Birational geometry of Fano hypersurfaces of index two

We prove that every non-trivial structure of a rationally connected fibre space on a generic (in the sense of Zariski topology) hypersurface V of degree M in the (M+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M+1)$$\end{document}-dimensional projective space for M≥16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\ge 16$$\end{document} is given by a pencil of hyperplane sections. In particular, the variety V is non-rational and its group of birational self-maps coincides with the group of biregular automorphisms and for that reason is trivial. The proof is based on the techniques of the method of maximal singularities and inversion of adjunction.


Statement of the main result
Fix an integer M ≥ 4. Denote by the symbol P the complex projective space P M+1 . Let V = V M ⊂ P be a non-singular hypersurface of degree M. Obviously, V is a Fano variety of index two: where H is the class of a hyperplane section. On the variety V there are the following structures of a non-trivial rationally connected fibre space: let P ⊂ P be an arbitrary subspace of codimension two, α P : P P 1 the corresponding linear projection, then its restriction Let χ : V Y be a birational map onto the total space of a rationally connected fibre space λ : Y → S. Then dim S ≤ r − 1 and if dim S = r − 1, then there is a linear subspace P ⊂ P of codimension r and a birational map β : P r −1 S such that λ • χ = β • π P , that is, the following diagram commutes Remark 1.1 For d ≤ M − 1 (that is, for r ≥ 3) one can certainly not expect that all structures of a rationally connected fibre space (or of a Fano-Mori fibre space) are linear projections. Already for a hypersurface of index 3 every pencil of quadrics defines a rational map onto P 1 , the fibre of which is a complete intersection of the type 2 · (M − 1) in P M+1 , that is, a Fano variety of index one. Conjectures, similar to Conjecture 1.1 have been discussed informally for some time, but until very recently never published, as there were no methods to approach them realistically. The situation is changing now-see the last section in [7]. The purpose of the present paper is to prove Theorem 1.1. As usual, its claim will be derived from a lot more technical and less visual description of maximal singularities of mobile linear systems on V . However, before explaining the structure of the proof of Theorem 1.1, let us give a precise meaning to the assumption of the hypersurface V being generic in the sense of Zariski topology.

The regularity conditions
Let F = P(H 0 (P, O P (M))) be the space parametrizing hypersurfaces of degree M in P. The local regularity conditions, given below, define an open subset F reg ⊂ F. A separate (but not difficult) problem is to show that for M ≥ 14 the set F reg is non-empty.
Let o ∈ P be an arbitrary point, (z 1 , . . . , z M+1 ) = (z * ) a system of affine coordinates with the origin at the point o and V o a non-singular hypersurface of degree M. It is given by an equation f = 0, where f = q 1 + q 2 + · · · + q M is a non-homogeneous polynomial in the variables z * , q i is its homogeneous component of degree i. Let ⊂ C M+1 be an arbitrary linear subspace of codimension c ∈ {0, 1, 2, 3}, on which q 1 does not vanish identically, that is, ⊂ T o V . We will need the following regularity conditions. where is an arbitrary linear subspace of codimension two in the tangent hyperplane, defines an irreducible reduced closed set. The last (forth) regularity condition is a global one. (R4) The intersection of the hypersurface V with an arbitrary linear subspace P ⊂ P of codimension two has at most isolated quadratic singularities.
The following claim is true.

Theorem 1.2 For M ≥ 16 there exists a non-empty
Zariski open subset F reg ⊂ F, such that every hypersurface V ∈ F reg is non-singular and satisfies the conditions (R1-R3) at every point, and also the condition (R4).
For the proof of Theorem 1.2 see Sect. 2.6.

Plan of the proof of Theorem 1.1
For an arbitrary subspace P ⊂ P of codimension two denote by the symbol V P the blow up of V along the subvariety V ∩ P. For a mobile linear system on V its strict transform on V P denote by the symbol P . Considering instead of its symmetric square, we can always assume that ⊂ |2n H|. Recall that c virt ( ) is the virtual threshold of canonical adjunction [24,Sec. 2.1]. Theorem 1.1 is an easy corollary from the technical fact formulated below. Theorem 1. 3 Assume that M ≥ 16 and V ∈ F reg . If the mobile system ⊂ |2n H| satisfies the inequality then there exists a unique linear subspace P ⊂ P of codimension two, such that the subvariety B = P ∩ V satisfies the inequality mult B > n, (2) whereas for the strict transform P the following equality holds: The integer M ≥ 16, the system and the integer n ≥ 1 are fixed throughout the paper. In its turn, Theorem 1.3 will be derived from the following two key facts. Theorem 1. 4 Assume that for some subvariety B ⊂ V of codimension two the inequality (2) holds. Then B = P ∩ V, where P ⊂ P is a linear subspace of codimension two. Theorem 1.5 Assume that the inequality (1) holds. Then for some irreducible subvariety B of codimension two the inequality (2) holds. Theorems 1.4 and 1.5 are given in the order in which they are shown. Theorem 1.5 (the exclusion of the infinitely near case) is the most difficult to prove. Further work is organized as follows.

