Varieties of minimal rational tangents on double covers of projective space
aa r X i v : . [ m a t h . AG ] M a r VARIETIES OF MINIMAL RATIONAL TANGENTS ON DOUBLECOVERS OF PROJECTIVE SPACE
JUN-MUK HWANG, HOSUNG KIM
Abstract.
Let φ : X → P n be a double cover branched along a smooth hypersurfaceof degree 2 m, ≤ m ≤ n −
1. We study the varieties of minimal rational tangents C x ⊂ P T x ( X ) at a general point x of X . We describe the homogeneous ideal of C x andshow that the projective isomorphism type of C x varies in a maximal way as x varies overgeneral points of X . Our description of the ideal of C x implies a certain rigidity propertyof the covering morphism φ . As an application of this rigidity, we show that any finitemorphism between such double covers with m = n − X . Keywords. double covers of projective space, Fano manifolds, varieties of minimal ratio-nal tangents
AMS Classification.
Introduction
Throughout the paper, we will work over the field of complex numbers. Let X be a Fanomanifold of Picard number 1. For a general point x ∈ X , a rational curve through x is calleda minimal rational curve if its degree with respect to K − X is minimal among all rationalcurves through x . Denote by K x the space of minimal rational curves through x . Theprojective subvariety C x ⊂ P T x ( X ) defined as the union of tangent directions to membersof K x is called the variety of minimal rational tangents (VMRT) at x . The projectivegeometry of C x plays a key role in understanding the geometry of X , often leading to acertain rigidity phenomenon (cf. the survey [3]) on X . This motivates the study of thegeometry of C x ⊂ P T x ( X ) for various examples of X .In the current article, we study the case when X is a double cover φ : X → P n , n ≥ , of projective space P n branched along a smooth hypersurface Y ⊂ P n of degree 2 m, ≤ m ≤ n −
1. Although this is one of the basic examples of Fano manifolds, its VMRT C x ⊂ P T x ( X ) has not been described explicitly. Our first result is the following descriptionof the defining equations of the VMRT. Theorem 1.1.
For a double cover X → P n , n ≥ , branched along a smooth hypersurfaceof degree m, ≤ m ≤ n − , the VMRT C x ⊂ P T x ( X ) at a general point x ∈ X is asmooth complete intersection of multi-degree ( m + 1 , m + 2 , . . . , m ) . It is enlightening to compare Theorem 1.1 with the case when X is a smooth hypersurfaceof degree m, ≤ m ≤ n , in P n +1 . In the latter case, it is classical that the VMRT at ageneral point is a smooth complete intersection of multi-degree (2 , , . . . , m ) (e.g. Example1.4.2 in [3] or [9]).In the course of proving Theorem 1.1, we will also prove the following partial converseto it. Both authors are supported by National Researcher Program 2010-0020413 of NRF and MEST.
Theorem 1.2.
Let Z ⊂ P n − , n ≥ , be a general complete intersection of multi-degree ( m + 1 , m + 2 , . . . , m ) with ≤ m ≤ n − . Then there exists a smooth hypersurface Y ⊂ P n of degree m such that a double cover X of P n branched along Y has a point x ∈ X with its VMRT C x ⊂ P T x ( X ) isomorphic to Z ⊂ P n − . Theorem 1.1 and Theorem 1.2 are proved by explicit computation for a certain choiceof Y , based on the fact that minimal rational curves of X correspond to lines of P n whichhave even contact order with Y ⊂ P n as recalled in Proposition 2.4. Then Theorem 1.1 forarbitrary smooth Y can be obtained by a flatness argument.Our explicit computation enables us to study also the variation of the VMRT C x as x varies over X . Describing the variation of VMRT is not an easy problem even for verysimple Fano manifolds, such as hypersurfaces in P n +1 . In [9], Landsberg and Robles provedthat when X is a general hypersurface of degree ≤ n in P n +1 , the VMRT at general pointsof X have maximal variation. We will prove the following analogue of their result in oursetting. Theorem 1.3.
Let Y ⊂ P n , n ≥ , be a general hypersurface of degree m, ≤ m ≤ n − ,and let X be a double cover of P n branched along Y . Then the family of VMRT’s {C x ⊂ P T x ( X ) | general x ∈ X } has maximal variation. More precisely, for a general point x ∈ X , choose a trivializationof P T ( U ) ∼ = P n − × U in a neighborhood U of x . Define a morphism ζ : U → Hilb( P n − ) by ζ ( y ) := [ C y ] for y ∈ U . Then the rank of dζ x is n and the intersection of the image of ζ and the GL ( n, C ) -orbit of ζ ( x ) is isolated at ζ ( x ) . The condition n ≥ n, m ) = (3 , C x often leads to a certain rigidity result.What rigidity phenomenon does our description of C x exhibit? The double covering mor-phism φ : X → P n sends members of K x to lines. We show that this property characterizes φ in the following strong sense. Theorem 1.4.
Let Y ⊂ P n , n ≥ , be a smooth hypersurface of degree m, ≤ m ≤ n − ,and let φ : X → P n be a double cover branched along Y . Let U ⊂ X be a neighborhood (inclassical topology) of a general point x ∈ X and ϕ : U → P n be a biholomorphic immersionsuch that for any member C of K y , y ∈ U , the image ϕ ( C ∩ U ) is contained in a line in P n . Then there exists a projective transformation ψ : P n → P n such that ϕ = ψ ◦ ( φ | U ) . As a consequence, we obtain the following algebraic version.
Corollary 1.5.
In the setting of Theorem 1.4, let ˆ X be an n -dimensional projective va-riety equipped with generically finite surjective morphisms g : ˆ X → X and h : ˆ X → P n such that for a minimal rational curve C through a general point of X , there exists anirreducible component C ′ of g − ( C ) whose image h ( C ′ ) ⊂ P n is a line. Then there existsan automorphism ψ : P n → P n such that h = ψ ◦ φ ◦ g . This is a remarkable property of the double cover φ : X → P n , because an analogousstatement fails drastically for many examples of Fano manifolds of Picard number 1 as thefollowing example shows. Example 1.6.
