Birationally rigid finite covers of the projective space
aa r X i v : . [ m a t h . AG ] J a n Birationally rigid finite coversof the projective space
A.V.Pukhlikov
In this paper we prove birational superrigidity of finite coversof degree d of the M -dimensional projective space of index 1,where d > M >
10, with at most quadratic singulari-ties of rank >
7, satisfying certain regularity conditions. Upto now, only cyclic covers were studied in this respect. Theset of varieties with worse singularities or not satisfying theregularity conditions is of codimension > ( M − M − Key words: maximal singularity, linear system, birational map, Fano variety, self-intersection, hypertangent divisor.14E05, 14E07
Introduction
Let us fix integers d > l >
2, where( d, l ) = (5 , M = ( d − l , so that M >
10. In the present paper we study d -sheeted covers of the complex projective space P = P M with at most quadraticsingularities of rank >
7, which are Fano varieties of index 1. Such covers have aconvenient presentation: let P = P (1 , . . . , | {z } M +1 , l ) = P (1 M +1 , l )be the weighted projective space with homogeneous coordinates x , . . . , x M , ξ , where x i are of weight 1 and ξ is of weight l . Furthermore, let F ( x ∗ , ξ ) = ξ d + A ( x ∗ ) ξ d − + . . . + A d ( x ∗ )be a (quasi)homogeneous polynomial of degree dl , that is, A i ( x , . . . , x M ) is a ho-mogeneous polynomial of degree il for i = 1 , . . . , d . The space F = d Y i =1 H ( P , O P ( il ))1arametrizes all such polynomials. If the hypersurface V = { F = 0 } ⊂ P has at most quadratic singularities of rank >
7, then the set Sing V of singularpoints is of codimension > V , so that by Grothendieck’s theorem [1] the variety V is factorial. Since the property to have at most quadratic singularities of rank > r is stable with respect to blow ups (see [2, Subsection 3.1]), the singularities ofthe variety V are terminal. NowPic V = Z H, K V = − H, where H is the class of a hyperplane section, so that V is a primitive Fano varietyof index 1.Let o ∗ = (0 : . . . : 0 : 1) = (0 M +1 : 1) ∈ P be the unique singular point of theweighted projective space P . Obviously, o ∗ V . Consider the projection π P : P \{ o ∗ } → P ,π P (( x : . . . : x M : ξ )) = ( x : . . . : x M ) “from the point o ∗ ”. Obviously, π = π P | V : V → P is a d -sheeted ramified cover of the projective space. (In particular, H is the π -pullback of the class of a hyperplane in P onto V .)Now let us state the main result of the paper. We identify a polynomial F ∈ F and the corresponding closed set { F = 0 } , which enables us to write V ∈ F . Theorem 0.1.
There is a Zariski open subset F reg ⊂ F , such that: (i) every V ∈ F reg is a factorial Fano variety of index 1 with terminal singulari-ties, (ii) the inequality codim(( F \F reg ) ⊂ F ) >
12 ( M − M −
5) + 1 , holds, (iii) every variety V ∈ F reg is birationally superrigid. Corollary 0.1.
Let V ∈ F reg . The following claims are true. (i) Every birational map χ : V V ′ onto a Fano variety V ′ with Q -factorialterminal singularities and Picard number 1 is a (biregular) isomorphism. (ii) There is no rational dominant map V S onto a positive-dimensionalvariety S , the fibre of which has negative Kodaira dimension. Therefore, on V thereare no structures of a rationally connected fibre space and Mori fibre space over apositive-dimensional base. In particular, V has no structures of a conic bundle and V is non-rational. The groups of birational and biregular automorphisms of the variety V co-incide: Bir V = Aut V. Proof.
