in Section 1. Then the base locus S of Φ is contained in a hyperplane Pn−1 of Pn, the subvariety S ⊂ Pn−1 is a special quadratic manifold and Φ coincides with the birational map φo for a special system of quadrics σ : Sym2U → W. Proof. By Proposition 2.3 (a) [ES], the base locus S ⊂ Pn of Φ is contained in a hyperplane. Since S is defined by quadratic equations, S ⊂ Pn−1 is also defined by quadratic equations. As S is the base locus of Φ, the rational map Φ is given by a linear subspace W∗ ⊂ H0(Pn−1,IS(2)) and the full linear system H0(Pn,IPn−1(2)). Then the map φo associated to W coincides with Φ. (cid:3) Example 2.3. We list some homogeneous examples of special quadratic manifolds in the next table. The data in this table can be found from Theorem 3.8 in Chapter III of [Z]. Note that in all these examples, the dimension a of W is equal to h0(Pn−1,I (2)), namely, we have S W∗ = H0(Pn−1,I (2)). S S Qd P1 ×Pd−1 Gr(2,5) S 5 S ⊂ Pn−1 O(1) Segre Plu¨cker Spinor Y point Gr(2,d) P4 Q8 Y ⊂ Pa−1 identity Plu¨cker identity O(1) Z Qd+2 Gr(2,d+2) S OP2 5 Z ⊂ Pn+a O(1) Plu¨cker Spinor Severi d = dimS d = n−2 d 6 10 n = dimZ n = d+2 2d 10 16 m = dimY 0 2(d−2) 4 8 a 1 d(d−1) 5 10 SPECIAL BIRATIONAL TRANSFORMATIONS OF TYPE (2,1) 5 Example 2.4. Here we give some examples of special birational transformations of type (2,1) with W∗ ( H0(Pn−1,I (2)). Recall from Ex- S ample 2.3 that the special quadratic manifold S = P1 ×Pd−1 ⊂ P2d−1 with d ≥ 6 is associated to the special system of quadrics σ : Sym2U → W = H0(PU,I (2))∗. S The corresponding birational transformation is φ◦ : P(T ⊕U) 99K Z := Gr(2,d+2) ⊂ P(T ⊕U ⊕W) with Y = Gr(2,d) ⊂ PW = P(∧2Cd). Note that Sec(Z) ∩ PW = Sec(Y). Take any linear subspace L ⊂ W such that PL∩Sec(Y) = ∅, then PL∩Sec(Z) = ∅. Let p : P(T ⊕U ⊕W) 99K P(T ⊕U ⊕W/L) L be the projection from PL. Then p sends Z (resp. Y) isomorphically L to a subvariety Z ⊂ P(T ⊕U ⊕W/L) (resp. Y ⊂ P(W/L)). L L The map φ◦ := p ◦φ◦ : P(T ⊕U) 99K Z L L L is a special birational transformation of type (2,1) associated to the special system of quadrics σ : Sym2U → W/L. L with Y(σ ) = Y ⊂ P(W/L) and B(σ ) ⊂ PU being P1×Pd−1 ⊂ P2d−1. L L L To discuss non-homogeneous examples, it is convenient to introduce the following notion. Definition 2.5. Let Z ⊂ PV be a nondegenerate submanifold and let W ⊂ V be a subspace such that PW ⊂ Z. Denote by (V/W)∗ ⊂ V∗ the set of linear functionals on V annihilating W such that P(V/W)∗ parameterizes the set of hyperplanes in PV containing PW. Then a general member of P(V/W)∗ is called a PW-general hyperplane in PV. Moregenerally, alinearsubspace ofcodimension-sinPV isPW-general if it is defined by a general member of Gr(s,(V/W)∗), i.e., it is general among subspaces of codimension-s containing PW. Proposition 2.6. Let S = B(σ) ⊂ PU be a special quadratic manifold defined by σ : Sym2U → W. Assume that dimS ≥ 2, dimPU > dimY(σ) and the intersection of Z(σ) with a PW-general hyperplane of P(T ⊕U ⊕W) is nonsingular. Then for a general subspace U′ ⊂ U of codimension 1, the restriction σ′ : Sym2U′ → W of σ is a special system of quadrics such that 6 BAOHUAFU AND JUN-MUKHWANG (i) the base locus scheme B(σ′) ⊂ PU′ coincides with the hyperplane section S′ = S ∩PU′ ⊂ PU of S = B(σ); (ii) Y(σ′) = Y(σ) ⊂ PW; and (iii) Z(σ′) = Z(σ)∩P(T ⊕U′ ⊕W). Proof. Firstly, we claim that for a general U′ ⊂ U, the hyperplane section Z(σ) ∩ P(T ⊕ U′ ⊕ W) is nonsingular. This is a consequence of the assumption