Let X ⊂ P N be a Fano manifold embedded in projective space of dimension N > n = dim X such that lines of P N lying on X cover X , i.e, minimal rational curves on MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 3 X are lines of P N lying on X . There are many such examples, e.g., rational homogeneousspaces under a minimal embedding or complete intersections of low degree in P N . We candefine a finite projection φ : X → P n to a linear subspace P n ⊂ P N by choosing a suitablelinear subspace P N − n − ⊂ ( P N \ X ) . Then minimal rational curves are sent to lines in P n by φ . There are many different ways to choose such P N − n − and projections. For mostexamples of X , different choices of φ need not be related by projective transformations of P n .What makes the difference between Corollary 1.5 and Example 1.6? The key point isthat the ideal defining the VMRT of the double cover, as described in Theorem 1.1, doesnot contain a quadratic polynomial. In fact, we will prove a general version, Theorem 5.4,of Theorem 1.4 where the double cover X is replaced by any Fano manifold whose VMRTat a general point is not contained in a hyperquadric. In this regard, we should mentionthat our double cover φ : X → P n is the first known example of a Fano manifold withPicard number 1 whose VMRT at a general point is not contained in a hyperquadric. Notethat the VMRT’s of Fano manifolds in Example 1.6 are contained in hyperquadrics comingfrom the second fundamental form of X ⊂ P N .Theorem 1.4 has an application in the study of morphisms between double covers. Therehave been several works, e.g., [1], [2], [6], [7] and [11], classifying finite morphisms betweenFano threefolds of Picard number 1. But there still remain a few unsettled cases. Onesuch case is finite morphisms between double covers of P branched along smooth quarticsurfaces. When the quartic surfaces do not contain lines, Theorem 1.5 (more precisely,in the subsection (4.2.2)) of [11] proves that such a morphism must be an isomorphism.However, the approach of [11] has technical difficulties when the quartic surfaces containlines. Using Theorem 1.4, we can settle this case. More precisely, we obtain the following. Theorem 1.7.
Let Y , Y ⊂ P n , n ≥ , be two smooth hypersurfaces of degree n − . Let φ : X → P n (resp. φ : X → P n ) be a double cover of P n branched along Y (resp. Y ).Suppose there exists a finite morphism f : X → X . Then f is an isomorphism. Another application of Theorem 1.4 is the following problem, which is Problem 7.9 in[4].
Problem 1.8 (Liouville-type extension problem) . Let X be a Fano manifold of Picardnumber 1. Let U and U be two connected open subsets (in classical topology) in X .Suppose that we are given a biholomorphic map γ : U → U such that for any minimalrational curve C ⊂ X , there exists another minimal rational curve C ′ with γ ( U ∩ C ) = U ∩ C ′ . Then does there exist Γ ∈ Aut( X ) with Γ | U = γ ?Problem 1.8 is called Liouville-type extension, because Liouville’s theorem in conformalgeometry gives an affirmative answer to Problem 1.8 when X is a smooth quadric hyper-surface in P n +1 . An affirmative answer to Problem 1.8 is known if dim K x > x ∈ X (this is essentially proved in [5]). However, when dim K x = 0, affirmative answersare known only in a small number of examples of X , such as hypersurfaces of degree n in P n +1 and Mukai-Umemura threefolds (see Section 7 of [4]).Theorem 1.4 enables us to give a stronger form of Liouville-type extension for our doublecover X as follows. Theorem 1.9.
Let Y , Y ⊂ P n , n ≥ , be two smooth hypersurfaces of degree m, ≤ m ≤ n − . Let φ : X → P n (resp. φ : X → P n ) be a double cover of P n branched along Y (resp. Y ). Let U ⊂ X and U ⊂ X be two connected open subsets. Suppose that we are JUN-MUK HWANG, HOSUNG KIM given a biholomorphic map γ : U → U such that for any minimal rational curve C ⊂ X ,there exists a minimal rational curve C ⊂ X with γ ( U ∩ C ) = U ∩ C . Then we canfind a biregular morphism Γ : X → X with Γ | U = γ . The organization of this paper is as follows. In Section 2, we will present some basicfacts concerning double covers of P n and their minimal rational curves. Theorem 1.1 andTheorem 1.2 will be proved in Section 3. In Section 4, the variation of VMRT is studiedand Theorem 1.3 will be proved. Finally, in Section 5, we review the notion of projectiveconnections to prove a general version of Theorem 1.4 and explain how Theorem 1.7 andTheorem 1.9 can be derived from Theorem 1.4.2. Minimal rational curves and ECO lines
Throughout, we will fix integers n ≥ ≤ m ≤ n −
1. Let Y ⊂ P n be a smoothhypersurface of degree 2 m . Let φ : X → P n be a double cover of P n ramified along Y . Such a double cover arises as a submanifold in the line bundle O P n ( m ) as explained inpp.242-244 of [10]. This implies the following uniqueness result, where the slightly awkwardappearance of the open subsets U and U are for our later use in Section 5. Lemma 2.1.
Given a smooth hypersurface Y ⊂ P n of degree m , let φ : X → P n and φ : X → P n be two choices of double covers of P n branched along Y . Let U ⊂ X and U ⊂ X be two connected open subsets (in classical topology) with φ ( U ) = φ ( U ) . Thenthere exists a biregular morphism Γ : X → X with Γ( U ) = U and φ = φ ◦ Γ . Definition 2.2.
Let φ : X → P n be a double cover branched along a smooth hypersurface Y ⊂ P n of degree 2 m . A rational curve C ⊂ X with φ ( C ) Y is a minimal rational curve if it has degree 1 with respect to φ ∗ O P n (1). For x ∈ X \ φ − ( Y ), we denote by K x the(normalized) space of minimal rational curves through x . It is known (e.g. II.3.11.5 in [8])that K x is a union of finitely many nonsingular projective varieties. From the adjunctionformula K X = φ ∗ ( K P n + 12 [ Y ]) = φ ∗ O P n ( − n + m − ,X is a Fano manifold of index n − m + 1 and dim K x = n − m − Definition 2.3.
Let Y ⊂ P n be an irreducible reduced hypersurface. A line ℓ ⊂ P n is an ECO (Even Contact Order) line with respect to Y if ℓ Y and the local intersectionnumber at each point of ℓ ∩ Y is even. For a point x ∈ P n \ Y , identify the space of linesthrough x with the projective space P T x ( P n ) and denote by E Yx ⊂ P T x ( P n ) the space ofECO lines through x with respect to Y .Next proposition is a direct generalization of well-known facts for ( n, m ) = (3 ,
2) (e.g.[12]).
