On Calabi--Yau threefolds associated to a web of quadrics
aa r X i v : . [ m a t h . AG ] J un ON CALABI–YAU THREEFOLDS ASSOCIATED TO A WEB OF QUADRICS
S LAWOMIR CYNK AND S LAWOMIR RAMS
Abstract.
We study the geometry of the birational map between an intersection of a web ofquadrics in P that contains a plane and the double octic branched along the discriminant of theweb. Introduction
It is a classical fact that there is a correspondence between the base locus S of a net of quadricsin P and the double sextic branched along the discriminant of the net. The latter is the modulispace of certain rank-2 sheaves on the former (see [26]). Moreover, if the base locus contains a line L , then the two surfaces are birational. More general conditions for the existence of a birationalmap were given by Nikulin and Madonna (see [22] and its sequels).A precise description of the birational map between the surface S and the double sextic can befound in [7]. In this case, S is the blow-up of the double sextic along rank-4 quadrics in the net.The latter results from the fact that the map defined by the linear system | H − L − P k L i | ,where H is the hyperplane section in P and L i are the lines on S that meet L (see [7, Thm 3.3]), ishyperelliptic. Moreover, one can show that the birational map factors through another K3 surface(a space quartic that contains a twisted cubic) and its geometry (e.g. the contracted curves) isgoverned by the behaviour of the lines L i . The birational map between the two surfaces can be alsoconstructed via an incidence variety ([18]). The latter construction was adopted in [24] to the caseof a generic web W = span( Q , Q , Q , Q ) in O P (2), such that its base locus X contains a fixedplane Π. More precisely, using Bertini-type and computer algebra arguments, Micha lek provedthat if we put S (resp. X ) to denote the discriminant surface of the web W (resp. the doublecover of the web W branched along the discriminant surface S ) and W is generic enough, thenthe Calabi-Yau varieties X and X are birational. However, the approach of [24] gives neitherexplicit sufficient condition for birationality of X and X nor a method to study the geometry ofthe map.In this paper, for the matrices q , . . . , q that give the quadrics Q , . . . , Q ∈ O P (2) such that Q ∩ . . . ∩ Q contains a plane Π we define two auxiliary matrices a , A and use them to obtain asurface B ⊂ P and a three-dimensional quintic X ⊂ P that contains the surface B . Then, underthe assumptions [A1]: X has exactly 10 singularities on Π and is smooth away from the plane Π, [A2]: no 4 singular points of X lie on a line, [A3]: the set { x ∈ B : rank( A ( x )) ≤ } consists of 46 points , [A4]: the discriminant surface S has only isolated singularities,we show that there is a birational map X X that factors as the composition X σ − ˜ X π −→ X ψ − ˜ X φ −→ X , where σ, ψ are certain blow-ups, π is resolution of the projection from Π and ˆ φ is obtained via Steinfactorization from restriction of the so-called Bordiga conic bundle to the blow-up of the quintic Mathematics Subject Classification.
Primary: 14J30; Secondary 14J32, 32S25.Research partially supported by MNiSW grant no. N N201 388834. . In particular, under the above assumptions B is the so-called (smooth) Bordiga sextic.Bordiga sextic and Bordiga conic bundle have been studied already by the Italian school (see [30],[2] and the bibliography in the latter), so the above factorization enables us to give a precisedescription of the geometry of the birational map in question. In particular, we are able to showthat the map has no two-dimensional fibers, describe the contracted curves (Thm 3.6), classify thesingularities of the discriminant of the web (and prove that all of them admit a small resolution)and give an upper bound of their number (see Cor. 4.7).Our considerations yield that the assumptions [A1], . . . ,[A4] are fulfilled by a generic web ofquadrics such that its base locus contains a fixed plane. Careful analysis of our arguments showsthat one can assume less in order to obtain a birational map X X , but once one omitsthe above assumptions the geometry of the birational map changes. For instance, if [A2] is notsatisfied, the surface in P one obtains as a result of the projection is no longer the Bordiga surface,without [A1] (resp. [A3]) the threefold X (resp. X ) has higher singularities etc. Still, the mainstrategy we use can be applied to study those degenerations - we do not follow this path in orderto maintain the paper compact.Our motivation is twofold. First, it seems a natural question to ask under what assumptions athree-dimensional Calabi-Yau analogue of the well-known result on K3 surfaces holds. Second, weobtain a very precise description of a map between certain Calabi-Yau manifolds that (with help ofa computer algebra system applied to a given example) could be of interest on its own, for instanceas a source of examples of small resolutions.The paper is organized as follows. In Sect. 1 we study the singularities of the threefold X and Hodge numbers of its blow-up ˜ X . Sect. 2 is devoted to properties of projection from theplane Π. In the next section we describe the behaviour of the restriction of Bordiga conic bundleto the blow-up of the quintic X we defined in Sect. 2. Finally, the last part (Sect. 4) contains aclassification of singularities of the discriminant of the web and proof of main results of the paper. Convention:
In this note we work over the base field C . By an abuse of notation we use the samesymbol to denote a homogeneous polynomial and its zero–set in projective space.1. Singularities of the intersection of four quadrics and a small resolution
Let Q , Q , Q , Q ⊂ P be linearly independent quadrics that contain a (fixed) plane Π and let X := Q ∩ Q ∩ Q ∩ Q be their (scheme-theoretic) intersection.Without loss of generality we can assume that Π := { ( x : . . . : x ) : x = . . . = x = 0 } , whichimplies that each Q i is given by the matrix q i = q i b Ti b i , here q i is a 5 × b i := l i m i n i and l i , m i , n i ∈ C are row–vectors. Moreover, in order tosimplify our notation we put b ( y ) := P i y i b i and c ( x , x , x ) := x (cid:2) l T l T l T l T (cid:3) + x (cid:2) m T m T m T m T (cid:3) + x (cid:2) n T n T n T n T (cid:3) . We have (compare [24, Prop. 1.8])
Lemma 1.1. sing( X ) ∩ Π = { (0 : . . . : x : x : x ) : rank( c ( x , x , x )) ≤ } In particular, if the set sing( X ) ∩ Π is finite, then it consists of at most points.Proof. Observe that the intersection X is singular at a point x , iff the differentials dQ i ( x ) = ( q i x ) T of quadratic forms Q i at x are linearly dependent, that is if there exists ( y : · · · : y ) ∈ P suchthat X i =0 y i q i x = 0 . For x = (0 : · · · : 0 : x : x : x ) ∈ Π the above condition reduces to P y i ( x l Ti + x m Ti + x n Ti ) = 0 . We can rewrite the latter as(1) b ( y ) T ( x , x , x ) T = 0 . For a fixed y ∈ P there exists a point in Π satisfying the above relation iff rank( b ( y )) ≤ x , x , x ) and y we have(2) c ( x , x , x ) y = b ( y ) T ( x , x , x ) T . Therefore, (0 , . . . , , x , x , x ) is a singularity of X iff there exist y ∈ P such that c ( x , x , x ) y =0 or equivalently rank( c ( x , x , x )) ≤ . Finally, suppose that the set sing( X ) ∩ Π is finite. Then, the number of its elements does notexceed the degree of the determinantal variety of 4 × ≤
3. The latter is 10 by[14, Ex. 14.4.14] (see also [19], [27]). (cid:3)
From now on we make the following assumption : [A1]: X has exactly singularities on Π and is smooth away from the plane Π , As an immediate consequence of [A1] we obtain
Remark . For each y ∈ P we have rank( b ( y )) ≥
2. Indeed, we assumed that X has onlyisolated singularities on Π. Therefore, for a fixed y ∈ P , there exists at most one point in Πsatisfying the relation (1), so rank( b ( y )) cannot be lower than 2.Lemma 1.1 and [6] support the following conjecture. Conjecture . a) A nodal complete intersection of four quadrics in P with at most nine nodes is Q -factorial.b) A nodal complete intersection of four quadrics in P with exactly ten nodes that is not Q -factorialcontains a plane Π. emma 1.4. Suppose that [A1] holds.a) The ideal of the set sing( X ) ∩ Π is generated by all × minors of the matrix c ( x , x , x ) . Inparticular, the ideal in question contains no cubics.b) For each x ∈ sing ( X ) there exists precisely one quadric in W such that x is its singularity.c) There exist three quadrics in the web W that meet transversally.d) The set { y ∈ P : rank( b ( y )) = 2 } consists of precisely points.Proof. a) Recall that the determinantal variety P ( V ) ⊂ P given by the conditionrank z . . . z ... ... z . . . z ≤ × C [ z , . . . , z ] / I( V ) is Cohen-Macaulay.The map ( x , x , x ) c ( x , x , x ) parametrizes a 3-plane P ⊂ C that meets V along ten lines.Since the ideal I( P ) in the ring C [ z , . . . , z ] / I( V ) is generated by 17 linear forms, it satisfiesthe assumptions of [13, Prop. 18.13]. Consequently, the quotient C [ z , . . . , z ] / (I( V ) + I( P )) is1-dimensional Cohen-Macaulay and the ideal I( V ) + I( P ) coincides with its radical.b) The plane P ( P ) ⊂ P meets the variety P ( V ) in exactly ten points, so none of the latterbelongs to sing( P ( V )). But, as one can check by direct computation (see also [30]), all points of V that satisfy the condition rank z . . . z ... ... z . . . z ≤ ∀ x ∈ sing( X ) rank( c ( x , x , x )) = 3 . Consequently, there exists precisely one y ∈ P that lies in the kernel of the matrix c ( x , x , x ).By (2), the latter is equivalent to the condition (0 : . . . : x : x : x ) ∈ sing( Q ( y )). In this way wehave shown the claim b).c) follows from b) by standard arguments.d) Suppose that a point y ∈ P satisfies the relation (1) for two various points in Π. Then, the linespanned by both points in question lies in the kernel of the matrix b ( y ) and rank( b ( y )) <
2, whichis impossible by Remark 1.2. In this way we have shown that { y ∈ P : rank( b ( y )) = 2 } ≥ X ) . The other inequality has been shown in the proof of part b). (cid:3)
Lemma 1.5.
