In this section, we study Fano manifolds of coindex 3. Because we already dealt with the case where n := dimX ≤ 5 in Theorem 5.1.2, we study the case where n≥6.
Proposition 5.6.1([KK04]). Let X be a projective variety and H a proper dominating family of rational curves such that none of the associated curves has a cuspidal singularity.
(i) For general x∈X, all curves inH passing through x are smooth at x and no two of them share a common tangent direction at x.
(ii) Assume that for general x ∈ X and any irreducible component H ′ ⊂ H, dimHx′ ≥ dim2X−1 holds. Then Hx is irreducible. In particular, H is irreducible.
Lemma 5.6.2([BS95, Corollary 1.4.3]). Let C be an integral curve andL a spanned line bundle of degree 1 on C. Then (C, L)∼= (P1,OP1(1)).
Notation 5.6.3. We denote by (d1)∩ · · · ∩(dk) ⊂ Pn a smooth complete intersection of hypersurfaces of degrees d1, . . . , dk, in particular, by (d)k if d = d1 = · · · = dk. We denote by G(k, n) a Grassmannian of k-planes in
Cn, by LG(k, n) a Lagrangian Grassmannian which is the variety of isotropic k-planes for a non-degenerate skew-symmetric bilinear form on Cn, by Sk the spinor variety which is an irreducible component of the Fano variety of k-planes in Q2k.
Theorem 5.6.4. Let X be a Fano manifold of coindex 3 with Pic(X) ∼= Z[OX(1)] and K a minimal rational component of X. Assume that n :=
dimX ≥ 6. Then (lK, dK) = (2,1) except the case X is a Lagrangian Grassmannian LG(3,6). In the case X = LG(3,6) we have (lK, dK) = (3,1).
Proof. We have an inequalityn+ 1 ≥p+ 2 = (n−2)dK. It follows from our assumption n≥6 that (p, dK) = (n−4,1).
By Iskovskikh Theorem [Isk80] or Mukai’s classification result of Fano manifolds of coindex 3 [Muk89, Mel99],X is
(i) a prime Fano manifold, which meansOX(1) is very ample,
(ii) a double coverπ :X →Pn with a branch divisor B ∈ |OPn(6)|, or (iii) a double coverπ :X →Qn with a branch divisor B ∈ |OQn(4)|. Claim 5.6.5. For a general pointx∈X, the variety of minimal rational tan-gentCx ⊂P∗(TxX)is an equidimensional disjoint union of smooth projective varieties.
When X is prime, this follows from Proposition 5.3.1. So we assume X is as in (ii) or (iii). We denote by Y the target of π which is Pn or Qn. By Proposition 5.2.1 it is sufficient to show that the tangent map τx :Kx→Cx
is isomorphic.
SinceOX(1) is spanned anddK = 1, Lemma 5.6.2 implies that anylinK is isomorphic toP1. Furthermore Proposition 5.6.1 implies thatτxis bijective.
For [l]∈Kx we haveOX(1).l= 1. Thereforeπ(l)⊂Y is a standard line and πl : l → π(l) is an isomorphism. According to the generality of x ∈ X, we may assume thatl is free and the natural morphism between normal bundles Nl/X →Nπ(l)/Y is generically surjective. Since Nl/X is semipositive,l ⊂X is a standard rational curve. Hence by Proposition 5.3.1, τx is an immersion.
As a consequence, we see τx is an embedding. So our claim holds.
Sincen ≥6, Proposition 5.3.1 implies thatCx ⊂P∗(TxX) is non-degenerate.
Whenn ≥7, we seeCx is irreducible. In fact, if there are distinct irreducible components Cx,1,Cx,2 of Cx, we see dimCx,1 + dimCx,2 −dimP∗(TxX) ≥0.
This implies that Cx,1 ∩Cx,2 ̸= ϕ. This contradicts the above claim. Ac-cording to Zak’s theorem on linear normality [Zak93] and Theorem 5.4.5, we
have lK = 2. So it remains to prove the case n = 6. If there exists an ir-reducible component ofCx whose secant variety coincides withP∗(TxX), we havelK = 2. Therefore we assume that the secant variety of any irreducible component ofCx does not coincide withP∗(TxX). IfCx is irreducible, then it is the Veronese surfacev2(P2)⊂P5. This follows from Zak’s classification of Severi varieties [Zak93]. Here remark that the Veronese surface is the variety of minimal rational tangents of the Lagrangian Grassmannian LG(3,6) at a general point (see [HM02, Eli02, LM03]). Thus in this case X is isomorphic to LG(3,6) by Theorem 5.2.4. Because the secant variety of the Veronese surface is a hypersurface, it implies that d2 = 4. Therefore we have lK = 3.