that a PW-general hyperplane section of Z(σ) ⊂ P(T ⊕ U ⊕ W) is nonsingular. To see this, associate to each vector v ∈ U the linear automorphism g of P(T ⊕U ⊕W) defined by v g : [t : u : w] 7→ [t : u+tv : w +2σ(u,v)+tσ(v,v)]. v For a general choice of v ∈ U and U′ ⊂ U, the automorphism g sends v P(T ⊕ U′ ⊕ W) to a PW-general hyperplane of P(T ⊕ U ⊕ W). The g -image of a general point [1 : u : σ(u,u)] ∈ Z(σ) is v g ([1 : u : σ(u,u)]) = [1 : u+v : σ(u,u)+2σ(u,v)+σ(v,v)] v = [1 : u+v : σ(u+v,u+v)] ∈ Z(σ). Thus g preserves Z(σ). Consequently, for a general choice of v, the v automorphism g sends the hyperplane section Z(σ)∩P(T ⊕U′ ⊕W) v to a PW-general hyperplane section of Z(σ). This proves the claim. Now we can choose a general subspace U′ ⊂ U of codimension 1 such that (1) dimPU′ ≥ dimY(σ) and dimU −dimU′ < dimS; (2) the (scheme-theoretic) linear section S′ := S∩PU′ is nonsingular; and (3) the restriction ψo|PU′ is dominant over Y(σ). Then σ′ : Sym2U′ → W is surjective, and S′ = B(σ′) ⊂ PU′ is a special quadratic manifold with Y(σ′) = Y(σ), and Z(σ′) = Z(σ) ∩ P(T ⊕U′ ⊕W). (cid:3) To check the condition in Proposition 2.6, we need the following. Proposition 2.7. In the setting of Definition 2.5, let Z ⊂ PV be a nondegenerate submanifold containing PW. Define Z∗ ⊂ PV∗ by W Z∗ = {[H] ∈ P(V/W)∗ | H ∩Z is singular at a point of PW}. W Assume that dimZ∗ < dimP(V/W)∗. Then for a PW-general hyper- W plane [H] ∈ P(V/W)∗, the intersection Z := Z ∩ H is a nonsingular subvariety containing PW and the submanifold Z ⊂ H satisfies dimZ∗ ≤ dimZ∗ . W W Proof. The intersection of Z with a PW-general H is nonsingular outside PW by Bertini. But we can choose [H] ∈ P(V/W)∗ outside Z∗ W SPECIAL BIRATIONAL TRANSFORMATIONS OF TYPE (2,1) 7 by the dimension condition dimZ∗ < dimP(V/W)∗. Thus Z = Z∩H W is nonsingular and it contains PW. It remains to check dimZ∗ ≤ W dimZ∗ . W By definition, Z∗ = {[L] ∈ P(H/W)∗ | L∩Z is singular at a point of PW}. W Given an element [L] of Z∗ , then L∩Z is singular at some point say W x ∈ PW. Take the hyperplane L ⊂ V to be the linear span of L and T Z, then L = L∩H and L∩Z is singular at x ∈ PW. This shows that x e the image of Z∗ under the projection P(V/W)∗ \ {H} → P(Hˆ/W)∗ We e contains Z∗ , where Hˆ is the hyperplane in V corresponding to H. W This implies that dimZ∗ ≤ dimZ∗ . (cid:3) W W By applying Proposition 2.7 repeatedly, we have the following. Corollary 2.8. In Proposition 2.7, let s be a positive integer satisfying s < dimZ and dimZ∗ ≤ dimP(V/W)∗ −s. W Then a PW-general linear section of Z with codimension s is nonsingular. Let us recall the following results from Proposition 2.19 and Remark 2.20 in Chapter III of [Z]. Proposition 2.9. (i) Let Y = PW ⊂ Z ⊂ PV be P2 ⊂ Gr(2,5) ⊂ P9 from S = P1 ×P2 of Example 2.3. Then Z∗ is isomorphic to a cone W over S. In particular, dimP(V/W)∗−dimZ∗ = 6−4 = 2. W (ii) Let Y = PW ⊂ Z ⊂ PV be P4 ⊂ S ⊂ P15 from S = Gr(2,5) of 5 Example 2.3. Then Z∗ is isomorphic to a cone over S. In particular, W dimP(V/W)∗−dimZ∗ = 10−7 = 3. W Lemma 2.10. Recall that a subvariety S ⊂ Pn−1 is arithmetically Cohen-Macaulay if Hi(Pn−1,I (l)) = 0 for all l and for all 0 < i < S dim(S)+1. Let S ⊂ Pn−1 be either the Segre embedding P1 ×P2 ⊂ P5 or the Plu¨cker embedding Gr(2,5) ⊂ P9. Let S′ ⊂ Pn−s−1 be any nonsingular linear section of S ⊂ Pn−1 with codimension s < dimS. Then S′ ⊂ Pn−s−1 is arithmetically Cohen-Macaulay and H0(Pn−1,I (2)) = S H0(Pn−s−1,IS′(2)). Proof. By [Z] (Chapter III, Theorem 1.2), S ⊂ Pn−1 is arithmetically Cohen-Macaulay. On the other hand, for a nonsingular hyperplane section, we have the exact sequence 0 → I (−1) → I → I → 0. S S S∩H⊂H Usingtheassociatedlongexact sequence, wededucethatS∩H isarith- metically Cohen-Macaulay. Repeating the same argument, we see that 8 BAOHUAFU AND JUN-MUKHWANG any nonsingular linear section of S is arithmetically Cohen-Macaulay. (cid:3) The last claim then follows easily. Now we can give some nonhomogeneous examples of special quadratic manifolds. Proposition 2.11. From Example 2.3, the Segre variety S = P1×P2 ⊂ P5 = PU is a special quadratic manifold associated with σ : Sym2U → W = H0(P5,I (2))∗ S and Z(σ) isomorphic to the Plu¨cker embedding Gr(2,5) ⊂ P9. A nonsingular linear section S′ of S by a general subspace U′ ⊂ U of codimension s ≤ 2 is a special quadratic manifold associated to a special system of quadrics σ′ : Sym2U′ → W = H0(P5,IS(2))∗ = H0(P5−s,IS′(2))∗ and Z(σ′) is equal to a nonsingular linear section of Gr(2,5) with codimension s. Proof. Applying Proposition 2.6 repeatedly in combination with Corol- lary 2.8 and Proposition 2.9, we see that a general linear section S′ of P1 × P2 with codimension s ≤ 2 is a special quadratic manifold associated to a special system of quadrics σ′ : Sym2U′ → W = H0(P5,I (2))∗ andZ(σ′)equal toa P2-generallinear section ofGr(2,5) S with codimension s. It is well-known (e.g. Remark 3.3.2 in [IP]) that all nonsingular sections of S with codimension s ≤ 2 are projectively equivalent and all nonsingular sections of Gr(2,5) with codimension s ≤ 2 are projectively equivalent. Thus we may say that Z(σ′) is any nonsingular linear section of Gr(2,5) of codimension s ≤ 2. Using Lemma 2.10, we see W∗ = H0(P5−s,IS′(2)). (cid:3) Proposition2.12. FromExample2.3, the GrassmannianS = Gr(2,5) ⊂ P9 = PU is a special quadratic manifold associated with σ : Sym2U → W = H0(P9,I (2))∗ S and Z(σ) is isomorphic to the Spinor variety S ⊂ P15. A nonsingular 5 linear section S′ of S by a general subspace U′ ⊂ U of codimension s ≤ 3 is a special quadratic manifold associated to a special system of quadrics σ′ : Sym2U′ → W = H0(P9,IS(2))∗ = H0(P9−s,IS′(2))∗ and Z(σ′) equals to a P4-general linear section of Z(σ) = S with codi- 5 mension s ≤ 3. SPECIAL BIRATIONAL TRANSFORMATIONS OF TYPE (2,1) 9 Proof. AsintheproofofProposition2.11, thisfollowsfromProposition 2.6, Corollary 2.8, Proposition 2.9 and Lemma 2.10, modulo the fact (seeRemark3.3.2in[IP])thatallnonsingularlinearsections ofGr(2,5) of a fixed codimension s ≤ 3 are projectively equivalent. (cid:3) Remark2.13. (1)LetS = S ⊂ PU,dimU = 16,bethe10-dimensional 5 Spinor variety. It is well-known that the dual variety S∗ ⊂ PU∗ is isomorphic to S ⊂ PU and the automorphism group Aut(S) acts tran- sitively on S and PU \S. This implies that all nonsingular hyperplane sections of S are projectively equivalent. 5 (2) All nonsingular linear sections of S with codimension 2 contain a 5 linearP4. Onewaytoseethisisusingthefactthatageneralhyperplane section S of S ⊂ P15 is isomorphic to a horospherical Fano manifold 1 5 of Picard number 1, the case 4 in Theorem 1.7 of [P] (this fact follows from Mukai’s classification [M]). From [P], the automorphism group Aut(S ) has two orbits, an open orbit and a closed orbit, say Q ⊂ 1 S , which is isomorphic to the 6-dimensional hyperquadric Q6. Let 1 π : Bl (S ) → S be the blow-up of S along Q and let E ⊂ Bl (S ) Q 1 1 1 Q 1 be the exceptional divisor. Then by the proof of Lemma 1.17 [P], there exists a morphism q : Bl (S ) → Q5 which is a P4-bundle. The Q 1 fibers of q are mapped to linear P4’s contained in S and any linear 1 P4 in S arises this way. The intersection of P4 in S with Q is a 1 1 3-dimensional hyperquadric Q3 ⊂ P4. Now suppose that a general nonsingular hyperplane section H ⊂ S does not contain any linear P4. 