Proposition 2.4.
In the setting of Definition 2.2, an irreducible reduced curve C ⊂ X with φ ( C ) Y is a minimal rational curve if and only if the image curve φ ( C ) ⊂ P n is an ECO line with respect to Y . Moreover, a minimal rational curve C is smooth and φ | C : C → φ ( C ) is an isomorphism.Proof. Let C ⊂ X be an irreducible curve such that ℓ := φ ( C ) is an ECO line with respectto Y . Suppose that φ C : C → ℓ is not birational, i.e., C = φ − ( φ ( C )). For a point z ∈ φ ( C ) ∩ Y , let t be a local uniformizing parameter on ℓ at z and let r z be the localintersection number of ℓ and Y at z . Then C is analytically defined by the equation s = t r z (cf. [10] pp.242-244). Let ˜ C be the normalization of C . Since r z is even for any choice of MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 5 z ∈ φ ( C ) ∩ Y , the composition of the normalization morphism ˜ C → C and the coveringmorphism φ | C : C → ℓ induces a morphism ˜ C → ℓ of degree 2 without ramification point, acontradiction. Thus φ | C : C → ℓ is birational and C has degree 1 with respect to φ ∗ O P n (1).Conversely, if C is a minimal rational curve, then ℓ := φ ( C ) is a line in P n with ℓ Y and φ | C : C → ℓ must be birational. Thus φ − ( ℓ ) has an irreducible component C ′ differentfrom C with φ ( C ∩ C ′ ) = ℓ ∩ Y . By the same argument as before, if the local intersectionnumber r z at z ∈ ℓ ∩ Y is odd, the germ of φ − ( ℓ ) over z , defined by s = t r z , is irreducible,a contradiction. Thus r z is even for all z ∈ ℓ ∩ Y and ℓ is an ECO line. Moreover, C mustbe smooth and the morphism φ | C : C → ℓ is an isomorphism. (cid:3) We have the following consequence.
Proposition 2.5.
In the setting of Proposition 2.4, let Y ′ ⊂ P n be an irreducible reducedhypersurface distinct from Y . Then a general ECO line with respect to Y intersects Y ′ transversally. In particular, a general ECO line with respect to Y cannot be an ECO linewith respect to Y ′ .Proof. On a Fano manifold X , for any subset Z ⊂ X of codimension ≥ D ⊂ X , a general minimal rational curve is disjoint from Z (e.g. Lemma2.1 in [3]) and intersects D transversally (the proof is similar to the proof of Lemma 2.1in [3]). Putting Z = φ − ( Y ∩ Y ′ ) and D = φ − ( Y ′ ) for our φ : X → P n branched along Y , we see that a general minimal rational curve C intersects φ − ( Y ′ ) transversally and φ ( C ) ∩ Y ∩ Y ′ = ∅ . Thus φ ( C ) intersects Y ′ transversally. (cid:3) Proposition 2.6.
In the setting of Proposition 2.4, let x be a general point of X . Let τ x : K x → P T x ( X ) be the tangent morphism associating each member of K x its tangentdirection at x . Then τ x is an embedding and the VMRT C x = Im( τ x ) ⊂ P T x ( X ) is anonsingular projective variety with finitely many components of dimension n − m − ,isomorphic to E φ ( x ) ⊂ P T φ ( x ) ( P n ) .Proof. The differential dφ x : P T x ( X ) → P T φ ( x ) ( P n ) sends C x ⊂ P T x ( X ) isomorphically to E Yφ ( x ) ⊂ P T φ ( x ) ( P n ) by Proposition 2.4. It follows that τ x is injective because lines on P n are determined by their tangent directions.Since we know that K x is nonsingular of dimension n − m −
1, to prove that τ x is anembedding, it remains to show that τ x is an immersion. By Proposition 1.4 in [3], this isequivalent to showing that for any member C ⊂ X of K x , the normal bundle N C/X satisfies N C/X = O P (1) n − m − ⊕ O m P . By the generality of x , we can write N C/X = O P ( a ) ⊕ · · · ⊕ O P ( a n − )for integers a ≥ · · · ≥ a n − ≥ P i a i = n − m −
1. Since φ is unramified atgeneral points of C and φ | C : C → ℓ := φ ( C ) is an isomorphism, we have an injective sheafhomomorphism φ ∗ : N C/X → N ℓ/ P n = O (1) n − . Thus a ≤
1. It follows that a = · · · = a n − m − = 1 and a n − m = · · · = a n − = 0. (cid:3) Defining equations of VMRT
Definition 3.1.
A polynomial A ( t , . . . , t m ) in m variables is said to be weighted homoge-neous of weighted degree k if it is of the form A ( t , ..., t m ) = X · i + ··· + m · i m = k c i ,...,i m t i · · · t i m m JUN-MUK HWANG, HOSUNG KIM with coefficients c i ,...,i m ∈ C . An equivalent way of defining it is as follows. We define the weighted degree of each variable t i by wt( t i ) := i and each monomial by wt( t i · · · t t N ) := P Nj =1 wt( t i j ). Then A is weighted homogeneous of weighted degree k if all monomial termsin A have weighted degree k . Definition 3.2.
A polynomial of degree 2 m, m ≥ , in one variable with complex coeffi-cients is an ECO polynomial if it can be written as the square of a polynomial of degree m . Proposition 3.3.
For any positive integer m , there exists a weighted homogeneous polyno-mial A k ( t , . . . , t m ) of weighted degree k for each k, m +1 ≤ k ≤ m , such that a polynomialin one variable λ of degree ma m λ m + a m − λ m − + · · · + a λ + 1 is an ECO polynomial if and only if a k = A k ( a , . . . , a m ) for each m + 1 ≤ k ≤ m. Remark 3.4.
Our proof below gives a recursive formula for A k , but an explicit expressionof the polynomials A k will not be needed in this paper. Proof.