Assume that Z P = { f ( y , . . . , y ) = 0 } ⊂ C is a three-dimensional isolated hyper-surface singularity that contains the germ of the plane { y = y = 0 } . If the ideal h ∂f∂y , . . . , ∂f∂y , f, y , y i ⊂ O C ,P is maximal, then Z P is a node.Proof. We are to show that hessian of f in P does not vanish. Let f , f ∈ O C ,P satisfy thecondition f = y · f + y · f . By direct computation we have(4) h f , f , y , y i = h y , y , y , y i . onsider the linear parts f (1) i = P j =1 f (1) i,j y j for i = 1 ,
2. Then hessian of f in P is given bydet f (1)1 , f (1)1 , + f (1)2 , f (1)1 , f (1)1 , f (1)1 , + f (1)2 , f (1)2 , f (1)2 , f (1)2 , f (1)1 , f (1)2 , f (1)1 , f (1)2 , = − det f (1)1 , f (1)2 , f (1)1 , f (1)2 , . To show that the right-hand side of the latter equality does not vanish put y = y = 0 in (4). (cid:3) Lemma 1.6.
If [A1] holds, then all singularities of X are nodes (i.e. A points).Proof. Without loss of generality we can assume that all singularities of X lie in the affine chart x = 0 and the variety Y := Q ∩ Q ∩ Q is smooth (see Lemma 1.4). By abuse of notation weuse the same symbol to denote a quadric and the dehomogenization of its equation (i.e. x = 1).Observe that putting x = x = · · · = x = 0 in the ideal h V Jac( Q , . . . , Q ) , Q , . . . , Q i weget the ideal in C [ x , x ] generated by 4 × c ( x , x , C -vector spacedim( C [ x , . . . , x ] / h V Jac( Q ,...,Q ) ,Q ,...,Q ,x ,...,x i ) = 10 . Moreover, the assumption [A1] yields an isomorphism M P ∈ sing( X ) O C ,P / h V Jac( Q ,...,Q ) ,Q ,...,Q ,x ,...,x iO C ,P ≃ C [ x , . . . , x ] / h V Jac( Q ,...,Q ) ,Q ,...,Q ,x ,...,x i Therefore, for each P ∈ sing( X ), we have(5) dim( O C ,P / h V Jac( Q ,...,Q ) ,Q ,...,Q ,x ,...,x iO C ,P ) = 1 . Fix a point P ∈ sing( X ) and assume that the germ of Y near P can be (analytically)parametrized as the graph of a map ( x ( x , . . . , x ) , . . . , x ( x , . . . , x )). Let ˜ Q be the compo-sition of the above parametrization with (the dehomogenized equation of) the quadric Q . Bydirect computation, (5) implies that the ideal h ˜ Q , ∂ ˜ Q ∂x , . . . , ∂ ˜ Q ∂x i + I(Π) ⊂ O Y,P is maximal. By Lemma 1.5 the point P is an A singularity of X . (cid:3) We introduce the following notation:(6) σ : ˜ X → X is the blow-up of X along the plane Π and S stands for the strict transform of the plane Π underthe blow-up σ . The variety ˜ X is smooth and the blow-up in question replaces the 10 nodes with10 disjoint smooth rational curves(7) E , . . . , E ⊂ S. Convention:
In the sequel, we shall identify smooth points of X with their images in ˜ X , i.e.write P instead of σ ( P ) whenever it leads to no ambiguity. In the next section we will use the following lemma. emma 1.7. The variety ˜ X is a projective Calabi–Yau manifold with the following Hodge dia-mond
10 00 2 01 56 56 10 2 00 01
Proof.
By Lemma 1.4.b we can assume that Y = Q ∩ Q ∩ Q is smooth. Let σ : ˜ Y −→ Y be theblow–up of Y along Π with exceptional divisor E . We have σ ∗ O ˜ Y ( kE ) = O Y , for k ≥ ,R σ ∗ O ˜ Y ( E ) = 0 ,R σ ∗ O ˜ Y (2 E ) = O Π ( − . Since O ˜ Y ( ˜ X ) = σ ∗ O Y ( X ) ⊗ O ˜ Y ( − E ) using the projection formula we get σ ∗ O ˜ Y ( − k ˜ X ) = O Y ( − kX ) , for k ≥ ,R σ ∗ O ˜ Y ( − ˜ X ) = 0 ,R σ ∗ O ˜ Y ( − X ) = O Π ( − . The Leray spectral sequence and the Kodaira vanishing imply H i ( O ˜ Y ( − ˜ X )) = 0 for i ≤ , H ( O ˜ Y ( − ˜ X )) ∼ = C . Since H i ( O Y ( − Y )) = 0 , for i ≤ ,H ( O Y ( − X )) ∼ = H ( O Y (2)) ∼ = C ,H ( O ˜ Y ( − X )) ∼ = H ( O ˜ Y ( ˜ X )) ∼ = H ( O Y ( X ) ⊗ I (Π)) ∼ = C ,H i ( R σ ∗ ( O ˜ Y ( − X ))) = 0 , for i = 0 , H ( R σ ∗ ( O ˜ Y ( − X ))) ∼ = H ( O Π ( − ∼ = C the Leray spectral sequence implies H i ( O ˜ Y ( − X )) = 0 , for i ≤ H i ( N ∨ ˜ X | ˜ Y ) = 0 for i ≤ . From the exact sequence 0 −→ σ ∗ Ω Y −→ Ω Y −→ Ω E/ Π −→ σ ∗ Ω Y = Ω Y , R σ ∗ Ω Y = O Π and so H Ω Y ∼ = C . Similarly, the exact sequence0 −→ σ ∗ (Ω Y ( − X )) ⊗ O ˜ Y ( E ) −→ Ω Y ( − ˜ X ) −→ Ω E/ Π ( − ⊗ σ ∗ O Y ( − X ) −→ σ ∗ Ω Y ( − ˜ X ) ∼ = Ω Y ( − X ) and R σ ∗ Ω Y ( − ˜ X ) ∼ = N Π | Y ⊗ O Y ( − X ) . wisting the exact sequence 0 −→ N Π | Y −→ N Π | P −→ N Y | P | Π −→ O Y ( − X ) ∼ = O Y ( −
2) we get H N Π | Y ⊗ O Y ( − X ) = H N Π | Y ⊗ O Y ( − X ) = 0 and H N Π | Y ⊗ O Y ( − X ) . Since H (Ω Y ( − X )) ∼ = H ( T Y ) = 36, while H (Ω Y ( − ˜ X )) ∼ = H ( T ˜ Y ) = 33 the Leray spectralsequence yields H i Ω Y ( − ˜ X ) = 0 , for i = 0 , , . From the exact sequence 0 −→ Ω Y ( − ˜ X ) −→ Ω Y −→ Ω Y ⊗ O ˜ X −→ H (Ω Y ⊗ O ˜ X ) ∼ = H Ω Y ∼ = C . Finally, the exact sequence0 −→ N ∨ ˜ X | ˜ Y −→ Ω Y ⊗ O ˜ X −→ Ω X −→ H Ω X ∼ = H (Ω Y ⊗ O ˜ X ) ∼ = C . The standard computation with help of [14, Example 3.2.12] yields that the Euler number e ( ˜ X ) = −
108 (see also [24, Prop. 1.14]), so we can compute h , ( ˜ X ). (cid:3) As another consequence of [A1] we obtain the following simple observation.