1 Then its proper image H′ ⊂ Bl (S ) is a P3-bundle over Q5 and the Q 1 exceptional divisor E∩H′ is a Q2-bundle over Q5. On the other hand, the exceptional divisor E ∩H′ is a P2-bundle over the blow-up center Q ∩ H, which is isomorphic to Q5. This leads to a contradiction by considering the Euler characteristic of E ∩H′. Thus any nonsingular hyperplane section of S must contain a linear P4. (As a matter of fact, 1 onecanshowthatallnonsingularlinearsectionsofS withcodimension 5 2 are projectively equivalent. But as this is not directly related to our main results, we will not discuss its proof, which is more involved.) (3)Ageneral linear sectionS ofS withcodimension 3doesnotcon- 3 5 tain a linear P4. To see this, use Proposition 2.19 of Chapter III in [Z] which says that the set of linear P4’s on S = S ⊂ PU is parametrized 5 by the dual variety S∗. This gives an embedding S∗ ⊂ Gr(5,U). Let U′ ⊂ U be a general linear subspace of codimension c > 0 and regard Gr(5,U′) as a submanifold of Gr(5,U) in a natural way. Then S∩PU′ contains a linear P4 if and only if Gr(5,U′) ∩ S∗ 6= ∅. By Theorem 10.8 of [Ha], the intersection Gr(5,U′)∩S∗ must be either empty or of 10 BAOHUAFU AND JUN-MUKHWANG dimension dimGr(5,U′)+dimS∗−dimGr(5,U) = (50−5c)+10−50 = 10−5c. Thus the intersection is empty if c > 2. This also shows that, when c = 2, the intersection Gr(5,U′) ∩ S∗, which is nonempty from (2), is finite, i.e., a general linear section S in (2) contains finitely many 2 linear P4’s. The following result from Corollary 3.21 in [AS] is a converse to Proposition 2.11 and Proposition 2.12. Proposition 2.14. A nonsingular linear section of the homogeneous special quadratic manifolds in Example 2.3 is a special quadratic manifold if and only if it is a hyperquadric or one provided by Proposition 2.11 and Proposition 2.12. Our aim is to show that Example 2.3, Example 2.4, Proposition 2.11 and Proposition 2.12 exhaust all special quadratic manifolds. More precisely, we have the following classification, which gives a complete classification of special birational transformations of type (2,1). Theorem 2.15. A special quadratic manifold Sd ⊂ Pn−1 is projectively equivalent to one of the following: (a) Qd ⊂ Pd+1 for d ≥ 1; (b) P1 ×Pd−1 ⊂ P2d−1 for d ≥ 3; (c) the 6-dimensional Grassmannian Gr(2,5) ⊂ P9; (d) the 10-dimensional Spinor variety S ⊂ P15; 5 (e) a nonsingular codimension ≤ 2 linear section of P1 ×P2 ⊂ P5; (f) a nonsingular codimension ≤ 3 linear section of Gr(2,5) ⊂ P9. The birational map ψ◦ is given by H0(Pn−1,I (2)) except for (b) with S d ≥ 6, whereψ◦ can begivenbysomepropersubspacesofH0(Pn−1,I (2)) S as in Example 2.4. The proof of Theorem 2.15 will be given in Section 6. 3. Classification of Y(σ) In this section, we classify varieties that can occur as the ψ-image Y(σ)ofaspecial quadraticmanifoldS ⊂ PU. Tosimplify thenotation, we will write Y for Y(σ) and Z for Z(σ). We will use the notation of Definition 2.1. Proposition 3.1. Let S ⊂ PU be a special quadratic manifold. Then Y is nonsingular.