Suppose that a m λ m + a m − λ m − + · · · + a λ + 1is an ECO polynomial. We can find ( σ , . . . , σ m ) ∈ C m such that a m λ m + a m − λ m − + · · · + a λ + 1 = ( σ m λ m + σ m − λ m − + · · · + σ λ + 1) . For convenience, define σ = 1 , σ m +1 = · · · = σ m = 0 , so that we can write, for each k, ≤ k ≤ m , a k = k X i =0 σ i σ k − i . Using a k = k X i =0 σ i σ k − i = k − X i =1 σ i σ k − i + 2 σ k , we have σ k = a k − P k − i =1 σ i σ k − i k = 1 , , ..., m . Thus σ = a , σ = a − σ a − a , · · · . Using induction on k , we see that σ k = G k ( a , . . . , a m ) for each k, ≤ k ≤ m, where G k ( t , . . . , t m ) is a weighted homogeneous polynomial of weighted degree k . Setting G = 1 , G m +1 = · · · = G m = 0, we see that a k = k X ℓ =0 G ℓ ( a , . . . , a m ) G k − ℓ ( a , . . . , a m )for all m + 1 ≤ k ≤ m . Define A k ( t , . . . , t m ) := k X ℓ =0 G ℓ ( t , . . . , t m ) G k − ℓ ( t , . . . , t m ) . MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 7
Then A k is a weighted homogeneous polynomial of weighted degree k such that a k = A k ( a , . . . , a m ) for each m + 1 ≤ k ≤ m .Conversely, given any ( a , . . . , a m ) ∈ C m , let a m + i = A m + i ( a , . . . , a m ) for each 1 ≤ i ≤ m where A m + i is defined above. Then for σ i = G j ( a , . . . , a m ), we see that( σ m λ m + · · · + σ λ + 1) = a m λ m + a m − λ m − + · · · + a λ + 1and a m λ m + a m − λ m − + · · · + a λ + 1is an ECO polynomial. (cid:3) Corollary 3.5.
Regard the affine space A m := { ( a m , a m − , . . . , a ) | a i ∈ C } as the set of polynomials a m λ m + a m − λ m − + · · · + a λ + 1 of degree m with the constant term 1. Then the set D ⊂ A m of ECO-polynomials isa smooth complete intersection of m divisors D , . . . , D m where D j is the smooth divisordefined by a m + j = A m + j ( a , . . . , a m ) where A m + j is the weighted homogeneous polynomialof weighted degree m + j defined in Proposition 3.3. Using Corollary 3.5, we will study the space of ECO lines defined in Definition 2.3. Forour computation, we introduce the following notation.
Notation 3.6.
Choose a homogeneous coordinate system t ,..., t n on P n . Let P n − ∞ ⊂ P n bethe hyperplane defined by t = 0. The restriction of t , . . . , t n on P n − ∞ will be denoted by z , . . . , z n . They provide a homogeneous coordinate system on P n − ∞ . Define the projectiveisomorphism υ y : P n − ∞ → P T y ( P n ) at each point y = [1 : y : · · · : y n ] ∈ P n \ P n − ∞ bysending [ z : · · · : z n ] ∈ P n − ∞ to the tangent direction of the line { ( y + λz , . . . , y n + λz n ) | λ ∈ C } at the point y . The collection { υ − y | y ∈ P n \ P n − ∞ } determines a canonical trivializationof the projectivized tangent bundle υ − : P T ( P n \ P n − ∞ ) ∼ = ( P n \ P n − ∞ ) × P n − ∞ . Definition 3.7.
For a homogeneous polynomial f ( t , . . . , t n ) of degree 2 m , 2 ≤ m ≤ n − k, ≤ k ≤ m, define a fk ( y ; z ) = a fk ( y , . . . , y n ; z , . . . , z n ) to be thepolynomial in 2 n variables satisfying f (1 , y + λz , . . . , y n + λz n ) = a f ( y ; z ) + a f ( y ; z ) λ + · · · + a f m ( y ; z ) λ m . Note that for a fixed y , a fk ( y ; z ) is a homogeneous polynomial in z of degree k . In particular, a f ( y ; z ) = f (1 , y , . . . , y n ) is independent of z . Proposition 3.8.
In Notation 3.6 and Definition 3.7, let Y ⊂ P n be the hypersurfacedefined by f ( t , . . . , t n ) = 0 . For any point y ∈ P n \ ( Y ∪ P n − ∞ ) , the variety υ − y ( E Yy ) ⊂ P n − ∞ is(set-theoretically) the common zero set of the homogeneous polynomials in z , B fk ( y ; z ) , m +1 ≤ k ≤ m , defined by B fk ( y ; z ) = B fk ( y , ..., y n ; z , ..., z n ) := a fk ( y ; z ) a f ( y ; z ) − A k a f ( y ; z ) a f ( y ; z ) , ..., a fm ( y ; z ) a f ( y ; z ) ! , JUN-MUK HWANG, HOSUNG KIM where A k is as in Proposition 3.3. Note that B fk ( y ; z ) is homogeneous in z of degree k because a fk ( y ; z ) is homogeneous in z of degree k and A k is weighted homogeneous of weighteddegree k . In particular, if E Yy is of pure dimension n − m − , then it is set-theoretically acomplete intersection of multi-degree ( m + 1 , m + 2 , . . . , m ) .Proof. A point [ z : · · · : z n ] ∈ P n − ∞ belongs to υ − y ( E Yy ) if and only if the polynomial in λf (1 , y + λz , . . . , y n + λz n ) = a f ( y ; z ) + a f ( y ; z ) λ + · · · + a f m ( y ; z ) λ m is an ECO polynomial. By Corollary 3.5, we see that υ − y ( E Yy ) is the common zero set of B fk ( y ; z ) , m + 1 ≤ k ≤ m . (cid:3) Proposition 3.9.