Remark . The web W contains no rank-4 quadrics. Proof.
Suppose that Q ∈ W is a rank-4 quadric. Then it is a cone through the 3-space sing( Q )over a smooth quadric in P . The latter contains no planes, so the 3-space sing( Q ) and the planeΠ meet. On the other hand, since each point in sing( Q ) ∩ Q ∩ Q ∩ Q is a singularity of X ,the assumption [A1] implies that sing( Q ) meets Π in exactly one point P ∈ sing( X ). Moreover,we have sing( Q ) ∩ Q ∩ Q ∩ Q = { P } .Lemma 1.4.b yields that the quadrics Q , Q , Q are smooth in P . By B´ezout the intersection mul-tiplicity of sing( Q ), Q , Q , Q in the point P is 8. The latter exceeds the product of multiplicitiesof the varieties in question in the point P . From [11, Thm 6.3] we obtain the inequality:(8) dim(sing( Q ) ∩ T P Q ∩ T P Q ∩ T P Q ) ≥ . To complete the proof, suppose that sing( Q ) is the zero set of the coordinates x , x , x , x .Recall that Π is given by vanishing of x , . . . , x , so we have P = (0 : . . . : 1 : 0 : 0) and only 12entries in the matrix q do not vanish.The point P is a node on X , so dim(T P Q ∩ T P Q ∩ T P Q ) = 4. Consider the affine chart x = 1. The inequality (8) implies that there exists a nonzero v := (0 , , v , v , v , ,
0) in the4-dimensional intersection of the tangent spaces. Furthermore, all quadrics in question contain Π,so the 4-space contains the vectors (0 , . . . , , ,
0) and (0 , . . . , , P Q ∩ T P Q ∩ T P Q is given by the map( λ , . . . , λ ) λ v + λ w + λ (0 , . . . , ,
0) + λ (0 , . . . , , where w := ( w , . . . , w , , P Q , T P Q , T P Q ,T P Q consists of two planes. The latter is impossible because we assumed the point P to be anode of X . Contradiction. (cid:3) . Projection from the plane
Here we maintain the notation of the previous section. Moreover, we assume that [A1] holds and [A2]: no singular points of X lie on a line. In view of Lemma 1.4.a it seems natural to ask whether the assumption [A1] implies [A2]. Theexample below shows that this is not the case.
Example . Consider the following 8 × q := − − − − −
24 4 4 1 − − − − − − − − − − − − − − q := − − − −
22 0 − − − − − − − − − − − − − − − − q := − − − − −
11 4 − − − − − − − − − − − − − − − − − − − − − q := − − − − − − − − − − − − − − − By direct computation with help of [15], the intersection in P of the quadrics defined by the abovematrices has 10 isolated singularities on the plane Π and is smooth elsewhere. In the same way onechecks that 4 singular points of the intersection in question lie on the line (0 : . . . : 0 : x : x ) andare given by the equation19 x + 102 x x + 189 x x + 137 x x + 27 x = 0 . In this section we study the projection X \ Π ∋ ( x : . . . : x ) ( x : . . . : x ) ∈ P from theplane Π. Observe that the map in question lifts to a regular map(9) π : ˜ X −→ P given by the linear system | H − S | , where H is the pullback of a hyperplane section under theblow-up σ : ˜ X → X , and S stands for the strict transform of Π. Lemma 2.2.
We have the following intersection numbers: H = 16 ,H · S = 1 ,H · S = − ,S = − , ( H − S ) = 5 . Proof.
The first two statements are obvious. The intersection number H · S equals the intersectionnumber in S of the restrictions H | S , S | S . Since S is a blow–up of the plane Π in 10 points, therestriction H | S is the pullback l of a line in Π. Moreover, S | S is the normal bundle of S in the alabi–Yau manifold ˜ X . Hence it is the canonical divisor K S = − l + P E i , where E , . . . , E are the 10 exceptional curves (see (7)). Finally, we have H · S = ( l · ( − l + X E i )) S = − . Similarly, S = (( − l + P E i ) ) S = 9 −
10 = −
1. The last statement follows from Newton’sformula. (cid:3)
To simplify our notation we put x := ( x : . . . : x ) ∈ P and define the following matrices :(10) a ( x ) := l x l x l x l x m x m x m x m x n x n x n x n x , A ( x ) := x T q xx T q xx T q xx T q x a ( x ) T . Observe that the following equality holds (cf. [2, p. 30])(11) a ( x ) y = b ( y ) x. Let Q i be the quadratic form associated to the matrix q i and let C i denote the cubic given bythe degree-3 minor of the matrix a ( x ) obtained by deleting its i -th column, e.g. C := det l x l x l x m x m x m x n x n x n x . Lemma 2.3. a) The image of ˜ X under π is the quintic X given by the equation (12) det( A ( x )) = C · Q − C · Q + C · Q − C · Q = 0 . b) The image of S under π is the (smooth) Bordiga sextic B ⊂ P given by vanishing of the cubics C , . . . , C (i.e. all × minors of the matrix a ( x ) ). Moreover, the map π | S : S → B is anisomorphism.Proof. Obviously, the restriction of the quadric P α i Q i to the 3-spacespan { x, Π } = { ( µ x : . . . : µ x : µ x : µ : µ : µ ) | ( µ : µ : µ : µ ) ∈ P } is given by the polynomial(13) ( X α i x T q i x ) µ + 2( X α i ( l i x )) µ µ + 2( X α i ( m i x )) µ µ + 2( X α i ( n i x )) µ µ . a) Observe that x ∈ P \ π ( S ) lies in the image of X under the projection from Π iff the planesresidual to Π in the intersections of the quadrics Q i with the 3-space span { x, Π } intersect. By (13),the latter is equivalent to the vanishing det( A ( x )) = 0. Laplace formula completes the proof.b) From (13) we obtain that the condition X α i ( l i x ) = X α i ( m i x ) = X α i ( n i x ) = 0is satisfied iff the restriction ( P α i Q i ) | span { x, Π } is the double plane 2Π. The latter holds preciselywhen x lies in the image of Π under the projection in question.It is well known that, for a generic 4 × × P blown-up in 10 points (see e.g. [2]).Still, it is not always the case (see e.g. [30]). To see that our surface is indeed the (smooth) ordiga sextic, observe that the linear system | H − S | restricts on S to the complete linear system | l − P i =10 i =1 E i | . We apply [4, Lemma 2.9.1] to show that the system in question embeds S into P as the (smooth) Bordiga sextic. By Lemma 1.4.a no cubic contains all singularities of X . Supposethat 8 singularities of X lie on a conic. Then its product with the line through the remaining twosingular points is a cubic containing sing( X ). Consequently the existence of such a conic is ruledout by Lemma 1.4.a. Finally no 4 singularities lie on a line by the assumption [A2]. (cid:3) Remark . a) Observe that, since the (scheme–theoretic) intersection B of the zeroes of thedegree-3 minors of the matrix a ( x ) is smooth, we haverank( a ( x )) = 2 for every x ∈ B . b) The rational curves E , . . . , E ⊂ ˜ X are mapped by π to lines in P contained in the Bordigasextic. Indeed, we have ( H − S ) · E j = ((4 l − P E i ) · E j ) S = 1 for j = 1 , . . . , ⊂ B correspond to the 3-spaces in the 4-space T P X , where P is a node of X , that contain the plane Π.We introduce the following notation: U := ˜ X \ ( S ∪ [ V linear , V ⊂ X ,V ∩ Π = ∅ σ − ( V )) . Lemma 2.5.