Historical remarks and acknowledgements
The result, completely similar to Theorem 1.1, has been shown for Fano double spaces of index two in [23], see also Chapter 8 in [24]. Prior to the paper [23], the only result giving a complete description of the structures of a rationally connected fibre space on a Fano variety of index two, was Grinenko's theorem [10,11] on the Veronese double cone, a very special Fano threefold.
A series of important results on birational geometry of Fano varieties of index two and higher was obtained by other methods: by the transcendent method of Clemens and Griffiths [3] and its subsequent generalizations (see [2]), and also by means of Kollár's technique [15,16]. For the details, see the introduction to the paper [23], where, in particular, the dramatic story of studying the birational geometry of the Veronese double cone and (not completed to this day) studying of the double space of index two is described.
Note that the problem of description of the birational type of Fano varieties of index higher than one was discussed already in the classical paper [13]; Fano himself also worked on the problem (for the cubic threefold V 3 ⊂ P 4 ) [6].
We would also like to draw the reader's attention to the note [25], where the problem of stable rationality was negatively solved for a large class of Fano hypersurfaces of higher index.
The author thanks the referee for numerous suggestions about improving the exposition and, especially, for spotting a miscalculation in the dimension count that proves that a general hypersurface satisfies the regularity condition (R2).
The results presented in this paper and the techniques used in it were discussed by the author in his talks given in 2009-2014 in Steklov Mathematical Institute. The author is grateful to the members of the divisions of Algebraic Geometry and Algebra and Number Theory for the interest in his work. The author also thanks his colleagues in the Algebraic Geometry research group at the University of Liverpool for the creative atmosphere and general support.

Pencils of hyperplane sections
In this section, we prove Theorem 1.1, assuming the claim of Theorem 1.3. After that, Theorem 1.3 is obtained from Theorems 1.4 and 1.5. Finally, we discuss the (routine) proof of Theorem 1.2.

Fano fibre spaces over P 1
Let us prove Theorem 1.1. Let ⊂ |2n H| be the strict transform on V of a free linear system on Y , which is the λ-pull back of a very ample linear system on the base S. Then the inequality (1) holds, because c virt ( ) = 0. The system ⊂ |2n H| is now fixed. Assuming the claim of Theorem 1.3, consider the subspace P ⊂ P of codimension two, such that for B = P ∩ V the inequality (2) holds. Let ϕ : V + → V be the blow up of the subvariety B and E B = ϕ −1 (B) ⊂ V + the exceptional divisor.

Lemma 2.1 (i)
The variety V + is factorial and has at most finitely many isolated double points (not necessarily non-degenerate). (ii) The linear projection π P : P P 1 from the subspace P generates the regular projection π = π P • ϕ : V + → P 1 , the generic fibre of which F t = π −1 (t), t ∈ P 1 is a non-singular Fano variety of index one, and a finite number of singular fibres have isolated double points. (iii) The following equalities hold: where H = ϕ * H for simplicity of notations, K + = K V + is the canonical class of the variety V + , F is the class of the fibre of the projection π, where Proof These claims are obvious by the regularity condition (R4) and the well known factoriality of an isolated hypersurface singularity in the dimension 4 and higher, see [1]. (Note that the smoothness of V is contained in the regularity condition (R1).) Indeed, it follows from (R4) that the section of V by a generic hyperplane, containing the subspace P, in non-singular, whereas it is well known that any hyperplane section of a non-singular hypersurface in a projective space can have at most isolated singularities. As the fibres of π are isomorphic to the sections of V by hyperplanes, containing the subspace P, we conclude that V + has at most finitely many isolated singular points, hence V + is factorial by Grothendieck's theorem [1]. Those isolated singularities are double points as they are double points on the corresponding fibres of π by the condition (R1). The rest is trivial. Q.E.D. for the lemma.
Let + be the strict transform of the system on V + .
Proof For some m ∈ Z + and l ∈ Z we have: where m = 2n−mult B and l = 2(mult B −n) ≥ 2. Thus the threshold of canonical adjunction is By Theorem 1.3, c( + , V + ) = c virt ( ) = 0, so that m = 0 and l = 2n, as we claimed. Q.E.D. for the lemma. Therefore, the mobile linear system is composed from the pencil of hyperplane sections, containing B, which completes the proof of Theorem 1.1.