All these claims are the standard implications of the birational super-rigidity, see [3, Chapter 2, Section 1].Note that every automorphism of the variety V ∈ F reg is induced by an auto-morphism of the ambient weighted projective space P . In Subsection 0.3 we give a list of knownresults about birational superrigidity of finite covers of index 1. All previous resultswere about cyclic covers (for cyclic covers the standard procedure of constructinghypertangent divisors worked well [3, Chapter 3], whereas in the case of an arbitrarycover this is not the case).In § F reg . This definition includes severalconditions, one of which is the condition to have at most quadratic singular pointsof rank >
7. Besides, we need at every point o ∈ V a certain regularity condition ,which is similar, but not identical to the usual regularity conditions on which thetechnique of hypertangent divisors is based.In § V , containing a fixed point o and notregular at that point. After that it is not difficult to globalize the estimate for thecodimension.In § V . Assumingthat the claim (iii) of Theorem 0.1 is not true, we obtain the existence of a mobilelinear system Σ ⊂ | nH | with a maximal singularity. In order to prove the birationalsuperrigidity, we have to exclude all possible types of maximal singularities. Themain technical ingredients are the 8 n -inequality for a non-singular point o of thehypersurface V and the recently discovered generalized 4 n -inequality for a completeintersection singularity, and, of course, the hypertangent divisors. It is not possibleto apply the well known technique of hypertangent divisors directly to non-cycliccovers; the essence of this paper is precisely to modify that technique, applying itnot to the variety V it self, but to its intersection with another hypersurface in theweighted projective space. The double covers ofthe projective space of index 1 are the ideal objects for the theory of birationalrigidity, due to their low degree. Soon after the classical paper [4], the birationalsuperrigidity of non-singular double covers of the projective space P , branchedover a sextic (“sextic double solids”) was shown in [5]. In the arbitrary dimensionthe birational superrigidity of non-singular double spaces of index 1 was proved in[6], and with certain types of singularities in the papers [7, 8, 9, 10, 11]. The cycliccovers of arbitrary degree were studied in [12] in a much more general context, triplecyclic covers with double points were considered in [13]. However, up to the presentpaper, non-cyclic covers were never studied. The reason was that the technique of3ypertangent divisors can not be applied directly to non-cyclic covers, because inthe weighted projective presentation there is a coordinate of weight >
2. The aimpf the present paper is to overcome this issue.The author thanks The Leverhulme Trust for the support of this work (ResearchProject Grant RPG-2016-279).The author is also grateful to the colleagues in the Divisions of Algebraic Geom-etry and Algebra of Steklov Mathematical Institute for the interest to his work, andalso to the colleagues-algebraic geometers at the University of Liverpool for generalsupport.
In this section we carry out some preliminary work, which we need to study finitecovers of the projective space. In Subsection 1.1 we consider the systems of affine andhomogeneous coordinates on P and hypersurfaces in P that “replace” hyperplanesof the ordinary projective space. In Subsection 1.2 we consider in more details thelocal equation of the hypersurface V ∈ F with respect to the affine coordinates ona suitable open subset in P . On that basis, in Subsection 1.3 we state the regularityconditions, defining the subset F reg ⊂ F . The claim (i) of Theorem 0.1 followsimmediately from the statement of these conditions. Let f ( z ∗ ) = q µ ( z ∗ ) + . . . + q N ( z ∗ )be a polynomial in the variables z , . . . , z M , decomposed into homogeneous compo-nents q i of degree i > f (0 , . . . ,
0) = 0). Set o = (0 , . . . , ∈ A Mz . Definition 1.1.
The affine hypersurface { f = 0 } is k - regular at the point o ,where µ k N , if the homogeneous polynomials q µ , . . . , q k form a regular sequence in O o, A M .Obviously, the condition of k -regularity at the point o means that the system ofequations q µ = . . . = q k = 0defines a closed subset (a cone) of codimension k − µ + 1 in A M . This conditions ismeaningful only for k − µ + 1 M .Now let us consider hypersurfaces in the weighted projective space P . Proposition 1.1.
For every homogeneous polynomial γ ( x , . . . , x M ) of degree l the equation ξ = γ ( x ∗ ) defines a hypersurface R γ ⊂ P that does not contain thepoint o ∗ = (0 M +1 : 1) . The projection π P | R γ is an isomorphism of R γ and P = P M . Proof:
This is obvious. 4n the affine chart { x = 0 } ⊂ P with the natural affine coordinates z i = x i /x and y = ξ/x l the projection π P takes the form of the usual projection A M +1 z ,...,z M ,y → A Mz ,...,z M , ( z , . . . , z M , y ) ( z , . . . , z M ) , where A Mz is the affine chart { x = 0 } in P . Obviously, the affine hypersurface R γ ∩{ x = 0 } is given by the equation y = g ( z , . . . , z M ), where g ( z ∗ ) = γ (1 , z , . . . , z M ).Now let us consider the singularities of the hypersurface V = { F = 0 } and itssections. Taking into account the (quasihomogeneous) Euler identity, we get thatthe closed set Sing V of the singular points of V is given by the system of equations ∂F∂x = . . . = ∂F∂x M = ∂F∂ξ = 0 . Proposition 1.2.