Suppose that [A1], [A2] hold.a) The map π | U is an isomorphism onto the image and we have the equality π ( U ) = ( X \ B ) . b) The inclusion sing( X ) ( B holds. In particular, the quintic X is normal.Proof. a) Fix P ∈ U . Then σ ( P ) / ∈ Π. Since X is an intersection of quadrics we have the equalityspan( σ ( P ) , Π) ∩ X = Π ∪ { σ ( P ) } , where σ ( P ) / ∈ Πwhich implies that π | U is injective and the linear map d P π is an isomorphism.We claim that π ( ˜ X \ U ) = B . Let V ⊂ X , V * Π be a linear subspace such that V ∩ Π = ∅ . Let σ ( P ) ∈ ( V \ Π) and let σ ( P ) ∈ ( V ∩ Π). By definition of π all points from span( σ ( P ) , σ ( P )) \ { σ ( P ) } lie in one fiberof π . On the other hand, the proper transform of the line span( σ ( P ) , σ ( P )) under σ meets S .Since π maps that proper transform of the line in question to one point and π ( P ) ∈ B we have π ( P ) ∈ B , and we obtain the claim.It remains to show the inclusion π ( U ) ⊂ ( X \ B ) . Suppose that π ( P ) = π ( P ), where P ∈ ˜ X \ U and P ∈ U . If σ ( P ) ∈ reg( X ), then the linespan( σ ( P ) , σ ( P )) is tangent to X in σ ( P ) and meets it in σ ( P ). In particular, it is containedin each quadric of the system W , so span( σ ( P ) , σ ( P )) ⊂ X and P / ∈ U . Contradiction.Similar argument yields contradiction when σ ( P ) ∈ sing( X ).b) By [A1] and part a) we know that sing( X ) ⊂ B . Suppose that sing( X ) = B . Since B issmooth, Lemma 2.3.a implies that det( A ( x )) ∈ I( B ) . The latter is impossible because the idealI( B ) is generated by the cubics C , C , C , C .Finally X is a 3-dimensional hypersurface with at most 1-dimensional singularities, so it is normal. (cid:3) After those preparations we can study higher-dimensional fibers of π . emma 2.6. a) The map π has no two-dimensional fibers and its only one-dimensional fibers areproper transforms of lines on X that meet Π but are not contained in Π .b) The following equality holds (14) sing( X ) := { x ∈ B : rank( A ( x )) ≤ } . c) The map π has only finitely many one–dimensional fibers.Proof. a) As we have already shown in the proof of Lemma 2.5 the proper transform of each lineon X that meets Π but is not contained in Π lies in a fiber of π .The regular map π is birational and its image is normal, so we can apply Zariski’s Main Theorem[17, Thm 5.2] to see that the map π has connected fibers. Moreover, by Lemma 2.3.b(15) each fiber of π meets the surface S in at most one point.Let F be a fiber of π such that dim( F ) ≥
1. Let P , P ∈ ( F \ S ). Then the 3-spaces span( σ ( P ) , Π),span( σ ( P ) , Π) coincide, so the line span( σ ( P ) , σ ( P )) meets the plane Π. Obviously, the in-tersection point does not coincide with P , P . Since X is intersection of quadrics, we havespan( σ ( P ) , σ ( P )) ⊂ X , which implies thatspan( σ ( P ) , σ ( P )) ⊂ σ ( F ) . Suppose that the fiber F contains a point P / ∈ S such that σ ( P ) / ∈ span( σ ( P ) , σ ( P )). Then,arguing as in (2), we show that span( σ ( P ) , σ ( P )) is a line contained in σ ( F ) and meeting theplane Π. But, (15) implies that the proper transforms (under the blow-up σ ) of two lines meetingΠ in different points cannot lie in the same fiber of π . Consequently, by (15), the image σ ( F ) isa plane in X that intersects Π in precisely one point. Observe that the planes σ ( F ), Π meet ina singularity of X . Let H be the pullback of a hyperplane section under the blow-up σ and let ] σ ( F ) denote the proper transform of σ ( F ). If we put ˜ l (resp. ˜ m ) to denote the proper transform ofa line in σ ( F ) (resp. in Π) that runs through no singularities of X , then we obtain the followingtable of intersection numbers. ] σ ( F ) S H ˜ l − m − H X is at least 3, which isimpossible by Lemma 1.7. This contradiction shows that the fiber F coincides with the propertransform of the line span( σ ( P ) , σ ( P )).b) As in the proof of Lemma 2.3, we see that the line through the points ( x, x , x , x ) and(0 , x ′ , x ′ , x ′ ) is contained in X iff for any λ ∈ C and i = 0 , . . . , x T q i x + 2( l i x, m i x, n i x )( x , x , x ) T + 2 λ ( l i x, m i x, n i x )( x ′ , x ′ , x ′ ) T = 0 . Fix x ∈ B . From Remark 2.4.a we know that rank( a ( x )) = 2. Consequently, there existpoints ( x , x , x ) and ( x ′ , x ′ , x ′ ) such that the line spanned by ( x, x , x , x ) and (0 , x ′ , x ′ , x ′ ) iscontained in X if and only if rank( A ( x )) = 2.c) Assume to the contrary that the map π contracts infinitely many lines. Then there is a ruledsurface G ⊂ ˜ X such that the fibers of G are contracted by π . Let l (resp. E i ) be the class ofa (general) fiber of G , (resp. of an exceptional curve of the blow-up σ ). We have the following ntersection numbers(16) S G Hl − E i − ν H and S are linearly independent in Pic( ˜ X ) ⊗ Q . Since h , ( ˜ X ) = 2, we can find d H , d S ∈ Q such that G ∼ num d H H + d S S . From (16) we obtain G ∼ num ( ν − H − νS. Therefore Lemma 2.2 yields the equality( H − S ) · G = 5 ν − . As the divisor G is contracted by π we conclude that ν = , which is impossible by (16). (cid:3) In particular, Lemma 2.6 implies that the map π : ˜ X −→ X is a resolution of singularitiesof the quintic X . As π contracts only finitely many curves (i.e. the singular locus of X iszero-dimensional), it is in fact a small resolution that introduces exactly one copy of P over eachsingularity.The lemma below gives a simple criterion when the quintic X is nodal. Lemma 2.7.