Mobile systems on the variety V
Assume the claims of Theorems 1.4 and 1.5. Let us prove Theorem 1.3. In the notations of Sect. 1.1 we have to show that for the mobile linear system holds. (This is precisely the claim of Theorem 1.3.) Assume the converse: then the pair (V + , 1 m + ) is not canonical, that is, the linear system + has a maximal singularity. Since a general fibre of the fibre space π : V + → P 1 is a non-singular birationally rigid variety (in fact, every fibre is birationally superrigid, see, for instance, [5]), the centre of every maximal singularity is contained in some fibre F t = π −1 (t). Restricting the linear system + onto such a fibre F = F t , we obtain an effective divisor D ∈ | − m K F |, such that the pair is not canonical (in fact, not log canonical, but we do not use that). For any curve C ⊂ F, C ∩ Sing F = ∅, it is known, see [24,Chapter 2], that mult C D ≤ m, which implies that the centre of every non canonical singularity of the pair (3) is either a point, or a curve, passing through a singularity of F. Furthermore, it is well known [24,Chapter 7], that a smooth point can not be the centre of a non canonical singularity, and the proof of that fact excludes also the case when the centre is a curve (since F has only isolated singularities). Therefore, we may assume that the centre of a maximal (non canonical) singularity of the pair (3) is a singular point o. At this moment, and up to the end of this section, it is convenient to slightly change the notations. We denote the variety F by the symbol W . It is a hypersurface of degree M in P M with an isolated quadratic point o ∈ W . On W there is an effective divisor D ∼ m H, where H is the class of a hyperplane section of W , such that the pair (W, 1 m D) is not canonical at the point o. We have to show that this is impossible, that is, to obtain a contradiction. We do it in several steps, modifying the proof in [22].

Step 1: Effective divisors on quadrics
Let Q ⊂ P M−1 be an irreducible quadric hypersurface of rank ≥5, H Q ∈ Pic Q = ZH Q the class of a hyperplane section and B ⊂ Q an irreducible subvariety, which is not contained entirely in Sing Q.
holds, where D ⊂ Q P is the strict transform of the divisor D.
Note that the divisor D is not assumed to be irreducible, and the integer m does not depend on the point p. It is assumed that the hyperplane F( p) depends algebraically on the point p. Let l ≥ 1 be the degree of the hypersurface in P M−1 , which cuts out D on Q, that is, D ∼ l H Q . Now, repeating the proof of Proposition 2.1 in [22] word for word, we obtain  [22]. Proof of the latter proposition works in our case without modifications.

Step 2: The germ of a quadratic singularity
In this subsection we consider o ∈ W as a germ of a quadratic singularity Consider an effective divisor D o and assume that for the pair (W, 1 m D) the point o is an isolated centre of a non-canonical singularity. Let D + ⊂ W + be the strict transform of the divisor D, so that D + = ϕ * D − l E for some l ≥ 1. Assume that l ≤ 2m, so that the pair (W + , 1 m D + ) is not log canonical. Finally, let S ⊂ E be the centre of a non log canonical singularity of that pair, which has the maximal dimension, in particular, S is not strictly contained in the centre of another non log canonical singularity, if they exist. Obviously, the inequality Proof is obtained partially by repeating the proof of Proposition 2.2 in [22] word for word, partially by reduction to that proposition via restricting the divisor D onto a generic section of the singularity o ∈ W by a linear subspace of dimension rk q 2 . More precisely, arguing as in [22], we obtain from the inequality (4), that if S ⊂ E is a prime divisor, then S ∼ H E is a hyperplane section of the quadric E, that is, the case (1) takes place. Therefore, we assume that codim(S ⊂ E) ≥ 2. Now, arguing as in [22] (replacing Proposition 2.1 in that paper by Proposition 2.1), we obtain that there exists a hyperplane section Z ⊃ S, which is uniquely determined by the log pair (W + , 1 m D + ), satisfying the description of the case (2) of Proposition 2.1. Now let us restrict the divisor D onto the section W = W ∩ of the variety W ⊂ C M by a generic linear subspace of dimension rk q 2 . In this way we obtain the pair (W , 1 m D ) satisfying the assumptions of Proposition 2.2 in [22] (the germ o ∈ W is a germ of a non-degenerate quadratic singularity), the subvariety S = S ∩ W + is the centre of a non log canonical singularity of the pair (W + , 1 m D + ), which has the maximal dimension, so that the hyperplane section Z = Z ∩ W + satisfies the inequality which by genericity of the linear subspace implies the required inequality (5). Proposition 2.2 is shown. Q.E.D.