Let R ∩ P be either the π P -preimage of a hyperplane in P , ora hypersurface R γ , defined above. Thendim Sing( V ∩ R ) dim Sing V + 1 . . Proof. If R is the π P -preimage of a hyperplane, we repeat word for word the wellknown argument for the usual projective space. Let R = R γ . Then the intersection V ∩ R is isomorphic to the hypersurface F ( x , . . . , x M , γ ( x , . . . , x M )) = 0in the projective space P , the singularities of which are given by the equations ∂F∂x i + ∂F∂ξ · ∂γ∂x i = 0 , i = 0 , . . . , M. Therefore the intersection of Sing( V ∩ R ) ⊂ P with the hypersurface n ∂F∂ξ = 0 o iscontained in Sing V . Q.E.D. for the proposition. Definition 1.2.
A non-singular point o ∈ V is a point of the first type , if ∂F∂ξ ( o ) = 0, and of the second type , if ∂F∂ξ ( o ) = 0.At a point of the second type the hypersurface, given by the equation M X i =0 x i ∂F∂x i ( o ) = 0 , is the natural “tangent hyperplane” T o V ⊂ P . At a point of the first (main) typethere is a whole family of candidates for the role of the tangent hyperplane, and theyare given by sections of the sheaf O P ( l ), because they include the coordinate ξ : theyare of the form R γ for suitable polynomials γ . In order to present the local equationof the hypersurface V more precisely, one needs to consider affine coordinates.5 .2. Affine coordinates. Let o ∈ V be a point. Let us choose projectivecoordinates on P in such a way that o = (1 : 0 : . . . : 0 : β ) = (1 : 0 M : β ) for some β ∈ C . Replacing the coordinate ξ by ξ ′ = ξ − βx l , we may assume that β = 0.Now in the affine coordinates z , . . . , z M , y we have o = (0 , . . . ,
0) and V ∩ { x = 0 } is given by the equation f = 0, where f = y d + a ( z ∗ ) y d − + . . . + a d − ( z ∗ ) y + a d ( z ∗ ) , the polynomial a i ( z ∗ ) has degree il . Write down a i ( z ∗ ) = a i, + a i, ( z ∗ ) + . . . + a i,il ( z ∗ ) , where a i,j is a homogeneous polynomial of degree j . In particular, a d, = 0. Thepoint o ∈ V is non-singular if and only if the linear form a d − , y + a d, ( z ∗ )is not identically zero, and in that case the point o is a point of the first type,if a d − , = 0, and of the second type, if a d − , = 0. If o ∈ V is a non-singularpoint of the first type, then z , . . . , z M is a coordinate system on the (affine) tangentspace T o V . In particular, every linear subspace Λ ⊂ T o V is given by a system oflinear equations that depend only on z ∗ , and for any non-zero linear form h ( z ∗ ) theintersection V ∩ { h = 0 } is non-singular at the point o . (The last intersection can be understood also as theintersection with the hyperplane { h ( x , . . . , x M ) = 0 } in P .)Now if o ∈ V is a non-singular point of the second type, then z , . . . , z M , re-stricted to T o V , are linearly dependent. A typical linear subspace Λ ⊂ T o V is givenby a system of linear equations, one of which is of the form y − h ( z ∗ ) = 0 (wherethe form h can be identically zero), and the rest depend only on z ∗ .Somewhat abusing the notations, we will use the same symbol V both for theoriginal hypersurface and for its affine part V ∩ { x = 0 } . For a linear form h ( z ∗ )(possibly, identically zero) the symbol V h means the intersection V ∩ R γ , where γ = h ( x , . . . , x M ) x l − in the system of coordinates ( x : . . . : x M : ξ ′ ),described above. For the affine part of this variety we will use the same symbol V h .Without special comments we consider V h to be embedded in P or A M , dependingon the situation.The intersection with a hyperplane { h = 0 } (for a non-zero form h ) we denoteby the symbol of restriction | { h =0 } ; again, we use this notation both in the affine andprojective context. Let us formulate the regularity conditionsfor a point o ∈ V . These conditions are slightly different, depending on whether thepoint o is a non-singular point of the first or second type or a singularity.6ssume that the point o ∈ V is non-singular.(R1) The hypersurface V h is p (3 dl ) / q -regular (at the point o ) for every linearpoint h ( z , . . . , z M ).(We use the notations and conventions of Subsection 1.2.)If the point o ∈ V is non-singular of the second type, that is, a d − , = 0, then,apart from the condition (R1), one more condition is needed. Recall that in thatcase the tangent hyperplane T o V is given in the affine coordinates by the equation a d, ( z ∗ ) = 0.(R1.2) For every linear form h ( z , . . . , z M ), which is linearly independent with a d, ( z ∗ ), the hypersurface V | { h =0 } is p (3 dl ) / q -regular.(The last hypersurface is contained in { h = 0 } = A M − .)Now assume that the point o ∈ V is singular. The affine equation of V at thepoint o starts with the quadratic form a d − , y + a d − , ( z ∗ ) y + a d, ( z ∗ ) . (1)(R2) The rank of the last form >
7, and the variety V is p dl/ q -regular at thepoint o . Definition 1.1.