All singularities of the quintic X are nodes iff the set sing( X ) consists of points.Proof. Let µ ( · ) stand for the Milnor number. Lemma 2.5 yields that the regular map π : ˜ X −→ X is birational. By Lemma 2.6 it contracts only the lines in X that intersect the plane Π. Thecontracted lines are pairewise disjoint, so we obtain − − X )) = e ( X ) = −
200 + X P ∈ sing( X ) µ ( P, X ) , where the second equality results from [10, Cor. 5.4.4]. To complete the proof recall that the Milnornumber of a singularity is 1 iff the singularity in question is an A point. (cid:3) Restriction of the Bordiga conic bundle
In this section we maintain the assumptions and notation of the previous one, i.e. we assumethat [A1], [A2] hold. In particular, the scheme-theoretic intersection of the zeroes of the degree- minors of the matrix a ( x ) is smooth (see (10)) and the locus { y ∈ P : rank( b ( y )) = 2 } consists of points . Moreover, we make the following assumption : [A3]: the set { x ∈ B : rank( A ( x )) ≤ } consists of points .One can show (see [2, Ex. 3 on p. 35]) that the rational map(17) P \ B ∋ x ( C ( x ) : −C ( x ) : C ( x ) : −C ( x )) ∈ P lifts to a regular map (so-called Bordiga conic bundle - see [2, Ex. 3 on p. 35])Φ : Bl B P → P . that is generically a conic-bundle ([ibid., Prop. 2.1]). The map Φ is the projection onto the secondfactor from the closure of the graph of the rational map defined by (17) (see also (11)) i.e. fromthe set(18) { ( x, y ) ∈ P × P : b ( y ) x = 0 } . y Lemma 1.4.d it has exactly ten 2-dimensional fibers over the points y ∈ P such that rank( b ( y )) =2. Such a fiber is the plane(19) Φ − ( y ) = { ( x, y ) : b ( y ) x = 0 } . Observe that restrictions of the cubics polynomials C i to the plane { b ( y ) x = 0 } are proportional,so the plane cuts B along a cubic curve (see also [2, Ex. 3 on p. 35]).The remaining fibers Φ − ( y ) are 3-secant lines to B . They are given by (19) with rank( b ( y )) = 3.In Sect. 1 we studied the map ˜ X −→ X . By Lemma 2.7 the quintic X admits another smallresolution of singularities(20) ψ : ˜ X −→ X obtained by blowing–up the Bordiga surface B . The strict transform S of B is a plane blown–up in56 points (some of the 46 points that are centers of the second blow-up may lie on the exceptionalcurves of the first blow–up). We put F , . . . , F to denote the exceptional curves of the smallresolution in question. Then, the two resolutions differ by flops of the 46 smooth rational curves L , . . . , L ⊂ ˜ X and F , . . . , F ⊂ ˜ X .The restriction of the conic bundle Φ induces the regular map φ : ˜ X −→ P . This regular map is given by the linear system | H − S | on ˜ X , where H is pullback of thehyperplane section O P (1). We have the following intersection numbers Lemma 3.1. H = 5 ,H · S = 6 ,H · S = − ,S = − , (3 H − S ) = 2 . Proof.
The first two statements follow from the fact that deg( X ) = 5 and deg( B ) = 6. The otherscan be obtained from the equalities(21) H | S = 4 l − X ψ ∗ ( π ( E i )) , S | S = − l + X ψ ∗ ( π ( E i )) + X F j . where l is the pull-back of O P (1) under both blow-ups. Recall (Remark 2.4.b) that the curves π ( E ), . . . , π ( E ) are lines on B . (cid:3) Since φ is surjective, as an immediate consequence of Lemma 3.1 we obtain Corollary 3.2.
The mapping φ is generically 2:1. In order to obtain a precise description of fibers of φ we will need the following lemma (compare[24]): Lemma 3.3.
A point z ∈ ˜ X is mapped by φ to y ∈ P iff the 3-space span(( ψ ( z ) : 0 : 0 : 0) , Π) iscontained in the quadric Q ( y ) := P i y i Q i .Proof. Observe that for any x = ( x : x : x : x ) ∈ span(( x : 0 : 0 : 0) , Π) we have(22) x T q ( y ) x = x T q ( y ) x + 2( x , x , x ) b ( y ) x ( ⇐ ): Put x = ψ ( z ) in (22) to obtain ψ ( z ) T q ( y ) ψ ( z ) = − x , x , x ) b ( y ) ψ ( z ) for all x , x , x ∈ C . he latter implies b ( y ) ψ ( z ) = 0 and (see (19)) the equality φ ( z ) = y .( ⇒ ): Suppose that z ∈ ˜ X \ S . From φ ( z ) = y we get b ( y ) ψ ( z ) = 0. By (22) we have x T q ( y ) x = ψ ( z ) T q ( y ) ψ ( z ) for all x = ( ψ ( z ) : x : x : x ) ∈ span( ψ ( z ) , Π) . But (see (17)), we can assume that y = ( C ( ψ ( z )) : . . . : −C ( ψ ( z ))). Therefore, Lemma 2.3.a yieldsthe equalities ψ ( z ) T q ( y ) ψ ( z ) = det( A ( ψ ( z ))) = 0. In this way we have shown the inclusion { ( x, y ) ∈ ˜ X : b ( y ) x = 0 } ⊂ { ( x, y ) ∈ P × P : span(( x : 0 : 0 : 0) , Π) ⊂ Q ( y ) } , which completes the proof. (cid:3) Recall, that we have the map ( ψ ◦ ( π | S ) − ◦ σ ) : S → B ⋍ S → Π. In the lemma below we putˆ l (resp. ˆ E , . . . , ˆ E ) to denote the pullback of O Π (1) (resp. of the exceptional divisors (7)) to S . Lemma 3.4.