Step 3: Exclusion of the maximal singularity
Let us come back to the hypersurface W ⊂ P M of degree M with an isolated quadratic singularity o ∈ W of rank ≤5. Let ϕ : W + → W be its blow up, E = ϕ −1 (o) the exceptional quadric. Consider an effective divisor D ∼ m H, where H is the class of a hyperplane section of W , and let D + ∼ m H − ν E be its strict transform on W + .
Proof Since the inequality to be shown is linear in D, without loss of generality we assume that D is a prime divisor. Assume the converse: ν > 3 2 m. To begin with, consider the first hypertangent divisor D 2 = {q 2 | W = 0}. Since by the regularity condition (R3) q 3 | E ≡ 0, we have D + 2 ∼ 2H − 3E, which implies that the divisor D 2 is reduced.

Lemma 2.3 The divisor D 2 is irreducible.
Proof Assume the converse. Then D 2 = 1 + 2 , where 1,2 are distinct hyperplane sections. Since the quadric E is irreducible, The contradiction proves the lemma. Q.E.D. Therefore, D and D 2 are distinct prime divisors, so that the scheme-theoretic intersection Y = (D • D 2 ) is well defined and satisfies the inequality Now, repeating the proof of Proposition 3.1 in [22] word for word, we obtain a contradiction by means of the method of hypertangent divisors. Note that here and everywhere else in this paper, whenever the method of hypertangent divisors is used, it is based on the regularity condition (R1), which makes it possible to apply the method to the hypersurface V and its sections by linear subspaces of codimension 1, 2 and 3. Proposition 2.3 is shown. Q.E.D. Now let us complete the proof of Theorem 1.3. Assume that the point o is an isolated centre of a non-canonical singularity of the pair (W, 1 m D). By linearity of the Noether-Fano inequality we may assume that D is a prime divisor. Since ν ≤ 3 2 m, the pair (W + , 1 m D + ) is not log canonical and a certain irreducible subvariety S ⊂ E is the centre of a non log canonical singularity of that pair. We assume that S has the maximal dimension among all centres of non log canonical singularities of the pair (W + , 1 m D + ), so that by [22, Proposition 1.1] we have dim S ≥ M − 5 (this is a well known consequence of the inversion of adjunction) and by the regularity condition (R2) the assumption of Proposition 2.2 is satisfied.

Proposition 2.4
The subvariety S has codimension at least 2 in the exceptional quadric E.
Proof repeats the proof of Proposition 3.2 in [22] word for word. Following the scheme of arguments in Sec. 3.2 in [22], we conclude that the second case of Proposition 2.2 takes place: there is a hyperplane section Z ⊃ S of the exceptional quadric E, satisfying the inequality (5). Let P ⊂ P M be the (unique) hyperplane, cutting out Z on E, that is, W + P ∩ E = Z , where W P = W ∩ P. Obviously, the prime divisors W P and D are distinct, so that the effective cycle D P = (D • W P ) of codimension 2 satisfies the inequality Now consider the pair (W P , 1 m D P ). Its strict transform (W + P , 1 m D + P ) is not log canonical. We may assume that the inequality mult o D P ≤ 4m holds, otherwise we obtain a contradiction, repeating the proof of Proposition 2.3 word for word. The subvariety S is contained in the maximal centre S of a non log canonical singularity of the pair (W + P , 1 m D + P ). It is easy to see that S ⊂ E P = Z (otherwise, dim ϕ(S ) ≥ 5, so that, as dim Sing W P ≤ 1, there is a curve C ⊂ ϕ(S ), C ∩ Sing W P = ∅, satisfying the inequality mult C D P > m, which is impossible for D P ∼ m H P ). For simplicity of notations we assume that S = S.
Applying Proposition 2.2 once again, we obtain that one of the following two cases takes place: 1. either S is a hyperplane section of the quadric E P , 2. or there is a hyperplane section Z * ⊃ S of the quadric E P , satisfying the inequality where D + P ∼ m H P −l * E P . By the inequality (6), the integer l * satisfies the inequality l * > 4 3 m. Now, repeating the arguments in the beginning of Sec. 3.3 in [22] word for word (using the regularity condition (R3) instead of the condition (R2.2) in [22]), we exclude the case (1). Now let us consider the hardest case (2). Since we can not use the strong regularity condition (R2.2) that was used in [22], we need to slightly modify the arguments of Sec. 3.3 in that paper; in particular, we have to assume that M ≥ 14. Let R ⊂ P = P M−1 be the unique hyperplane, cutting out Z * on the exceptional quadric E P , that is, 8 3 , so that by linearity of the inequality (6) and linearity of the condition of non log canonicity of the pair (W + P , 1 m D + P ) at S, we may assume that D P does not contain the hyperplane section W R as a component (in other words, removing that component, we only make the inequality (6) and the log Noether-Fano inequality stronger). Therefore, we can take the effective cycle D R = (D P • W R ) of codimension 2 on W P , which satisfies the inequality Since by the regularity condition (R3) the quadric q 2 | R = 0 is irreducible and q 3 | R∩{q 2 =0} ≡ 0, we may repeat the proof of Lemma 2.3 and conclude that the divisor D 2 | R is irreducible and has the multiplicity precisely 6 at the point o. Therefore, This is impossible for M ≥ 14.
Proof of Theorem 1.3 is complete. Q.E.D.