Given a general smooth complete intersection Z ⊂ P n − of multi-degree ( m + 1 , . . . , m ) , there exist a smooth hypersurface Y ⊂ P n of degree m and a point y ∈ P n \ Y , such that E Yy ⊂ P T y ( P n ) is projectively equivalent to Z ⊂ P n − . In particular,for a general hypersurface Y ⊂ P n of degree m and a general y ∈ P n \ Y , the variety ofECO lines E Yy ⊂ P T y ( P n ) is a smooth complete intersection of degree ( m + 1 , ..., m ) .Proof. Denote by { b k ( z , . . . , z n ) | m + 1 ≤ k ≤ m } homogeneous polynomials withdeg b k = k defining Z . By the generality of Z , we may assume that(i) the affine hypersurface1 + b m +1 ( t , . . . , t n ) + · · · + b m ( t , . . . , t n ) = 0in C n = { ( t , . . . , t n ) , t i ∈ C } is smooth and(ii) the projective hypersurface b m ( z , . . . , z n ) = 0in P n − with homogeneous coordinates ( z : z : · · · : z n ) is smooth.Let Y ⊂ P n be the hypersurface of degree 2 m defined by the polynomial f ( t , t , . . . , t n ) := t m + t m − b m +1 ( t , . . . , t n ) + · · · + t b m − ( t , . . . , t n ) + b m ( t , . . . , t n ) . Then Y is a smooth hypersurface because it has no singular point on its intersection withthe hyperplane t = 0 by the assumption (ii), while it has no singular point on the affinespace t = 0 by the assumption (i). In Notation 3.6, consider the point y = [1 : 0 : 0 : · · · :0] ∈ P n \ ( P n − ∞ ∪ Y ). Then f (1 , y + λz , . . . , y n + λz n ) = f (1 , λz , . . . , λz n ) = 1 + λ m +1 b m +1 ( z ) + · · · + λ m b m ( z ) . Comparing with Definition 3.7, we obtain a f ( y ; z ) = 1 , a f ( y ; z ) = · · · = a fm ( y ; z ) = 0 , a fk ( y ; z ) = b k ( z ) for m + 1 ≤ k ≤ m. In the notation of Proposition 3.8, B fk ( y ; z , . . . , z n ) = a fk ( z , . . . , z n ) = b k ( z , . . . , z n )for m + 1 ≤ k ≤ m. This implies that E Yy , y := [1 : 0 : · · · : 0], is projectively equivalent to Z ⊂ P n − . (cid:3) Proof of Theorem 1.1 and Theorem 1.2.
Theorem 1.2 is a direct consequence of Proposi-tion 2.6 and Proposition 3.9.To prove Theorem 1.1, it suffices to show by Proposition 2.6 that for any smooth Y ⊂ P n and a general point x ∈ P n \ Y , the subvariety E Yx ⊂ P T x ( P n ) is a smooth completeintersection of multi-degree ( m + 1 , . . . , m ). Proposition 3.9 says that this is O.K. if Y isa general hypersurface. MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 9
To check it for any smooth Y ⊂ P n , choose a deformation { Y t | | t | < ǫ } of Y = Y such that for a Zariski open subset U t ⊂ P n \ Y t , E Y t x ⊂ P T x ( P n ) is a smooth completeintersection for any t = 0 and any x ∈ U t . By shrinking ǫ if necessary, the intersection ∩ t =0 U t is nonempty. By Proposition 2.6, we have a Zariski open subset U ⊂ P n \ Y suchthat E Yx is smooth for any x ∈ U . Pick a point x ∈ ( ∩ t =0 U t ) ∩ U .We can construct a smooth family { φ t : X t → P n | | t | < ǫ } of double covers of P n branched along Y t ’s. Choose z t ∈ φ − t ( x ) in a continuous way. The proof (e.g. II.3.11.5 in[8]) of the smoothness of K x mentioned in Definition 2.2 works for the family K z t , i.e., thefamily {K z t | | t | < ǫ } is a flat family of nonsingular projective subvarieties. Via Proposition2.6, this implies that {E Y t x | | t | < ǫ } is a flat family of nonsingular projective varieties in P T x ( P n ). By our choice of x , E Y t x is (scheme-theoretically) a smooth complete intersectionfor t = 0, while E Y x is a nonsingular variety which is set-theoretically a complete intersectionof the same multi-degree as E Y t x , t = 0 , by Proposition 3.8. We conclude that E Y x is also asmooth complete intersection of multi-degree ( m + 1 , . . . , m ). (cid:3) Variation of VMRT
Notation 4.1.
Let V k be the vector space of homogeneous polynomials of degree k in z ,..., z n . Each polynomial h ∈ V k is of the form h ( z , ..., z n ) = X i + ··· + i n = k e i ,...,i n z i · · · z i n n . Regarding V k as a complex manifold, take { e i ,..,i n } i + ··· + i n = k as linear coordinates on V k and (cid:26) ∂∂e i ,...,i n (cid:12)(cid:12)(cid:12) i + · · · + i n = k (cid:27) as a basis for the tangent spaces T h ( V k ) of V k at each h ∈ V k . There is a canonicalisomorphism between V k and T h ( V k ) identifying a polynomial X i + ··· + i n = k E i ,...,i n z i · · · z i n n ∈ V k with the tangent vector X i + ··· + i n = k E i ,...,i n ∂∂e i ,...,i n ∈ T h ( V k ) . Notation 4.2.
For a homogeneous polynomial f ( t , . . . , t n ) of degree 2 m, ≤ m ≤ n − f ( t , ..., t n ) = t m f ( t , ..., t n ) + t m − f ( t , ..., t n ) + · · · + f m ( t , ..., t n )where f k ( t , ..., t n ) is a homogeneous polynomial of degree k = 0 , ..., m in t ,..., t n . Com-paring with Definition 3.7, we have f (1 , , ...,
0) = f ( z , ..., z n ) = a f (0; z ) and a fi (0; z ) = f i ( z ) for i = 1 , ..., m. Definition 4.3.
For a homogeneous polynomial f ( t , . . . , t n ) of degree 2 m, ≤ m ≤ n − Y ⊂ P n be the hypersurface defined by f ( t , . . . , t n ) = 0 and define a morphism µ : P n \ ( P n − ∞ ∪ Y ) → V m +1 by sending y = [1 : y : · · · : y n ] to the polynomial in zµ ( y ) := [ B fm +1 ( y ; z )] ∈ V m +1 with B fm +1 ( y ; z ) as in Proposition 3.8. Proposition 4.4.
In Notation 4.2 and Definition 4.3, assume that f (1 , , ...,
0) = f ( z ) = 1 and f ( z ) = · · · = f m ( z ) = 0 . Then for x = [1 : 0 : · · · : 0] ∈ P n \ ( P n − ∞ ∪ Y ) and P ni =1 v i ∂∂y i ∈ T x ( P n ) , dµ x ( n X i =1 v i ∂∂y i ) = [ n X i =1 v i ∂f m +2 ∂t i ( z )] ∈ T µ ( x ) ( V m +1 ) = V m +1 . We will use the following lemma.
Lemma 4.5.