An irreducible curve D ⊂ S is contracted by φ iff (up to a relabelling of the divisors ˆ E , . . . , ˆ E and F , . . . , F ) it belongs to one of the following linear systems a) | ˆ E − F − F − F − F | , b) | ˆ l − ˆ E − ˆ E − ˆ E − F − F − F | , c) | l − ˆ E − . . . − ˆ E − F − F | , d) | l − E − ˆ E − . . . − ˆ E − F − . . . − F | .In the cases (a)–(c) the curve in question is the proper transform of a line in B , whereas the case (d)corresponds to a conic in the intersection of B with the plane { b ( y ) x = 0 } , where rank( b ( y )) = 2 .In particular, if the intersection B ∩ { b ( y ) x = 0 } is an irreducible cubic, then its proper transformis not contracted by φ .Proof. Recall that φ = Φ | ˜ X and the fibers of Φ are lines and planes given by (19).Before we prove the claim, we study two-dimensional fibers of Φ. Let sing( X ) = { P , . . . , P } .By (3) for each singularity P i there exists a unique point y ( i ) ∈ P such that c ( P i ) y ( i ) = 0. Then,by (2), we have rank( b ( y ( i ) )) = 2.Lemma 1.4.a yields that for each i ∈ { , . . . , } there is a unique degree-three curve C i ⊂ Π suchthat P j ∈ C i , for j = i . Let ˜ C i := σ ∗ C i − P j = i E j ∈ | l − P j = i E j | be the corresponding curve on S . By direct computation the following equality holds(23) π ( ˜ C i ) = B ∩ { x ∈ P : b ( y ( i ) ) x = 0 } In general, cubics C i are smooth, and the curves π ( ˜ C i ) ⊂ B are also smooth planar cubics. Wehave the following possible degenerations:(i) The curve C i is irreducible, but sing( C i ) = { P j } for a j = i . Then the exceptional curve E j is a component of the curve ˜ C i := σ ∗ C i − P j = i E j and the curve ˜ C i − E j is irreducible. ByRemark 2.4.b the image π ( E j ) is a line on B , whereas π ( ˜ C i ) is a smooth conic. In this way weobtain a decomposition of B ∩ { x ∈ P : b ( y ( i ) ) x = 0 } . Observe that for a given integer i = j thereexists at most one cubic in |O Π (3) − P j = i E j − E j | .(ii) The cubic ˜ C i is union of a line and a smooth conic. Then, by [A2] and Lemma 1.4.a the linecontains two (resp. three) singularities of X and the conic contains 7 (resp. 6) of them.(iii) The curve ˜ C i can be union of three lines. The assumption [A2] yields that each line containsthree singularities of X .In this way (up to a permutation of the points in P , . . . , P ), we obtain the following possibilities or the decomposition of the cubic (23) for i = 10:(3 l − E − E − · · · − E ) + E , ( l − E − E ) + (2 l − E − · · · − E ) , ( l − E − E − E ) + (2 l − E − · · · − E ) , (24) ( l − E − E − E ) + (2 l − E − · · · − E ) + E , ( l − E − E − E ) + ( l − E − E − E ) + ( l − E − E − E ) . After those preparations we can prove the lemma. Assume that an irreducible curve D ⊂ S iscontained in φ − ( y ) for a point y ∈ P . The map φ | S : S → P is given by the linear system(25) | l − X ˆ E i − X F j | , so D = F j for each j ≤ b ( y )) = 2. We can assume that D ⊂ φ − ( y (10) ). Then ψ ( D ) ⊂ B is acomponent of (23). If ψ ( D ) is image under π of a curve from the system | l − E − E − · · · − E | ,then we havedeg( ψ ( D )) = (3 l − E − E − · · · − E ) · (4 l − X E i ) = 12 − − . Let sing( X ) ∩ ψ ( D ) = { ψ ( F ) , . . . , ψ ( F p ) } . Since D coincides with the proper transform of ψ ( D )under the blow-up ψ , we have D ∈ | l − E − ˆ E − · · · − ˆ E ) − F − · · · − F p | . and, by (25), the degree of φ ( D ) is (5 − p ). Consequently, the curve D is contracted by φ iff p = 5.In the following table we collect data on each curve considered in (24). In particular, the integerin the last column is the number of singularities of X that lie on ψ ( D ) provided D is contractedby the map φ : | π − ( ψ ( D )) | deg( ψ ( D )) X ) ∩ ψ ( D ))3 l − E − E − · · · − E l − E − . . . − E l − E − . . . − E l − E − E l − E − E − E E y ( i ) ∈ P , where i = 1 , . . .
10, the intersection(26) X ∩ { x ∈ P : b ( y ( i ) ) x = 0 } is a degree–5 planar curve, so it is union of the cubic considered above and a conic (possibly re-ducible) that does not lie on B . The points ψ ( F j ) are singular points of X , so they are also singularpoints of the quintic curve (26), which yields some extra constrains on the possible arrangements.Since a line contained in (26) intersects the residual quartic in four points, the line of the type( l − E − E ) is never contracted. Similar argument rules out the conic (2 l − E − . . . − E ). Inthis way we arrive at the cases (a)–(d) of the lemma.Assume that rank( b ( y )) = 3. Then D is the strict transform of a line l y ⊂ B . In particular,there exist d, m i , n j ∈ Z such that D ∈ | d ˆ l − P m i ˆ E i − P n j F j | . Since the curve D is smooth nd rational, we have n j = 0 or 1. Moreover, by the genus formula( d ˆ l − X m i ˆ E i − X n j F j ) · (( d − l − X ( m i −
1) ˆ E i − X ( n j − F j ) = d − d − X ( m i − m i ) = − . Furthermore, the equality 4 d − P m i = 1 holds because l y is a line on B (see also Lemma 2.3.b).Finally, since D is contracted by the map given by the linear system | H − S | we have(15ˆ l − X ˆ E i − X F j ) · ( d ˆ l − X m i ˆ E i − X n j F j ) = 15 d − X m i − X n j = 0 . From the above we obtain the following equations X m i = d + d + 1 , X m i = 4 d − , − d = X n j , where n j = 0 ,
1. The solution d = 3, m = 2, m i = 1 for i > (cid:3) Now we are in position to prove
Lemma 3.5.
Let y ∈ P be a point such that rank( b ( y )) = 3 . Then the fiber φ − ( y ) is -dimensionaliff rank( q ( y )) = 6 .Proof. By abuse of notation we put ψ to denote the blow-up Bl B P → P .Assume that the line Φ − ( y ) is contracted by φ . Then the set ψ (Φ − ( y )) = { x ∈ P : b ( y ) x = 0 } is a line on X . Observe that the linear space span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π) is4-dimensional. By Lemma 3.3 the quadric Q ( y ) contains the 4–space span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π), which yields rank( q ( y )) ≤
6. Finally rank( q ( y )) = 6, because rank( b ( y )) = 3.On the other hand, if rank( q ( y )) = 6, then sing( Q ( y )) is a line. Since rank( b ( y )) = 3, the linesing( Q ( y )) does not meet the plane Π. Put L to denote the image of the line sing( Q ( y )) under theprojection from the plane Π. Then span(( x : 0 : 0 : 0) , Π) ⊂ Q ( y ) for every x ∈ L . From Lemma 3.3we obtain that the the proper transform of the line L under the blow-up ψ is contracted by φ . (cid:3) In the theorem below we identify curves in P with their proper transforms under the blow-up ψ : whenever we say a line (resp. a conic) we mean its proper transform. Theorem 3.6.
There are four types of fibers φ − ( y ) of the map φ : ˜ X −→ P : a) union of the conic residual to the cubic B ∩ Φ − ( y ) in the planar quintic X ∩ Φ − ( y ) with thecomponents of the cubic that satisfy the conditions of Lemma 3.4 iff rank( q ( y )) ∈ { , , } and rank( b ( y )) = 2 (i.e. a singularity of Q ( y ) lies on Π ), b) a line in P iff rank( q ( y )) = 6 and rank( b ( y )) = 3 (equivalently sing( Q ( y )) ∩ Π = ∅ ), c) one point iff rank( q ( y )) = 7 and rank( b ( y )) = 3 , d) two points iff rank( q ( y )) = 8 .Proof. Suppose that rank( b ( y )) = 3. Then the linear space span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π)is 4-dimensional and sing( Q ( y )) ∩ Π = ∅ . In view of Lemma 3.5, we can assume that rank( q ( y )) ≥ ψ (Φ − ( y )) = { x : b ( y ) x = 0 } is not contained in X .Moreover, by (22), for every point x = ( x, x , x , x ) ∈ span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π)we have(27) x T q ( y ) x = x T q ( y ) x . bserve, that the quadratic form given by q ( y ) does not vanish identically on the line { x : b ( y ) x =0 } because the latter is not contained in X . Consequently, intersection of Q ( y ) with the linear4-space span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π) consists of either one or two 3-spaces.Lemma 3.3 implies that the fibre φ − ( y ) consists of a unique point iff the restriction(28) Q ( y ) | span( { ( x :0:0:0) : x ∈ ψ (Φ − ( y )) } , Π) is a full square.Suppose that the fibre in question is one point. From (27) there exists a point v ∈ P , such that b ( y ) v = 0 and q ( y ) v = 0which means that ( v : 0 : 0 : 0) ∈ sing( Q ( y )) and rank( q ( y )) < q ( y )) <
8. Then Q ( y ) is a cone with the unique vertex ( v : v : v : v ) awayfrom the plane Π. The latter yields v = 0. Moreover, since the tangent space to Q ( y ) in each pointcontains the vertex we have b ( y ) v = 0 and( v : v : v : v ) ∈ span( { ( x : 0 : 0 : 0) : x ∈ ψ (Φ − ( y )) } , Π)Now ( v : v : v : v ) is a singularity of the restriction (28), so the polynomial x T q ( y ) x has a uniquedouble root on the line { x : b ( y ) x = 0 } and (28) is a full square.Assume that y ∈ P is a point such that rank( b ( y )) = 2, and maintain the notation of the proofof Lemma 3.4. Then y = y ( i ) for an i ∈ { , . . . , } . By definition of the map φ , the propertransform under the blow-up ψ of the (possibly reducible) conic residual to (23) in the quintic (26)is always contracted by φ . Moreover, a component of (23) is contracted iff it satisfies the conditionsof Lemma 3.4.Observe that rank of the quadric Q ( y ( i ) ) does not exceed 7 because we have rank( b ( y ( i ) )) = 2. (cid:3) Remark . By Lemma 1.4.d there are exactly ten fibers of φ of the type a). The number of fibersof type b) will be discussed in the next section (see Cor. 4.7).4. Discriminant of the web W In this section we maintain the notation and the assumptions of the previous ones. In particularwe assume that [A1], [A2], [A3] hold. Let S stand for the discriminant surface of the web W .From now on we assume that [A4]: the discriminant surface S has only isolated singularities .To simplify notation we put I l := [ a i,j ] i,j =0 ,..., , where a i,i = 1 for i = 1 , . . . , l and a i,j = 0 otherwise.At first we give conditions when a singularity of S is a node: Lemma 4.1.