Regular Fano hypersurfaces
Let us consider Theorem 1.2. We may prove that a generic Fano hypersurface V satisfies the conditions (R1-R4) separately for each of these conditions. First, we outline the proof for the condition (R1). It is convenient to start with the following general situation.

Proposition 2.5
The following equality holds:

Remark 2.2
The claim can be made more precise: the closed subset P non−reg is reducible and only one of its components has the codimension given above, namely, the component, consisting of such tuples ( p * ) that the closed subset { p 1 = · · · = p N = 0} contains a line in P N . The codimensions of the other components of the set P non-reg are higher. However, we do not need this more precise claim.
Proof of Proposition 2.5 is obtained by means of the methods of the papers [17,18] (see also [24,Chapter 3]): it is completely similar to the arguments of [17, Section 1], when regularity of the sequence of polynomials p 1 , . . . , p N is violated for the first time for the i-th polynomial, i = 1, . . . , N −2, and to the arguments of [18,Section 3] in the case when regularity is for the first time violated at one of the last two steps, that is, either the set is a curve and the polynomial p N vanishes identically on one of its components, or the set is a surface and the polynomial p N −1 vanishes identically on one of its components. If the regularity fails for the first time for the i-th polynomial, where i ≤ N − 2, the computations are identical to [17, Section 1] and quite elementary, so we leave them to the reader.
In the remaining two cases, we give a sketch of the arguments, based on [18,Section 3]. We consider the case when the regularity first fails at the last step, that is, for p N , the other case being similar.
Taking into account Remark 2.2 above, we can assume that the set { p 1 = · · · = p N −1 = 0} is a union of irreducible curves, and p N vanishes on one of them, say C, where deg C ≥ 2. Set C = P k to be the linear span of C, k ∈ {1, . . . , N }. If k = N , then the condition p N | C ≡ 0 imposes at least N (N + 1) + 1 independent conditions on the coefficients of p N (see [18,Section 3]), which is more than we need, so we can discard this option and assume that k ∈ {1, . . . , N − 1}. Now, following [18, Section 3]), let us fix C , a k-plane in P N ; then there exists a good sequence of polynomials p i 1 , . . . , p i k−1 , for which C is an associated subvariety (we use the terms and facts from [18,Section 3]). Fixing these polynomials, we get the following set of independent conditions for the remaining ones: and p N | C ≡ 0. The worst estimate for the codimension of the irregular set corresponds to the choice and (taking into account the dimension of the Grassmannian of k-planes in P N ) is given by the expression Calculating the derivative f (t), we see that on the interval [2, N − 1] the function f (t) is first increasing (until t = t * ≈ N 3 ), then decreasing. Therefore, its minimum on that interval is equal to and elementary calculations show that both numbers are higher than 1 2 The case when the regularity fails first for p N −1 , is completely similar. Q.E.D. for Proposition 2.5.