IfCx is reducible, there exists disjoint irreducible componentsV1and V2. Re-mark that we assumed thatS1Vi does not coincide withP∗(TxX) fori= 1,2.
If dimS(V1, V2)≤4, we have a point q∈P5\S(V1, V2)∪S1V1∪S1V2. For a projectionπqfrom a pointq,πq(Vi)⊂P4 is a surface. Hence it turns out that πq(V1)∩πq(V2)⊂P4 is non-empty. This contradictsq ∈S(V1, V2). Therefore we have S(V1, V2) = P∗(TxX). In particular, S1Cx =P∗(TxX) andlK = 2.
Here we remark a relation between 2-connectedness by lines and conic-connectedness.
Definition 5.6.6 ([KS02, IR07]). For a projective manifold X ⊂ PN, we call X conic-connected if there exists an irreducible conic passing through two general points on X.
Lemma 5.6.7 (cf. [IR07]). Let X ⊂ PN be a projective manifold which is covered by lines. Then
(i) if two general points on X are connected by two lines, X is conic-connected;
(ii) if X is conic-connected, then the Fano index iX is at least n+12 .
(iii) Assume that X is conic-connected. Then two general points on X are not connected by two lines if and only if iX = n+12 .
Proof. (i) is well-known to the experts (see [KMM92I], [Deb01, Proof of Proposition 5.8]). Suppose that two general pointsx1, x2 ∈X are connected by two lines l1, l2. Then, without loss of generality, we may assume such two lines are free. By the gluing lemma, there exists a smoothing (π : C → (T,0), F :C →X, s1) of l1∪l2 ⊂X fixing x1, where s1 :T →C is a section of π such that s1(0) = x1 ∈ π−1(0) ∼= l1 ∪ l2 and F ◦s1(T) = {x1} (see [Kol96, Chapter II.7]). According to a suitable base change, we may assume
that there exists a section s2 of π such that s2(0) = x2 ∈ π−1(0) ∼= l1∪l2. LetZ ⊂X×X be the set of points (y1, y2)∈X×X which can be joined by an irreducible conic in X. Then for a point t̸= 0 in T, (s1(t), s2(t))∈ Z. It turns out that (x1, x2) is contained in the closure of Z. By the generality of (x1, x2)∈ X×X, we see Z is dense in X ×X. Consequently our assertion holds.
(ii) is in [IR07]. If X is conic-connected, then there exists a smooth conic C such thatTX|C is ample. This implies that 2iX = degTX|C ≥n+ 1. Hence (ii) holds.
(iii) Suppose that Xis conic-connected and it is not 2-connected by lines.
Then for general two points x, y ∈ X there exists a smooth conic f : P1 ∼= C ⊂X passing through x and y such that TX|C is ample. This implies that H1(P1, f∗TX(−2)) = 0. Hence there is no obstruction in the deformation of f fixing the marked points x, y. It turns out that
dim[f]Hom(P1, X :f(0) =x, f(∞) = y) = 2iX −n. (5.1) If 2iX −n ≥ 2, Mori’s Bend and Break implies C degenerates into a union of two lines containing x and y. This is a contradiction. Hence 2iX−n ≤1.
By combining (ii), we have iX = n+12 . Conversely if the Fano index satisfies iX = n+12 , it turns out from the same argument as in Theorem 5.4.5 (i) that X is not 2-connected by lines.
Example 5.6.8. Let S4 ⊂P15 be the 10-dimensional spinor variety and let X be S4 or its linear section of dimensionn ≥6. ThenX is a Fano manifold of coindex 3 with the genusg := H2n+ 1 = 7, where H is the ample generator of the Picard group of S4. There exists a 6-dimensional smooth quadric passing through two general points on S4 [ES89]. So X is conic-connected and 2-connected by lines. Hence two geneal points on X can be connected by a chain of two lines which is obtained as a degeneration of a conic.
Example 5.6.9. LetXbe a GrassmaniannG(2,6)⊂P14or its linear section of dimension n ≥ 6. Then X is a Fano manifold of coindex 3 with the genus g = 8. For two distinct points x, y ∈ G(2,6), they correspond to 2-dimensional vector subspacesLx, Ly ⊂C6. Then there exists a 4-dimensional vector subspace V ⊂ C6 which contains the join < Lx, Ly >. This implies that x, y is contained in a 4-dimensional quadricQ4 ∼=G(2,4)⊂G(2,6). So X is conic-connected and 2-connected by lines.
Remark 5.6.10. X :=G(2,6)∩(1)3 ⊂P14is a 5-dimensional Fano manifold of index 3. According to Theorem 5.5.8,X is 3-connected by lines. However X is conic-connected. This example shows that our chain of minimal rational curves connecting two general points is not necessary a chain with minimal total degree.