Let Q be a rank- quadric in the web W . a) The quadric Q is a smooth point of S iff sing( Q ) / ∈ X . b) The quadric Q is a node of S iff sing( Q ) ∈ X .Proof. Let q k =: [ q ( k ) i,j ] i,j =0 ,..., and let Q ( k ) := ( q ( k )0 , , . . . , q ( k )6 , ). After an appropriate change ofcoordinates we can assume that q = I . In particular, sing( Q ) = { (0 : . . . : 0 : 1) } .Let G := [ g i,j ] i,j =1 , , , where g i,j := hQ ( i ) , Q ( j ) i and h· , ·i stands for the bilinear form defined bythe identity matrix. By direct computation we havedet( q + X k =1 µ k · q k ) = ( X k =1 µ k · q ( k )7 , ) − (( µ , µ , µ ) G ( µ , µ , µ ) T ) + (terms of degree ≥ . ) Obviously, (1 : 0 : 0 : 0) is a smooth point of S iff the vector ( q (1)7 , , q (2)7 , , q (3)7 , ) does not vanish.The latter holds iff (1 : 0 : 0 : 0) / ∈ X , which concludes the proof. b) ( ⇒ ): the implication in question results immediately from the part a).( ⇐ ): Assume that ( q (1)7 , , q (2)7 , , q (3)7 , ) = 0. Then, Q = (1 : 0 : 0 : 0) ∈ sing( S ) is a node iff thematrix G has maximal rank, i.e. Q (1) , Q (2) , Q (3) are linearly independent. Moreover, we have(0 : . . . : 0 : 1) ∈ sing( X ).Suppose that rank( G ) <
3. Then, the last row in a matrix obtained as a non-trivial linearcombination of the matrices q , q , q vanishes, which means that the point (0 : . . . : 0 : 1) is asingularity of a quadric that belongs to span( { Q , Q , Q } ). In particular, the quadric in questiondoes not coincide with Q . The latter is impossible by Lemma 1.4.b. Contradiction. (cid:3) In the rank-6 case we have the following characterization.
Lemma 4.2.
Let Q be a rank- quadric in the web W . a) The quadric Q is a node of S iff sing( Q ) * Q for all Q = Q , Q ∈ W . b) Q is an A m singularity, where m ≥ , iff sing( Q ) ∩ Π = ∅ and there exists a quadric Q ∈ W , Q = Q such that sing( Q ) ⊂ Q . c) The quadric Q is a double point of the surface S .Proof. As in the proof of Lemma 4.1 we change the coordinates in such a way that q = I . Then,the line sing( Q ) is the set of zeroes of the coordinates x , . . . , x . Let h· , ·i − be the bilinear formon C given by the formula:(29) h ( q (1)6 ,, , q (1)6 , , q (1)7 , ) , ( q (2)6 ,, , q (2)6 , , q (2)7 , ) i − := 1 / · ( q (1)6 ,, · q (2)7 , + q (1)7 , · q (2)6 ,, − q (1)6 , q (2)6 , )and let H := [ h i,j ] i,j =1 , , , where h i,j := h ( q ( i )6 ,, , q ( i )6 , , q ( i )7 , ) , ( q ( j )6 ,, , q ( j )6 , , q ( j )7 , ) i − . By direct computationwe have(30) det( q + X k =1 µ k · q k ) = (( µ , µ , µ ) H ( µ , µ , µ ) T ) + (terms of degree ≥ . a) Observe that, by (30), the quadric Q is a node of S iff rank( H ) = 3.( ⇒ ): Suppose that there exists a quadric Q = Q , Q ∈ W such that sing( Q ) ⊂ Q . If Q is givenby the matrix [ q i,j ] i,j =0 ,..., , then q , , q , , q , vanish, which yields that rank( H ) < ⇐ ): If rank( H ) <
3, then we can find a matrix q = [ q i,j ] i,j =0 ,..., such that q ∈ span( { q , q , q } )and the entries q , , q , , q , vanish. The latter means that the quadric Q given by q contains theline sing( Q ). We have Q = Q because Q / ∈ span( { Q , Q , Q } ). b) By part a) we can assume that sing( Q ) ⊂ Q , which implies that the entries q (1)6 , , q (1)6 , , q (1)7 , ofthe matrix q vanish. Moreover, by (30), the quadric Q is an A m singularity, where m ≥
2, iffrank( H ) = 2 (see e.g. [9, Prop. 8.14]).( ⇒ ): Suppose that P ∈ sing( Q ) ∩ Π. Then P ∈ sing( X ) and there exists a quadric in the pencilspan( { Q , Q } ) that meets the line sing( Q ) only in the point P . In particular we can assume that Q ∩ sing( Q ) = { P } and P := (0 : . . . : 0 : 1). The latter yields q (2)6 ,, = 1 and q (2)6 ,, = q (2)7 ,, = 0 . Furthermore, since P ∈ Q we have q (3)7 , = 0. Then(( µ , µ , µ ) H ( µ , µ , µ ) T ) = − ( q (3)6 , ) · µ , which implies that Q is not an A m singularity of the octic surface S .( ⇐ ): By Lemma 4.2.a we have rank( H ) ≤
2, so it suffices to show that rank( H ) / ∈ { , } . ssume that rank( H ) = 1. This means that(31) rank (cid:20) h , h , h , h , (cid:21) = 1 . Suppose that the vectors ( q (2)6 ,, , q (2)6 , , q (2)7 , ), ( q (3)6 ,, , q (3)6 , , q (3)7 , ) are linearly independent. By replacing q with an appropriate linear combination of q , q we can assume that the first column of thematrix (31) vanishes. Then, from (29) and h , = 0 we obtain the equality rank([ q (2) i,j ] i,j =6 , ) = 1.Performing an appropriate change of coordinates on the line sing( Q ) we arrive at(32) q (2)6 ,, = 1 and q (2)6 ,, = q (2)7 ,, = 0 . Then, the equality h , = 0 yields q (3)7 , = 0. The latter implies that(0 : . . . : 0 : 1) ∈ sing( Q ) ∩ sing( X ) . Finally, the assumption [A1] gives P ∈ sing( Q ) ∩ Π.Suppose that (31) holds and the vectors ( q (2)6 ,, , q (2)6 , , q (2)7 , ), ( q (3)6 , , q (3)6 , , q (3)7 , ) are linearly dependent.Then, we can assume that the entries q (2)6 , , q (2)6 , , q (2)7 , vanish, which implies sing( Q ) ⊂ Q . Finally,since the line sing( Q ) is contained in the quadrics Q , Q , each point in the intersection Q ∩ sing( Q ) is a singularity of X . By [A1] we have sing( Q ) ∩ Π = ∅ .In the same way the equality rank( H ) = 0 implies sing( Q ) ∩ Π = ∅ . We omit the details. c) By parts a) and b) we can assume that sing( Q ) ⊂ Q and sing( Q ) ∩ Π = ∅ . Suppose that H = 0. From h , = 0 we obtain (32). Then h , = 0 yields q (3)7 , = 0, and by h , = 0 the entry q (3)6 , vanishes. By replacing q with ( q − q ) we obtain the inclusion sing( Q ) ⊂ Q .To complete the proof we assume, as in Section 1 (see the proof of Remark 1.8), that the planeΠ (resp. the line sing( Q )) is given by vanishing of the coordinates x , . . . , x (resp. x , . . . , x and x , x ). Observe that the point P = (0 : . . . : 0 : 1 : 0 : 0) ∈ sing( Q ) ∩ Π is a singularityof X . Therefore, Lemma 1.4.b yields that the quadrics Q , Q , Q are smooth in P . By directcomputation, there exist v , . . . , v ∈ C such that the intersection of the tangent spaces T P Q ,T P Q , T P Q is parametrized by the map( λ , λ , λ , λ ) ( λ v , λ v , λ v , λ v , λ , λ , λ ) . Substituting the above parametrization to (dehomogenized) Q we see that the tangent cone C P X is contained in union of two 3-planes, so the point P ∈ X is not a node. Contradiction (seeLemma 1.6). (cid:3) Remark . Direct computation with help of [15], gives examples of webs of quadrics such thatthe assumptions [A1], [A2], [A3] [A4] are fulfilled and the quadric Q satisfies the conditions ofLemma 4.2.b. One can check that for generic choice of the quadrics one obtains an A singularityof the discriminant octic S .To complete the description of singularities of S we prove the following lemma. Lemma 4.4.