Corollary 2.1 A generic (in the Zariski sense) hypersurface V satisfies the condition
Proof In the notations of the condition (R1) it is sufficient to consider the worst case c = 3. Taking into account the dimension of the Grassmannian of subspaces of codimension 3 in C M+1 and the fact that the point o ∈ V is arbitrary, by Proposition 2.5 we get that the hypersurface V satisfies the condition (R1), if the inequality holds. It is easy to check that the latter inequality is true for M ≥ 13. Proof of the corollary is complete. Now let us look at the remaining conditions (R2-R4).
In order to show that a general Fano hypersurface V satisfies (R2), it is sufficient to demonstrate that for an integer the condition rk q 2 | {q 1 =0} ≤ M − a imposes on the coefficients of the quadratic form q 2 (with q 1 fixed) at least M + 1 independent conditions. One checks easily that which is at least 6 (for M = 14). Projectivizing the set q 2 | = 0, we get a quadric G of rank at least r M ≥ 6 in P( ) ∼ = P M−3 . Now G is a factorial variety with the Picard group generated by the class of a hyperplane section. Therefore, the restriction q 3 | G = 0 defines a reducible or non-reduced set if and only if this divisor is a sum of a hyperplane section and a section of G by a quadric in P( ). As we see that violation of the condition (R3) imposes (taking into account the dimension of the Grassmannian of subspaces ) independent conditions on the coefficients of the cubic form q 3 , which is higher (and much higher) than M. Thus the claim of Theorem 1.2 is shown for the condition (R3). Finally, let us consider the condition (R4). Unlike the previous three conditions, it is a global one. First, we recall that since V is a smooth hypersurface, dim Sing V ∩ P ≤ 1 for any linear subspace P ⊂ P of codimension 2. Thus we need to show that the set of non-singular hypersurfaces V ∈ F such that for some subspace P ⊂ P of codimension 2 the hypersurface V ∩ P in P has at least one irreducible curve C of singular points, has a positive codimension in F. For the curve C we have three options: C is a plane curve of degree at least 2 in the plane C ∼ = P 2 , 3. the linear span is a k-subspace in P, k ≥ 3.
It is easy to check that the set of hypersurfaces V such that for some P the hypersurface V ∩ P is singular along a line, is of codimension M 2 − 3M + 3 in F, so the option (1) is excluded for a general V . It is not much harder to exclude the option (2). By the condition (R1), we may assume that the plane C ⊂ V (through every point o ∈ V there are only finitely many lines on V ). Now the set of hypersurfaces V such that for some 2-plane the intersection V ∩ contains a double curve C of degree at least 2 is of codimension 5M − 13 (this particular value corresponds to the case deg C = 2; for higher degrees the codimension is much higher). So we may assume that of singular points of the hypersurface {F = 0}.

Lemma 2.4
Assume that m ≥ 3. For any set of k linearly independent points P 1 , . . . , P k ∈ P N , where k ≤ N + 1, the condition Proof We may assume that P 1 = (1 : 0 : 0 : · · · : 0), P 2 = (0 : 1 : 0 : · · · : 0) etc. correspond to the first k vectors of the standard basis of C N +1 . Taking into account that m ≥ 3, the claim of the lemma becomes obvious. Q.E.D. Now let us consider a pencil of hyperplanes in P N through a linear subspace ⊂ P N of codimension 2, Bs = . Fix an integer m ≥ 3 and take any l = m − 2 distinct hyperplanes Further, consider a set of kl distinct points such that for every i = 1, . . . , l the points P i j ∈ H i \ and {P i1 , . . . , P ik } is a set of linearly independent points in H i (so that k ≤ N ).

Lemma 2.5 The condition
Proof It is easier to see this claim in the affine setting: where λ 1 = 0 and λ 1 , . . . , λ l are distinct. Every (non-homogeneous) polynomial g(u, v) can be written in a unique way as where {λ 1 , . . . , λ m } is a set of distinct values, extending the given set {λ 1 , . . . , λ l }, and (Once again, when g m (u) = g + m (u) is fixed.) Continuing in this way, for fixed polynomials . . , u N −1 , whose underlying linear subspace is given by the condition so is of codimension k N . The last polynomial, to which Lemma 2.4 can be applied, is g 3 (u). Proof of Lemma 2.5 is complete.

Corollary 2.2
For a fixed pencil and hyperplanes H 1 , . . . , H l as above, the condition that there exist points P i j ∈ H i , i = 1, . . . , l, j = 1, . . . , k as described above, such that (that is, the set of points depends on the polynomial F), defines a subset of codimension Proof Indeed, this is an obvious dimension count, as for the hyperplanes H i being fixed, every point P i j varies in a (N − 1)-dimensional family. Q.E.D. Now let us prove the condition (R4) in the case (3). Fix the linear subspace P ⊂ P of codimension 2 and consider the set X P of smooth hypersurfaces V ⊂ P of degree M such that Sing (V ∩ P) contains a curve C with C at least 3-dimensional. The degree of C and the number of such curves in the set Sing (V ∩ P) are bounded by a constant depending on M. For a pencil of hyperplanes in P and distinct hyperplanes be the subset defined by the condition that Sing (V ∩ P) contains a curve C with the linear span C at least 3-dimensional and such that for every i = 1, . . . , M − 2 the intersection C ∩ H i contains dim C linearly independent points. It is obvious that is Zariski open in X P (the curve C depends algebraically on V ). By Corollary 2.2 we have As P varies in a 2M-dimensional family, the dimension count tells us that in the case (3) the codimension is at least M − 6.
Proof of Theorem 1.2 is complete.

Remark 2.3
More precise arguments, taking into consideration every possible value of the dimension dim C , lead to a much stronger estimate in the case (3) of the condition (R4), which is quadratic in M.