In Notation 4.2, set f m +1 = 0 for convenience. Assume that f (1 , , ...,
0) = f ( z ) = 1 . Then B fk ( y ; z ) of Proposition 3.8 satisfies ∂B fk ( y ; z ) ∂y i (cid:12)(cid:12)(cid:12) (0; z ) = ∂f k +1 ∂t i ( z ) − f k ( z ) ∂f ∂t i ( z ) − m X j =1 ∂A k ∂x j ( f ( z ) , ..., f m ( z )) (cid:18) ∂f j +1 ∂t i ( z ) − f j ( z ) ∂f ∂t i ( z ) (cid:19) for all k = m + 1 , ..., m and i = 1 , ..., n .Proof. In the equality ∂f ( t ) ∂t i (cid:12)(cid:12)(cid:12) t =(1 ,λz ,...,λz n ) = ∂f (1 , y + λz , ..., y n + λz n ) ∂y i (cid:12)(cid:12)(cid:12) y = ··· = y n =0 , the left hand side can be written, via Notation 4.2, ∂f ∂t i ( z , ..., z n ) + ∂f ∂t i ( z , ..., z n ) λ + · · · + ∂f m +1 ∂t i ( z , ..., z n ) λ m . On the other hand, the right hand side is, by Definition 3.7, ∂a f ∂y i (0; z ) + ∂a f ∂y i (0; z ) λ + · · · + ∂a f m ∂y i (0; z ) λ m . Therefore for each i = 1 , . . . , n , ∂a f m ∂y i (0; z ) = 0 and ∂a fk ∂y i (0; z ) = ∂f k +1 ∂t i ( z ) for k = 0 , ..., m. From this and the assumption that a f (0; z ) = f (1 , , ...,
0) = 1, we obtain ∂∂y i a fk ( y ; z ) a f ( y ; z ) !(cid:12)(cid:12)(cid:12) ( y ; z )=(0; z ) = a f (0; z ) ∂a fk ∂y i (0; z ) − a fk (0; z ) ∂a f ∂y i (0; z ) a f (0; z ) = ∂f k +1 ∂t i ( z ) − f k ( z ) ∂f ∂t i ( z ) MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 11 for all k = 0 , ..., m and i = 1 , ..., n . Thus ∂B fk ( y ; z ) ∂y i (cid:12)(cid:12)(cid:12) (0; z ) = ∂∂y i a fk ( y ; z ) a f ( y ; z ) !(cid:12)(cid:12)(cid:12) (0; z ) − m X j =1 ∂A k ∂x j a f (0; z ) a f (0; z ) , ..., a fm (0; z ) a f (0; z ) ! ∂∂y i a fj ( y ; z ) a f ( y ; z ) !(cid:12)(cid:12)(cid:12) (0; z ) = ∂f k +1 ∂t i ( z ) − f k ( z ) ∂f ∂t i ( z ) − m X j =1 ∂A k ∂x j ( f ( z ) , ..., f m ( z )) (cid:18) ∂f j +1 ∂t i ( z ) − f j ( z ) ∂f ∂t i ( z ) (cid:19) for all k = m + 1 , ..., m and i = 1 , ..., n . (cid:3) Proof of Proposition 4.4.
Since A k ( x , ..., x m ) is weighted homogeneous of weighted degree k , if k ≥ m + 1, then the linear part of A k ( x , ..., x m ) does not contain variables x ,..., x m .Therefore for all k = m + 1 , ..., m and j = 1 , ..., m,∂A k ∂x j (cid:12)(cid:12)(cid:12) (0 ,..., = 0 . Thus putting f = · · · = f m = 0 in Lemma 4.5, we obtain ∂B fm +1 ∂y i (0; z ) = ∂f m +2 ∂t i ( z ) . It follows that dµ x ( n X i =1 v i ∂∂y i ) = n X i =1 v i ∂B fm +1 ∂y i (0; z ) = n X i =1 v i ∂f m +2 ∂t i ( z ) . (cid:3) Notation 4.6.
Denote the action of A ∈ GL ( n, C ) on C n by ( z , . . . , z n ) A ( z , . . . , z n ) . We have the natural induced action on V k given by( A.h )( z , ..., z n ) := h ( A − ( z , ..., z n )) , h ∈ V k . Denote the orbit of h ∈ V k by GL ( n, C ) .h := { A.h (cid:12)(cid:12) A ∈ GL ( n, C ) } ⊂ V k . Proposition 4.7.
We use the terminology of Notation 4.1 and Notation 4.6. A tangentvector X i + ··· + i n = k E i ,...,i n ∂∂e i ,...,i n ∈ T h ( V k ) is tangent to the orbit GL ( n, C ) .h if and only if there exists an ( n × n ) matrix ( s ij ) i,j =1 ,...,n such that X i + ··· + i n = k E i ,...,i n z i · · · z i n n = ddt h ( z + t n X i =1 s i z i , ..., z n + t n X i =1 s in z i ) (cid:12)(cid:12)(cid:12) t =0 . Proof.
Define a morphism α h : GL ( n, C ) → V k sending A to A.h . Then GL ( n, C ) .h is the image of α h and α h ( I ) = h where I is the ( n × n )identity matrix. The tangent space of GL ( n, C ) .h at h is the image of the differential d ( α h ) I : T I ( GL ( n, C )) → T h ( V k ) . Let us identify T I ( GL ( n, C )) with the vector space M n of all ( n × n ) matrices so that A ∈ M n corresponds to the tangent vector at I of the curve c ( t ) = I + tA which is indeed a curve on GL ( n, C ) for sufficiently small t . Since α h ◦ c ( t ) is the polynomial h (( I + tA ) − ( z , . . . , z n )),the differential d ( α h ) I sends A = ddt c ( t ) (cid:12)(cid:12)(cid:12) t =0 to ddt h (( I + tA ) − ( z , . . . , z n )) (cid:12)(cid:12)(cid:12) t =0 which is ofthe form on the right hand side of the equation in the proposition. (cid:3) Proposition 4.8.
In the setting of Proposition 4.4, T µ ( x ) ( GL ( n, C ) .µ ( x )) = ( n X i,j =1 s ij z i ∂f m +1 ∂t j ( z ) (cid:12)(cid:12)(cid:12) s ij ∈ C ) ⊂ T µ ( x ) ( V m +1 ) = V m +1 . Proof.