Theorem 5.6.11. Let X be a prime Fano n-fold with iX = 23n and K a minimal rational component of X. Then lK = 2 except the following cases:
(i) (3)⊂P4 a hypersurface of degree 3.
(ii) (2)∩(2)⊂P5 a complete intersection of two hyperquadrics.
(iii) G(2,5)∩(1)3 ⊂P6 a 3-dimensional linear section of G(2,5).
(iv) LG(3,6) a Lagrangian Grassmannian.
(v) G(3,6) a Grassmannian.
(vi) S5 a spinor variety.
(vii) E7(ω1) a rational homogeneous manifold of type E7. Furthermore in the cases (i)−(vii) we have lK = 3.
Proof. According to the assumption that 3iX = 2n, n is 3, 6, or at least 9.
If n = 3, X is a del Pezzo 3-fold. So Remark 5.4.7 implies that (lK, dK) = (3,1). Hence X is isomorphic to one of the manifolds listed in (i),(ii) or (iii) by the Fujita-Iskovskikh’s classification result [Fuj80I, Fuj80II, Fuj81I]. In the case where n= 6, we have lK = 2 or X is LG(3,6) by Theorem 5.6.4.
From here, we make the assumption n ≥ 9. In this case, we have 2iX > n + 1. So dK = 1, that is, X is covered by lines. By Proposi-tion 5.3.1 the variety of minimal raProposi-tional tangentsCx ⊂P∗(TxX) is smooth irreducible and non-degenerate. It follows from our assumption 2p≥ n−1.
Hence Cx ⊂ P∗(TxX) is a non-degenerate irreducible projective manifold of dimension 23n−2. By Zak’s theorem on linear normality, a classification of Severi varieties [Zak93] and the assumption that n ≥ 9, S1Cx = P∗(TxX) or Cx ⊂ P∗(TxX) is isomorphic to the Segre product P2 ×P2 ⊂ P8, the Grassmann varietyG(2,6)⊂P14 or E6-variety E6(ω1)⊂ P26. In the former case Theorem 5.4.5 implies that lK = 2. So we assume the latter holds.
Remark that the above Segre variety, Grassmann variety and E6-variety are varieties of minimal rational tangents of G(3,6), S5 and E7(ω1) respectively (For example, see [HM02, Eli02, LM03]). By Theorem 5.2.4,X is isomorphic to one of these varieties. In this case, sinceCx ⊂P∗(TxX) is a Severi variety, S1Cx ⊂P∗(TxX) is a hypersurface [Zak93]. This implies d2 =n−1 [HK05, Theorem 3.14]. Hence lK = 3.
Corollary 5.6.12. Let X be a prime Fano n-fold of Picard number 1 with iX = 23n and K a minimal rational component of X. Assume that n ≥ 6.
Then the following are equivalent.
(i) lK ̸= 2.
(ii) lK = 3.
(iii) X ⊂P(H0(X,OX(1))) is not conic-connected.
(iv) The dimension of the secant variety S1X ⊂P(H0(X,OX(1))) is2n+ 1.
(v) The variety of minimal rational tangents Cx ⊂ P∗(TxX) at a general point is a Severi variety.
(vi) X ⊂P(H0(X,OX(1)))is projectively equivalent to one of the manifolds listed in Theorem 5.6.11 (iv)−(vii).
Proof. By the above theorem and its proof, (i),(ii),(v) and (vi) are equivalent to each other. In general, if X ⊂PN is conic-connected, then the dimension of the secant variety S1X is smaller than 2n+ 1 (see [IR08, Proposition 3.2]
and [Rus09, Theorem 2.1]). Hence (iv)⇒(iii) holds. (iii)⇒(i) follows from Lemma 5.6.7. Finally, (vi)⇒(iv) comes from [Kon88].
Remark 5.6.13. Corollary 5.6.12 and Theorem 5.1.1 implies that iX = 23n is also a boundary of conic-connectedness and defectiveness of the secant variety (c.f. Remark 2.3.3):
Property iX > 23n iX = 23n iX = 23n
lK 2 2 3
Conic-connectedness Yes Yes No
Defectiveness of the secant variety Yes Yes No
5.7 Lengths of Fano manifolds admitting the structures of double covers
Let X be a Fano n-fold with Pic(X) ∼= Z[OX(1)], where OX(1) is ample and n := dimX ≥ 3. In this section, we assume that X is a double cover of a projective manifold π : X → Y. Barth-type Theorem [Laz80] implies Pic(X)∼= Pic(Y) andπ∗OY(1)∼=OX(1), whereOY(1) is the ample generator of the Picard group of Y. It follows from the ramification formula of the branched cover that Y is a Fano manifold. We denote by B ∈ |OY(b)| the branch divisor of π and by R1 the family of rational curves of degree 1 RatCurvesn1(X). We assume that R1 is a dominating family. Then we can define the k-th locus lockR1(x) and the length with respect to R1 as in Definition 5.4.1 and Definition 5.4.2.