Subvarieties of codimension two
In this section we prove Theorem 1.4: if B is a maximal subvariety of codimension two for the system , then B is a section of the hypersurface V by a linear subspace of codimension two. The proof makes use of the cone technique, see [24,Chapter 2]. The main idea of our arguments is to consider two-dimensional cones, swept out by secant lines of the subvariety B.

The secant space of the subvariety B
Assume that the inequality (2) holds. We need to show that B = P ∩ V , where P ⊂ P is a linear subspace of codimension two. If B is contained in a hyperplane, B ⊂ , then the claim of the theorem is almost obvious: the hyperplane section V = V ∩ is a factorial variety, Pic V = ZH , where H is the class of a hyperplane section, so that B ∼ m H on V for some m ≥ 1. The restriction of the linear system onto V is a non-empty system of divisors, ⊂ |2n H |, whereas mult B > n, that is, B is a fixed component of the system of multiplicity (mult B ). This implies that m = 1, so that B ∈ |H | is a hyperplane section of the variety V ⊂ , which is what we need.
Starting from this moment, we assume that B is not contained in a hyperplane, that is, B = P. Let us show that the inequality (2) is impossible for the mobile linear system ⊂ |2n H|. In order to do this, we assume that this inequality is true and show that this assumption leads to a contradiction.
Define the secant space as the closure of the set is the line, connecting the distinct points p, q. Let π B and π P be the projections of the irreducible variety Sec(B) onto B × B and P, respectively.

Proposition 3.1
The projection π P is surjective.
Proof is given below in Sect. 2.3. Proposition 3.1 implies that the image of the restriction of the projection π P onto the set Sec * (B) contains an open subset in P. In the sequel, speaking about a point x of general position in P, we will always mean, in particular, that x / ∈ V , so that the restriction of the projection from the point x onto V is a finite morphism V → P M . Let Sec(B, x) and Sec * (B, x) be the fibres of the projection π P and its restriction onto Obviously, Sec(B, x) can be considered as a closed subset in B × B, invariant with respect to the involution τ : ( p, q) → (q, p), and Sec * (B, x) as a closed subset in B × B\ B , where for a sufficiently general point x ∈ P) where π 1,2 : B × B → B are the projections onto the first and second direct factors, respectively. By construction, C − is contained in the cone with the vertex x and the base C + and the other way round. The properties of that cone (swept out by the lines [ p, q] x, for p, q ∈ ) and of the curves C ± can be made more precise.
The following fact is true.

Proposition 3.2 For some positive integers d C and d R there is an algebraic family
, where C is an effective 1-cycle of degree d C on P, R is an effective 1-cycle of degree d R on P and x ∈ P is a point, satisfying the following conditions: 1. the projection π P : A → P, for the cone C(x) ⊂ P with the vertex x and the base C + we have: C ± are sections of the cone and the equality has at the point p a simple tangency with the hypersurface V : that is, p / ∈ R, 7. the components of the curves R sweep out V : Proof, which makes use of the construction, immediately preceding the statement of Proposition 3.2, is given below in Sect. 2.2.
Let us complete the proof of Theorem 1.4. In the notations of Proposition 3.2, consider an arbitrary irreducible component of the residual curve R, which we for simplicity denote by the same symbol. Let D ∈ be a generic divisor. By the property (7), we may assume that R ⊂ D. Since B ⊂ D and, moreover, mult B D > n, the inequality holds, where the last sum is taken over the usual and infinitely near points of intersection of the curve R and the subvariety B: the set of those points is denoted by the symbol R ∧ B (see [20]). Since C ± ⊂ B and, by the property (6), the curve R does not contain the points of the intersection C + ∩ C − , we have means a disjoint union. By the property (5), the curve R meets C ± at non-singular points of those curves. The following lemma is a version of a very well known claim [20].

Lemma 3.1
The following equality holds: In the last formula we mean any choice of the sign + or −. Now from the inequality (9), taking into account (10), we obtain: 2ndeg R > n(deg R + deg R), which is impossible. Q.E.D. for Theorem 1.4.

Remark 3.1
Repeating the previous arguments word for word, we exclude the possibility of two maximal subvarieties of codimension two for the system . Therefore, the section B = V ∩ P is uniquely determined.