A quadric Q ∈ W is a point of multiplicity at least on S iff rank( q ) = 5 .Proof. ( ⇒ ): Lemmata 4.1, 4.2 imply that rank( Q ) ≤
5. Remark 1.8 completes the proof.( ⇐ ): Assume that q = I and compute the determinant det( q + P k =1 µ k · q k ). (cid:3) The example below shows that the bound of Remark 1.8 is sharp, and the discriminant octic S can have triple points. xample . We define the following matrices: q := − − −
40 0 − − − − − q := − − − − − − − − −
12 0 0 − − − − − −
20 4 2 3 − − − − − − − − − − − q := − − − − − − − − − − − − − − − − − − − − − − − q := − − − − − − − − − − − − − − − − − − − − − − − − − − − By direct computation with help of [15], the intersection in P of the quadrics defined by the abovematrices satisfies the assumptions [A1], . . . , [A4]. As one can easily see, we have rank( q ) = 5.We put π : X → W to denote the double cover of the web W branched along the discriminantsurface S . We have the following theorem (compare [24, Thm 3.1]). Theorem 4.6.
Assume that [A1], . . . , [A4] hold. a) There exists a (small) resolution ˆ φ : ˜ X → X of singularities of the double octic X suchthat the following diagram commutes: ˜ X φ / / ˆ φ AAAAAAAA P X π > > }}}}}}}} b) Let π be the map induced by the projection from the plane Π (see (9) ) and let σ (resp. ψ )be the blow up defined by (6) (resp. (20) ). Then the composition X σ − ˜ X π −→ X ψ − ˜ X φ −→ X is a birational map between the base locus of the web W and its double cover branched alongthe discriminant surface S . In particular, the base locus X and the discriminant doubleoctic X are birational to the quintic -fold X (see (12) ) that contains Bordiga sextic.Proof. a) Consider Stein factorization of the map φ : ˜ X → P : φ = ˆ φ ◦ φ ′ where φ ′ is finite and ˆ φ has connected fibers. By Cor. 3.2 the map φ ′ is a (ramified) double coverof P . Thm 3.6 and the assumption [A4] imply the equality φ ′ = π . Then the map ˆ φ : ˜ X −→ X is birational (see e.g. [8, p. 11]). Thm 3.6 implies that the set of 1-dimensional fibers of the lattermap coincides with ˆ φ − (sing( X )). This completes the proof.b) We have just shown that the map ˆ φ is birational. The claim follows from Lemma 2.5.a. (cid:3) n the case of the double sextic defined by a net of quadrics that contain a (fixed) line thediscriminant curve has only nodes as singularities (see [7, Thm 3.3]).In the corollary below we discuss the singularities of the discriminant surface S . Corollary 4.7.
Assume that [A1], . . . , [A4] hold. a) The equality P P ∈ sing( X ) ( µ ( P, X ) + 1) = 188 holds, where µ ( P, X ) stands for the Milnornumber of X in the point P . b) A quadric Q ∈ W is a singularity of S of the type given in the first column of the tablebelow iff it satisfies the conditions listed in the other columnType of singularity Conditions rank( q ) smooth point Q ) ∩ X = ∅ A Q ) ∩ X = ∅ { Q ∈ W : Q = Q , sing( Q ) ⊂ Q } = ∅ A m , m ≥ , m odd Q ) ∩ Π = ∅ { Q ∈ W : Q = Q , sing( Q ) ⊂ Q } 6 = ∅ double point of corank Q ) ∩ Π = ∅ { Q ∈ W : Q = Q , sing( Q ) ⊂ Q } 6 = ∅ k -fold point, k ≥ Proof. a) To compute the sum of Milnor numbers of singularities of X we compare topologicalEuler numbers of ˜ X and X . By the assumption [A3] and Lemma 2.7 we have e ( ˜ X ) = − P is 304, so [10,Cor. 5.4.4] implies e ( X ) = −
296 + P P ∈ sing( X ) µ ( P, S ). Observe that in our set-up the equality µ ( P, S ) = µ ( P, X ) holds. From Thm 4.6.a we get(33) −
108 + X ) = −
296 + X P ∈ sing( X ) µ ( P, X ) . that yields the claim.b) By Thm 4.6.b and [28, Cor. 1.16] the octic S has no A m points with m even. The claim followsnow directly from Lemmata 4.1, 4.2, and Lemma 4.4. (cid:3) Remark . Under the assumptions [A1], . . . , [A4] the following inequality holds { P ∈ sing( S ) : P is not an A m point, where m ≥ } ≤ . Proof.
By Lemmata 4.1, 4.2 each double point Q ∈ sing( S ) that is not an A m singularity is asingular quadric and its singular locus meets the plane Π. The same holds for rank-5 quadrics inthe web W (see Thm 3.6). Therefore, the inequality results from Remark 3.7. (cid:3) Final remarks: a) According to [21, Thm 4.1] the normal bundle a smooth rational curve that iscontracted on a 3-fold is one of the following: ( O P ( − ⊕ O P ( − O P ( − ⊕ O P ), ( O P ( − ⊕O P (1)). Remark 4.3 and Ex. 4.5 show that all such bundles can come up in our set-up. For theconditions imposed on the equation of a (smooth) 3-fold quintic in P by the normal bundle of acontracted curve the reader should consult [20, App. A, B].b) Assume that all singularities of S are A-D-E points. By [3, Thm 1.1] the Hodge diamond of anysmall K¨ahler resolution of the double octic X coincides with the one given in Lemma 1.7. In view of[29, Cor. 5.1] and [ibid., Prop. 6.1], the latter implies that the assumptions [A1], . . . , [A4] determineposition of singularities of S with respect to sections of O P (8) (compare [24, Prop. 2.13]). ) In Thm 3.6 we describe components of Φ − ( y ) when rank( b ( y )) = 2. Since all singularities of X admit a small resolution, [25, Thm 5.5] can be applied to obtain a more precise description ofsuch fibers. We omit details because of lack of space. Acknowledgement.
Part of this paper was written during the second author’s stay at the Depart-ment of Mathematics of N¨urnberg-Erlangen University. The second author would like to thankProf. W. P. Barth for his help and encouragement.
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