From µ ( x ) = [ B m +1 (0; z )] ∈ V m +1 and Proposition 4.7, we get T µ ( x ) ( GL ( n, C ) .µ ( x )) = ( ddt B fm +1 (0; z + t n X i =1 s i z i , ..., z n + t n X i =1 s in z i ) (cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12) s ij ∈ C ) . From Notation 4.2, we have a f (0; z ) = f (1 , , ...,
0) = 1 and a fi (0; z ) = f i ( z ) = 0 for i = 1 , ..., m . Thus B fm +1 (0; z ) = a fm +1 (0; z ) a f (0; z ) − A m +1 a f (0; z ) a f (0; z ) , ..., a fm (0; z ) a f (0; z ) ! = f m +1 ( z ) . So the following equalities hold ddt B fm +1 (0; z + t n X i =1 s i z i , ..., z n + t n X i =1 s in z i ) (cid:12)(cid:12)(cid:12) t =0 = ddt f m +1 ( z + t n X i =1 s i z i , ..., z n + t n X i =1 s in z i ) (cid:12)(cid:12)(cid:12) t =0 = n X i,j =1 s ij z i ∂f m +1 ∂t j ( z ) . Putting it in the above expression for T µ ( x ) ( GL ( n, C ) .µ ( x )), we obtain the result. (cid:3) Proposition 4.9.
There exists a smooth hypersurface Y ⊂ P n , n ≥ , defined by a homoge-neous polynomial f of degree m, ≤ m ≤ n − , such that, for a general x ∈ P n \ ( P n − ∞ ∪ Y ) ,using the terminology of Definition 4.3, rank( dµ x ) = n, dim C T µ ( x ) ( GL ( n, C ) .µ ( x )) = n and Im( dµ x ) ∩ T µ ( x ) ( GL ( n, C ) .µ ( x )) = 0 . Proof.
First, consider the case m = 2. Set f ( t , ..., t n ) = t + b ( t + · · · + t n ) t + ( t + · · · + t n ) + X ≤ i
3. Pick f ( t , . . . , t n ) = t m + b ( t m +11 + · · · + t m +1 n ) t m − + ct t t ( t m − + · · · + t m − n ) t m − + t m + · · · + t mn with some constants b, c ∈ C ∗ . Using Notation 4.2, we have f = · · · = f m = f m +3 = · · · = f m − = 0 , f m +1 = b ( t m +11 + · · · + t m +1 n ) ,f m +2 = ct t t ( t m − + · · · + t m − n ) and f m = t m + · · · + t mn . From the smoothness of the Fermat hypersurface in P n defined by t m + t m + · · · + t mn = 0,we can see that the hypersurface Y defined by f = 0 is smooth for general b and c . Set x := [1 : 0 : · · · : 0]. Propositions 4.4 and 4.8 show thatIm( dµ x ) = ( n X i =1 v i ∂f m +2 ∂t i ( z ) (cid:12)(cid:12)(cid:12) v i ∈ C ) = ( ( v z z + v z z + v z z )( z m − + · · · + z m − n ) + n X i =4 v i z z z z m − i (cid:12)(cid:12)(cid:12) v i ∈ C ) and T µ ( x ) ( GL ( n, C ) .µ ( x )) = ( n X i,j =1 s ij z i ∂f m +1 ∂t j ( z ) (cid:12)(cid:12)(cid:12) s ij ∈ C ) = ( n X i,j =1 s ij z i z mj (cid:12)(cid:12)(cid:12) s ij ∈ C ) . The condition m ≥ dµ x ) = n . It is easy to see thatdim C T µ ( x ) ( GL ( n, C ) .µ ( x )) = n and Im( dµ x ) ∩ T µ ( x ) ( GL ( n, C ) .µ ( x )) = 0 . (cid:3) Proof of Theorem 1.3.
By Proposition 2.6, we may prove the corresponding statement forthe morphism η : W → Hilb( P n − ) defined on a neighborhood W of a general pointin P n by η ( y ) := [ E Yy ] for y ∈ W . Since E Yx is a complete intersection of multi-degree( m + 1 , . . . , m ) for general Y and general x ∈ P n \ Y , the equation B m +1 of degree m + 1is uniquely determined up to GL ( n, C )-action by the projective equivalence class of E Yx .Thus it suffices to show that rank( dµ x ) = n and dµ x ( T x ( P n )) ∩ T µ ( x ) ( GL ( n, C ) .µ ( x )) = 0for a general Y and general x . This follows from Proposition 4.9. (cid:3) Projective connections and rigidity of maps
Definition 5.1.
Given a complex manifold M of dimension n , the projectivized tangentbundle π : P T ( M ) → M is equipped with the tautological line bundle ξ ⊂ π ∗ T ( M ) whosefiber at α ∈ P T ( M ) is given by ˆ α ⊂ T π ( α ) ( M ), the 1-dimensional subspace correspondingto α ∈ P T π ( α ) ( M ) . We have the vector subbundle
T ⊂ T ( P T ( M )) of rank n whose fiber at α ∈ P T ( M ) is given by T α := dπ − α ( ˆ α )where dπ α : T α ( P T ( M )) → T π ( α ) ( M ) is the differential of the projection π. A projectiveconnection on M is a homomorphism p : ξ → T of vector bundles which splits the exactsequence of vector bundles on P T ( M )0 −→ T π −→ T −→ T /T π ∼ = ξ −→ T π ⊂ T ( P T ( M )) is the relative tangent bundle of π . Given a projective connection p : ξ → T , the image p ( ξ ) ⊂ T ⊂ T ( P T ( M )) is a line subbundle in the tangent bundle of P ( T ( M )) and defines a foliation of rank 1 on P T ( M ). Example 5.2. On P n , we have a canonical projective connection p : ξ → T such thatthe leaves of the foliation p ( ξ ) are exactly the tangent directions of lines on P n . We callthis the flat projective connection and denote it by p flat . Let U be a connected complexmanifold of dimension n and let ϕ : U → P n be an immersion. Via the biholomorphicmorphism P T ( U ) ∼ = P T ( ϕ ( U )), we have an induced projective connection ϕ ∗ p flat on U .By the affirmative answer to Problem 1.8 when X = P n (see the remark after Problem1.8), two immersions ϕ i : U → P n , i = 1 , , are related by a projective transformation, i.e.,there exists an automorphism ψ : P n → P n such that ϕ = ψ ◦ ϕ , if and only if the twoprojective connections ϕ ∗ p flat and ϕ ∗ p flat coincide. Proposition 5.3.