The hypersurface V ∈ F is regular , if at every non-singularpoint the condition (R1) holds, and at every non-singular point of the second typealso the condition (R1.2), and t every singular point the condition (R2) holds.The set of regular hypersurfaces is denoted by the symbol F reg . Since everyhypersurface V ∈ F reg is either non-singular, or has at most quadratic singularitiesof rank >
7, the claim (i) of Theorem 0.1 holds. The claim (ii) of Theorem 0.1 isshown in § The aim of this section is to prove the claim (ii) of Theorem 0.1. In Subsection 2.1we localize the task: we reduce it to a similar problem for a fixed point o ∈ P . InSubsection 2.2 we recall the methods of estimating the violations of the regularitycondition. In Subsection 2.3 we prove the local estimates, completing the work. Fix a point o ∈ P , o = o ∗ . Let F ( o ) ⊂ F be thesubset (hyperplane) of polynomials that vanish at the point o , and F reg ( o ) ⊂ F ( o ) isthe subset of polynomials, satisfying the corresponding regularity condition at thatpoint. Set F non − reg ( o ) = F ( o ) \F reg ( o ) . Obviously,
F \F reg ⊂ [ o ∈ P \{ o ∗ } F non − reg ( o ) ,
7o that, taking into account the equality codim( F ( o ) ⊂ F ) = 1, the claim (ii) ofTheorem 0.1is implied by the following local fact. Proposition 2.1.
The following inequality holds: codim( F non − reg ( o ) ⊂ F ( o )) > ( M − M − M + 1 . It is the last inequality that we will show. For each of the reguarity conditions,stated in Subsection 1.3, we have to check that a violation of that condition imposeson the coefficients of the polynomial
F ∈ F ( o ) at least ( M − M −
5) + M + 1independent conditions. As we will see below, the minimal number of independentconditions correspond to the violation of the condition (R2). We will use two wellknown methods. The first method was used many times, see [3, Chapter 3, Section1]. Let, as in Subsection 1.1, f ( z ∗ ) = q µ ( z ∗ ) + . . . + q N ( z ∗ )be a polynomial in the affine coordinates z , . . . , z M and µ k M + 1 − µ . Bythe symbol P i,M we denote the space of homogeneous polynomials of degree i ∈ Z + in z , . . . , z M , and for i < j by the symbol P [ i,j ] ,M we denote the product j Y a = i P a,M , so that f ∈ P [ µ,N ] ,M . Repeating the arguments of Subsection 1.3 in [3, Chapter 3]word for word, we get that the codimension of the set of polynomials f that do notsatisfy the condition of k -regularity, with respect to the space P [ µ,N ] ,M is not smallerthan the number min µ i k (cid:18) M − µi (cid:19) . Taking into account the well known behaviour of the binomial coefficients, weconclude that this minimum is realized either for i = µ , or for i = k . It is easy tochoose, at precisely which value of these two ones it is realized. Note that if it isknown that the point (0 , . . . ,
0) is non-singular on the hypersurface { f = 0 } , thatis, µ = 1 and q
0, then we may fix q and restrict q i , i >
2, onto the hyperplane { q = 0 } . In that case the codimension is not lower than the number (cid:18) M (cid:19) for k M −
2. We will need only this estimate.The second method is the well known fact that the codimension of the set ofquadratic forms of rank r in the space of all quadratic forms in N variables is( N − r )( N − r + 1)2 . .3. Proof of Proposition 2.1. In the notations of Subsections 1.2, 1.3 letus consider one by one the violations of each regularity conditions at the point o .Consider first the hypersurfaces, non-singular at that point.The hypersurface V = { f = 0 } ⊂ A M +1 is uniquely determined by the set ofpolynomials a ( z ∗ ) , . . . , a d ( z ∗ ) , where deg a i il . The hypersurface V h ⊂ A M for some linear form h ( z ∗ ) is given bythe equation f h = 0, where f h = h ( z ∗ ) d + a ( z ∗ ) h ( z ∗ ) d − + . . . + a d ( z ∗ ) . Fix a form h , such that V h is non-singular at the point o . Since the polynomial a d ( z , . . . , z M ) of degree dl is arbitrary, in the presentation f h = q ( z ∗ ) + q ( z ∗ ) + . . . + q dl ( z ∗ ) , where deg q i = i , the homogeneous components q i are arbitrary and do not depend oneach other. Therefore we may apply the first method of estimating the codimension,described in Subsection 2.2. By assumption, q
0, so that we fix the hyperplane { q = 0 } and restrict the polynomial f h onto this hyperplane: f h | { q =0 } = q + . . . + q dl . Since p (3 dl ) / q 6 M −
2, the codimension of the set of polynomials f h | { q =0 } that donot satisfy the condition of p (3 dl ) / q -regularity, is (cid:0) M (cid:1) . The same is the codimensionof the set of non-regular polynomials f h . Since h varies in an M -dimensional family,the codimension of the set of polynomials f , violating the condition (R1), is (cid:18) M (cid:19) − M. In a similar way we estimate the codimension of the set of polynomials, violatingthe condition (R1.2): the non-regularity of the hypersurface V | { h =0 } gives (cid:18) M − (cid:19) − M + 1independent conditions for f . (The additional codimension +1 comes from theequality a d − , = 0 for a non-singular point of the second type.)Now let us consider the hypersurfaces V which are singular at the point o . Sincein that case a d − , = 0 and a d, ( z ∗ ) ≡
0, we have M + 1 additional independentconditions for f . If the rank of the quadratic form (1) does not exceed 6, we obtain,therefore, (cid:18) M − (cid:19) + M + 19ndependent conditions for f .Obviously, p dl/ q 6 M −
2. Thus if V is a non-regular hypersurface, we obtain (cid:18) M + 12 (cid:19) + M + 1independent conditions for f .Comparing the results obtained above and choosing the smallest one for M > Remark 2.1.
We excluded the option ( d, l ) = (5 ,
2) from consideration for theonly reason: it violates the uniformity of the statement of the claim (ii) of Theorem0.1. For M = 8 the minimum of the codimension corresponds to the violation of thecondition (R1.2) and is equal to 14. With this modification the claims of Theorem0.1 are true for the values ( d, l ) = (5 , In this section we prove the birational superrigidity of a regular hypersurface V . InSubsection 3.1 we recall the key concept of a maximal singularity and exclude certaintypes of maximal singularities. The remaining types of singularities are classified,after that we start to exclude maximal singularities of “general position” (Proposi-tion 3.2). In order to complete this work, we need the technique of hypertangentdivisors, which is recalled in Subsection 3.2. Finally, in Subsection 3.3 we excludemaximal singularities of all remaining types, which completes the proof of Theorem0.1. Let V ∈ F reg be a fixed regular variety. Assumethat V is not birationally superrigid. It is well known that in this case on V thereis a mobile linear system Σ ⊂ | nH | with a maximal singularity: for some birationalmorphism ϕ : e V → V , where e V is a non-singular projective variety, and some ϕ -exceptional prime divisor Q ⊂ e V the Noether-Fano inequality holds:ord Q ϕ ∗ Σ > n · a ( Q, V ) , where a ( Q, V ) is the discrepancy of Q with respect to V (see, for instance, [3,Chapter 2]). Let B = ϕ ( Q ) ⊂ V be the centre of the maximal singularity Q . If B Sing V , then the inequality mult B Σ > n (2)holds. Proposition 3.1.