Proposition 5.7.1. LetX andR1 be as in above. Then the following holds.
(i) For general x1, x2 ∈ X, π(loc1R
1(x1))∩π(loc1R
1(x2))̸= ϕ if and only if X is 2-connected by R1.
(ii) Under the assumptionOY(1) is spanned, X is2-connected byR1 if and only if for general points y1, y2 ∈ Y there exists curves l1 ∋ y1, l2 ∋ y2 on Y such that l1 ∩l2 ̸=ϕ, OY(1).li = 1 and lengthq(B∩li)≡ 0 mod 2 for any q ∈Y and i= 1,2.
Proof. (i) The ”only if” part is trivial. We show the converse. Let x1, x2
be points on X which are not on the ramification locus of π and we set y2 := π(x2). Then we have π−1(y2) = {x2, x2′}. We assume there exists a point z ∈ π(loc1R1(x1))∩π(loc1R1(x2)). Then there exists a curve [lxi] ∈ R1
such that xi ∈lxi and z ∈π(lxi) for i= 1,2. Since π(lx2) ⊂Y is a curve of degree 1,π−1(π(lx2)) is a curve of degree 2. It follows from the inclusionlxi ⊂ π−1(π(lxi)) that there exists a curve [lx2′] ∈ R1,x2′ such that π−1(π(lx2)) = lx2∪lx2′. Our assumption implies thatlx1∩lx2 ̸=ϕorlx1∩lx2′ ̸=ϕ. Sox2orx2′ is contained in loc2R1(x1). This meansπ|loc2R1(x1): loc2R1(x1)→Y is dominant.
Since π|loc2R1(x1) is proper, it is surjective. Hence we see X = loc2R1(x1).
(ii) Suppose that OY(1) is spanned. Let l be a rational curve on Y satisfying OY(1).l = 1. π−1(l) is denoted by C. From (i), it is sufficient to show the following claim.
Claim 5.7.2. C is reducible if and only iflengthq(B∩l)≡0 mod 2 for any q∈Y.
For the double cover π:X →Y, we have π∗OX ∼=OY ⊕L−1, whereL is an ample line bundle onY which satisfies L⊗2 ∼=OY(B). Furthermore there exists a morphism X ,→ L := Spec(SymL−1) over Y. Since X is a divisor on L, we can obtain the defining equation of X on L. In particular, we see that there exists a global section s ∈ Γ(C, πC∗Ll) such that s2 = πC∗ϕ, where ϕ ∈ Γ(P1,OP1(b)) and (ϕ = 0) = l ∩B as divisors of l. We may assume that πC is unramified at ∞ ∈P1. Then we see C is reducible if and only if π−C1(A1) is reducible. Without loss of generality, we may assume that ϕ= (x−a1y)· · ·(x−aby), where ai ∈C and Γ(P1,OP1(b))∼=⊕b
i=0Cxiyb−i. Furthermore we may assume πC−1(A1) = (s2 = (x−a1)· · ·(x−ab)) ⊂ A2. Thus C is reducible if and only if the cardinality #{j|aj = ai} ≡ 0 mod 2 for any i. Hence we obtain our assertion.
Corollary 5.7.3. Let X, Y and R1 be as in above. If Y =Pn and n ≥ b, then X is 2-connected by R1.
Proof. There exists a standard rational curve f :P1 →X such thatf∗TX ∼= O(2)⊕O(1)p⊕On−1−p. By the ramification formula, the Fano index iX of X is equal to n+ 1−2b. It follows from the assumptionn ≥b thatiX > n+12 . Hence we have deg f∗(P1) = 1 and p = n− b2 −1. For general two points x1, x2 ∈X,
dimπ(loc1R1(x1)) + dimπ(loc1R1(x2))−dimPn= 2(n− b
2)−n =n−b≥0.
(5.2) Hence Proposition 5.7.1 implies that X is 2-connected by R1.
Corollary 5.7.4. Let X, Y and R1 be as in above. If Y ⊂ Pn+1 is a hypersurface of degree d and n ≥2d+b−1, then X is 2-connected by R1. Proof. By the same argument as in Proposition 5.7.3, we see that there exists a standard rational curve f : P1 → X such that degf∗(P1) = 1 and p = n− 2b −d. For general two points x1, x2 ∈X,
dimπ(loc1R1(x1)) + dimπ(loc1R1(x2))−dimPn+1 ≥0. (5.3) ThusX is 2-connected by R1.
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