In the setting of Definition 5.1, let
C ⊂ P T ( M ) be a closed subvari-ety dominant over M such that for a general point x ∈ M , the fiber C x ⊂ P T x ( M ) isnot contained in a quadric hypersurface. Suppose that p , p : ξ → T are two projectiveconnections on M such that p | C = p | C . Then p = p .Proof. Since p and p split the exact sequence in Definition 5.1, the difference p − p determines an element σ ∈ H ( P T ( M ) , T π ⊗ ξ − ) . For a general x ∈ M, σ x is a section of T ( P T x ( M )) ⊗ ξ − on the projective space P T x ( M ). The condition p | C = p | C implies that σ x vanishes on the subvariety C x . In term of a homogeneous coordinate system on projectivespace P n − , a nonzero section of T ( P n − ) ⊗ O P n − (1) is represented by a homogeneouspolynomial vector field with quadratic coefficients. In particular, the zero set of such asection must be contained in some quadric hypersurface. By the assumption that C x isnot contained in a quadric hypersurface, we see that σ x = 0. Since it is true for a general x ∈ M , we obtain p = p . (cid:3) We have the following general version of Theorem 1.4. In fact, Theorem 1.4 is a corollaryof Theorem 5.4 by Theorem 1.1.
MRT ON DOUBLE COVERS OF PROJECTIVE SPACE 15
Theorem 5.4.
Let X be a Fano manifold. For a general point x ∈ X , we denote by K x the space of minimal rational curves through x and by C x ⊂ P T x ( X ) the VMRT at x .Assume that C x is not contained in a quadric hypersurface in P T x ( X ) . Let U ⊂ X be aconnected neighborhood of a general point x ∈ X and ϕ , ϕ : U → P n be two biholomorphicimmersions such that for any y ∈ U and any member C of K y , both ϕ ( C ∩ U ) and ϕ ( C ∩ U ) are contained in lines in P n . Then there exists a projective transformation ψ : P n → P n such that ϕ = ψ ◦ ϕ .Proof. Let
C ⊂ P T ( X ) be the closure of the union of C x ⊂ P T x ( X ) as x varies overgeneral points of X . For a member C of K x and its smooth locus C o ⊂ C , the curve P T ( C o ) ⊂ P T ( X ) lies in C . In fact, by the definition of C such curves cover a dense opensubset in C .Consider the projective connections ϕ ∗ i p flat on U . Let C be a general minimal rationalcurve intersecting U . Since ϕ ( C ∩ U ) and ϕ ( C ∩ U ) are contained in lines in P n , thedifference ϕ ∗ p flat − ϕ ∗ p flat ∈ H ( P T ( U ) , T π ⊗ ξ − ) , in the notation of Definition 5.1 with M = U , vanishes along the Riemann surface P T ( C o ) ∩ P T ( U ). Since such Riemann surfaces cover a dense open subset in C ∩ P T ( U ), the twoprojective connections must agree on C ∩ P T ( U ). Applying Proposition 5.3 with M = U ,we conclude ϕ ∗ p flat = ϕ ∗ p flat . As mentioned in Example 5.2, this implies the existence of aprojective transformation ψ satisfying ϕ = ψ ◦ ϕ . (cid:3) Proof of Theorem 1.7.
Putting m = n − X i , i = 1 , , have trivial normal bundles and rational curves throughgeneral points with trivial normal bundles are minimal rational curves. By Proposition 6of [6] (also cf. Theorem 3.1 (iv) in [11]), for a general minimal rational curve C ⊂ X ,each irreducible component of f − ( C ) is a minimal rational curve in X . In other words, f sends minimal rational curves of X through a general point to those of X . Puttingˆ X = X , X = X , g = f, φ = φ , and h = φ in Corollary 1.5, we see that φ = ψ ◦ φ ◦ f for some projective transformation ψ . Thus f must be birational, and hence an isomorphism. (cid:3) Proof of Theorem 1.9.
Applying Theorem 1.4 to ϕ := φ ◦ γ : U → φ ( U ) ⊂ P n , we havea projective transformation ψ ∈ Aut( P n ) such that ψ ◦ φ | U = φ ◦ γ . By the assumptionon γ and Proposition 2.4, we have dψ ( E Y x ) = E Y ψ ( x ) for x ∈ φ ( U ). By Proposition 2.5,this implies ψ ( Y ) = Y . Thus replacing Y by ψ ( Y ) and φ by ψ ◦ φ , we may assumethat Y = Y and φ ( U ) = φ ( U ). From Lemma 2.1, there exists a biregular morphismΓ : X → X with Γ | U = γ . (cid:3) References [1] E. Amerik, Maps onto certain Fano threefolds, Documenta Math. (1997) 195-211[2] E. Amerik, Some remarks on morphisms between Fano threefolds, Documenta Math. (2004) 471-486[3] J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, in School on vanishing Theoremsand Effective Results in Algebraic Geometry (Trieste, 2000) , ICTP Lect. Notes, vol. 6, Abdus SalamInt. Cent. Theoret. Thys., Trieste, 2001, pp. 335-393.[4] J.-M. Hwang, Geomery of varieties of minimal rational tangents, to appear in
Current Developmentsin Algebraic Geometry , MSRI Publications vol. 59, Cambridge University Press, 2012[5] J.-M. Hwang and N. Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds ofPicard number 1. Journal Math. Pures Appl. (2001) 563-575[6] J.-M. Hwang and N. Mok, Finite morphisms onto Fano manifolds of Picard number 1 which haverational curves with trivial normal bundles. J. Alg. Geom. (2003) 627-651 [7] A. Iliev and C. Schuhmann, Tangent scrolls in prime Fano threefolds. Kodai Math. J. (2000) 411-431[8] J. Koll´ar, Rational curves on algebraic varieties , Ergebnisse der Mathematik und ihrer Grenzgebiete,vol. 32, Springer, 1996[9] J. M. Landsberg and C. Robles, Lines and osculating lines of hypersurfaces. J. Lond. Math. Soc. (2) (2010) 733-746[10] R. Lazarsfeld, Positivity in algebraic geometry I , Springer, 2004[11] C. Schuhmann, Morphisms between Fano threefolds, J. Alg. Geom. (1999) 221-244[12] A. Tihomirov, The geometry of the Fano surface of the double cover of P branched in a quartic,Math. USSR Izv. (1981) 373-397 Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea
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