The codimension of the subvariety B ⊂ V it at least 5. roof. Assume the converse:codim( B ⊂ V ) . Then B Sing V , so that the inequality (2) holds. For a general polynomial g ( x , . . . , x M ) of degree l by Bertini’s theoremcodim(Sing g ⊂ V g ) = codim(Sing V ⊂ V ) > . Set B g = B ∩ V g and Σ g = Σ | V g , where Σ g ⊂ | nH g | is a mobile linear system on V g ,and H g is the class of a hyperplane section of the hypersurface V g ⊂ P . Let P ⊂ P be a general 6-plane. The variety V P = V g ∩ P is a non-singular hypersurface in P ∼ = P . On V P there is a mobile linear systemΣ P = Σ g | V P ⊂ | nH P | , where Pic V P = Z H P , and moreover, mult B ∩ P Σ P > n .However, B ∩ P is positive-dimensional, so that the last inequality can not be true(this is a well known fact for any non-singular hypersurface in the projective space,and in fact it is sufficient for the linear system Σ P to be non-empty, see, for instance,[3, Chapter 2, Section 2]). We obtained a contradiction which completes the proofof the proposition.Starting from theis moment, we assume that codim( B ⊂ V ) >
5. Fix a point ofgeneral position o ∈ B . There are three options:(1.1) the point o Sing V is non-singular of the first type,(1.2) the point o Sing V is non-singular of the second type,(2) the point o ∈ Sing V is a quadratic singularity.We must exclude each of them.Let us consider first the nonsingular cases. Set Z = ( D ◦ D ) to be the self-intersection of the linear system Σ, where D , D ∈ Σ are general divisors. Theeffective cycle Z ∼ n H of codimension 2 satisfies the (classical) 4 n -inequalitymult o Z > n and the 8 n -inequality mult o Z + mult Λ Z + > n , (3)here Λ ⊂ E is some linear subspace of codimension 2, E = ε − ( o ) ⊂ V + is theexceptional divisor of the blow up ε : V + → V of the point o , E ∼ = P M − , see, forinstance, [3, Chapter 2, Sections 2,4]. Proposition 3.2.
The case (1.1) does not realize.
Proof.
Assume the converse: the case 1.1 takes place. In the affine coordinates(see Subsection 1.2) the tangent hyperplane T o V is given by the equation a d − , y + a d, ( z ∗ ) = 0, where a d − , = 0, so that z , . . . , z M is a system of coordinates on T o V and ( z : . . . : z M ) is a system of homogeneous coordinates on E = P ( T o V ). LetΛ = { h ( z ∗ ) = h ( z ∗ ) = 0 } , h , h are linearly independent forms. Let h = λ h + λ h be a general formin the pencil.Since deg Z = deg H Z = dn , the inequality (3) can be re-written in the formmult o Z + mult Λ Z + > d deg Z. (4)This inequality is linear in Z . Therefore, there is an irreducible component ofthe cycle Z , satisfying this inequality. In order not to make the notations toocomplicated, let us simply assume that the cycle Z itself is an irreducible subvariety. Lemma 3.1.
The subvariety V ∩ { h ( z ∗ ) = h ( z ∗ ) = 0 } is irreducible, non-singular at the point o and not equal to Z . Remark 3.1.
In the statement of the lemma, the coordinates z ∗ are consideredas affine coordinates on A M +1 z ∗ ,y . We also used z ∗ above as homogeneous coordinateson E . The subvariety V ∩ { h = h = 0 } is understood as a projective subvarietyin P , that is, the closure of the corresponding affine set. These changes from affinenotations to projective ones are obvious and do not require special explanations. Proof of the lemma.
Non-singularity at the point o is obvious, irreducibilityfollows from Proposition 1.2 and the assumption on the rank of the quadratic points(the codimension of the singular set Sing V is at least 6). Finally, the subvariety V ∩ { h = h = 0 } has degree d and multiplicity 1 at the point o , and its stricttransform on V + has multiplicity precisely 1 along Λ, so that this subvariety doesnot satisfy the inequality (4). Therefore, it is not equal to Z . Q.E.D. for the lemma.Set g ( z ∗ ) = − a d − , a d, ( z ∗ ) + h ( z ∗ ) . By the lemma, Z V g , so that the effective cycle of scheme-theoretic intersection( Z ◦ V g ) is well defined. It satisfies the inequalitymult o ( Z ◦ V g ) > dl deg( Z ◦ V g )(since V g ∼ lH and V + g contains Λ by the choice of the form h ). By the linearityof the last inequality there is an irreducible subvariety Y ⊂ V g of codimension 2 (acomponent of the effective cycle ( Z ◦ V g )), satisfying the inequalitymult o deg Y > dl (5)(the symbol mult o / deg means, as usual, the ratio of the multiplicity at the point o to the H -degree). The subvariety Y is contained in V g , which is an irreducible hy-persurface of degree dl in the projective space P . This hypersurface by the condition(R.1) is k -regular at the point o , where k = p (3 dl ) / q .12 .2. The technique of hypertangent divisors. We continue our proof ofProposition 3.2. Considering V g as a hypersurface in the projective space P , let usdecompose its equation into components, homogeneous in z ∗ : f g = q + q + . . . + q dl . Let us construct the hypertangent linear systems on V g at the point o :Λ i = ( i X j =1 s i − j ( q + . . . + q j ) | V g = 0 ) , where s a independently from each other run through the space P a,M , see the detailsand examples in [3, Chapter 3]. By the condition (R1),codim o (Bs Λ i ⊂ V g ) = i for i = 1 , . . . , k −
1. Now, applying the technique of hypertangent divisors in theusual way, let us construct a sequence of irreducible subvarieties Y , Y , . . . , Y k − ofcodimension codim( Y i ⊂ V g ) = i , where Y = Y and the last variety in this sequencesatisfies the inequalitymult o deg Y k − > dl · · · . . . · kk − > , since 8 k > dl by assumption. This gives the required contradiction and completesthe proof of Proposition 3.2. In order to exclude the cases(1.2) and (2), we repeat the arguments in the proof of Proposition 3.2 with somemodifications. We only consider in detail those modifications.
Proposition 3.3.
The case (1.2) does not realize.
Proof.
Assume the converse. The tangent hyperplane T o V is given by theequation a d, ( z ∗ ) = 0. For the subspace Λ ⊂ E there are two options:(1.2.1) Λ is given by the equations y − h ( z ∗ ) = 0 , h ( z ∗ ) = 0 , where the forms h and a d, are linearly independent. If h , h are linearly dependent,then we may assume that h ≡ h ( z ∗ ) = 0 , h ( z ∗ ) = 0 , where the forms h , h and a d, are linearly independent.Assume first that the case (1.2.1) takes place.13 emma 3.2. The subvariety V ∩ { y − h ( z ∗ ) = h ( z ∗ ) = 0 } is irreducible, non-singular at the point o and not equal to Z . Proof is completely similar to the proof of Lemma 3.1 and we do not give it.Now set g ( z ∗ ) = h ( z ∗ ) + λh ( z ∗ ) for a sufficiently general value λ ∈ C . Now thecontradiction is obtained by word for word the same arguments as in the case (1.1).We have shown that the option (1.2.1) does not realize.Assume now that the case (1.2.2) takes place. Let h = λ h + λ h ∈ h h , h i be a general form. Again it is easy to check that Z V | { h =0 } , so that the effective cycle Z | { h =0 } = ( Z ◦ V | { h =0 } ) has codimension 2 on the irre-ducible hypersurface V | { h =0 } ⊂ V and satisfies the inequalitymult o Z | { h =0 } > d deg Z | { h =0 } . By the linearity of this inequality we may assume that the cycle Z | { h =0 } is an irre-ducible subvariety. Now if Z | { h =0 } V , then set Y to be the irreducible component of the effective cycle ( Z | { h =0 } ◦ V ) withthe maximal value of the ratio mult o / deg. If, on the contrary, Z | { h =0 } ⊂ V , thenset Y = Z | { h =0 } . In any case, Y ⊂ V | { h =0 } is a subvariety of codimension 1 or 2,satisfying the inequality (5). Now we argue in the word for word the same way asin the proof of Proposition 3.2, using the condition (R1.2). Q.E.D. for Proposition3.3. Proposition 3.4.
The case (2) does not realize.
Proof.
Assume the converse: B ⊂ Sing V , so that the point o is a quadraticsingularity of the variety V . By the condition (R2) the point o ∈ V is a quadraticsingularity of rank >
7, so that we can apply the generalized 4 n -inequality [14] andconclude that mult o Z > n · mult o V = 8 n , so that mult o Z > d deg Z. Now we argue as in the proof of Proposition 3.3: by the linearity of the last inequalityin Z we may assume that Z is an irreducible subvariety of codimension 2. If Z V ,then we set Y to be a component of the effective cycle ( Z ◦ V ) with the maximalvalue of the ratio mult o / deg. If Z ⊂ V , then we set Y = Z . Now we complete14he proof in the word for word same way as the proof of Proposition 3.3. The onlydifference is that now the variety V is sibgular at the point o : it has a quadraticsingularity (of rank > V ⊂ P is given (in the affinecoordinates) by the equation q + q + . . . + q dl = 0 , then the hypertangent systems are of the formΛ i = ( i X j =2 s i − j ( q + . . . + q j ) | V = 0 ) , where i > V to be k -regular, where k = p dl/ q , leads in the notations of the proof of Proposition 3.2 to the inequalitymult o deg Y k − > dl · · · . . . kk − > , since 2 k > dl . Q.E.D. for Proposition 3.4.Proof of Theorem 0.1 is complete. Remark 3.2.
From the very beginning we assumed that d >
5. The doublecovers are cyclic covers and their superrigidity is well known, see Subsection 0.3. If d ∈ { , } , then the birational superrigidity follows just from the condition, that thevariety V has at most quadratic singularities of rank >
7. Indeed, if B ⊂ V is thecentre of an infinitely near maximal singularity and codim( B ⊂ V ) >
3, then either B Sing V and the usual 4 n -inequality holds:mult B Z > n , or B ⊂ Sing V and the generalized 4 n -inequality holds, which in this case takes theform of the estimate mult B Z > n . In any case, mult B Z > deg H Z , which is impossible (the linear system | H | is freeand defines the finite morphism π : V → P ). Thus for d = 3 or 4 the superrigidityholds in essentially weaker assumptions for the variety V . References [1] Call F. and Lyubeznik G., A simple proof of Grothendieck’s theorem on theparafactoriality of local rings, Contemp. Math. (1994), 15-18.[2] Pukhlikov, A. V., Birationally rigid Fano fibrations. II. Izv. Ross. Akad. NaukSer. Mat. (2015), no. 4, 175-204; Izvestiya: Mathematics (2015), no. 4,809–837. 153] Pukhlikov Aleksandr, Birationally Rigid Varieties. Mathematical Surveys andMonographs , AMS, 2013.[4] Iskovskikh V. A. and Manin Yu. I., Three-dimensional quartics and counterex-amples to the L¨uroth problem, Math. USSR Sb. (1971), no. 1, 140-166.[5] Iskovskikh V. A., Birational automorphisms of three-dimensional algebraic va-rieties, J. Soviet Math. (1980), 815-868.[6] Pukhlikov A. V., Birational automorphisms of a double space and a doublequadric, Math. USSR Izv. (1989), 233-243.[7] Pukhlikov A. V., Birational automorphisms of double spaces with singularities,Journal of Math. Sciences (1997), No. 4, 2128-2141.[8] Cheltsov I. A., A double space with a double line. Sbornik: Mathematics (2004), No. 9-10, 1503-1544.[9] Cheltsov I. A., On nodal sextic fivefold. Math. Nachr. (2007), No. 12,1344-1353.[10] Cheltsov I. A. and Park J., Sextic double solids. In: Cohomological and geo-metric approaches to rationality problems. Progr. Math. , Birkh¨auser, 2010,75-132.[11] Mullany R., Fano double spaces with a big singular locus, Math. Notes (2010), No. 3, 444-448.[12] Pukhlikov A. V., Birational geometry of algebraic varieties with a pencil ofFano cyclic covers. Pure and Appl. Math. Quart. (2009), No. 2, 641-700.[13] Cheltsov I. A., Birationally super-rigid cyclic triple spaces. Izvestiya: Mathe-matics (2004), No. 6, 1229-1275.[14] Pukhlikov A. V., The 4 n -inequality for complete intersection singularities.Arnold Math. J., (2017), No. 2, 187-196.Department of Mathematical Sciences,The University of Liverpool [email protected]@liverpool.ac.uk