Proof of technical facts
Let us show Lemma 3.1. By genericity of the curve C = C + + C − , each of the curves C ± is a section of the cone C(x). The normalizations C ± of these curves are naturally isomorphic. Let C + (x) be the blow up of the vertex of the cone C(x) and the non-singular ruled surface over C ± , where the smooth curves C + and C − are realized as its sections. Set R to be the strict transform of R on C(x). By the properties (4)-(6) at each point p ∈ (R ∩ C + ) (R ∩ C − ) the corresponding curve C + or C − is non-singular and transversal to the generator of the cone [ p, x], so that where the subset R ∧C ± ( p) ⊂ R ∧C ± consists of the point p and infinitely near points over it, where the point of the surface C(x), corresponding to the point p of intersection of the curves R and C + (or C − ), is denoted by the same symbol p. Therefore, but the last number is equal to deg R, see [20]. Q.E.D. for the lemma.
Proof of Proposition 3.2 The construction, immediately preceding the statement of Proposition 3.2, gives an algebraic family A, satisfying the property (1) by Proposition 3.1. Let us show that, somewhat shrinking the family A (that is, taking a Zariski open subset in that family), one can ensure that the remaining properties (2)-(7) hold. Indeed, our construction yields in the general case distinct irreducible curves C ± = C − , so that the property (2) can be assumed. Proof of Proposition 3.1 implies easily that a generic secant line [ p, q] of the variety B is not a 3-secant (see Remark 3.2 in Sect. 2.3 below), that is, C ± are sections of the cone C(x), whereas C ± come into the 1-cycle (C(x) • V ) with multiplicity 1, which gives the property (3).
For any point p ∈ B we have B ⊂ T p V (since B is not contained in a hyperplane by assumption), and for that reason for a general point x ∈ P the direction of the line [ p, x] defines a field of directions on a proper closed subset of the set Sec(B, x) (consisting of the points p ∈ B, at which [ p, x] ⊂ T p B) and for that reason for a general curve its projections C ± are nowhere tangent to the lines [ p, x] (that is, at no point p ∈ C ± ), that is, the property (4) is satisfied (one should also take into account that for a general point p ∈ B the set Sing B ∩ T p V has dimension at most M − 4, since the hypersurface V is non-singular and the linear system M+1 i=0 λ i (∂ F/∂ x i ) defines a finite morphism P → P). The property (5) again follows from the fact that B is not contained in a hyperplane: obviously, so that the point ( p, q) ∈ is a singularity of that curve (for a general curve ) if and only if p or q belongs to Sing B. Since the pairs of points ( p, q) ∈ B × B such that p ∈ Sing B and q ∈ T p V , form a subset of codimension al least 2, a general curve does not contain such pairs. This proves the property (5).
Let us consider the property (6). The subset π −1 B ( B ) is a closed subset of codimension 1 in Sec(B), which may consist of several irreducible components of different codimensions. A general curve does not intersect the components of codimension two, so we are only interested in the divisorial components.
It is easy to see that the closure of the set is a prime Weil divisor on Sec(B). For a non-singular point p ∈ B\Sing B we have Let (z 1 , . . . , z M+1 ) be a system of affine coordinates with the origin at the point p ∈ P and f = q 1 + q 2 + · · · + q M the equation of the hypersurface V . By the condition on the rank of the quadratic form has in Sec(B) codimension 2, which is what we need. Therefore, it is sufficient to prove the property (6) for singular points and q 2 | T p V ≡ 0, we obtain once again, that the set of triples ( p, p, x) ∈ Sec(B), such that the inequality (11) holds, is of codimension at least two in Sec(B), which is what we need. Therefore, it is sufficient to consider a divisorial component Q ⊂ Sing (B), codim(Q ⊂ B) = 1. We will explain briefly how these limit sets are obtained, leaving it to the reader to perform the computations.

Lemma 3.2 There exists a non-empty Zariski open subset U Q ⊂ Q such that for any point p ∈ U Q the set
As a model example, let us consider the following problem: determine the limit set of all secant lines [q, r ] as q, r → o, where C ⊂ P N , N ≥ 3, is a (possibly reducible but reduced) curve with a singular point o ∈ C, mult o C ≥ 2. Even if irreducible, the curve C can have several (possibly singular) branches at o. We have therefore to consider the limits of secant lines [q, r ], as (i) q, r → o along the same singular branch C 1 of C (if the branch is non-singular, then the limit is obviously the tangent line to the branch, T o C 1 = T o C 2 (one or both of these branches may be non-singular), (iii) the same as (ii), but the tangent lines coincide, Let us show that in each of the three cases the limit set of secant lines [q, r ] sweeps out a union of 2-planes containing the point o. In the case (ii) this is obvious: this set is the plane T o C 1 , T o C 2 . Let us consider the case (i).
Making a linear change of coordinates and re-parameterizing the branch C 1 , we may assume that with respect to a system (z 1 , . . . , z N ) of affine coordinates, where o = (0, . . . , 0), the branch C 1 has the following parameterization at the point o: where the sum may be finite or infinite, for all i in the sum above and gcd(d 0 , d 1 , . . . ) = 1 (so that the parameterization can not be factored through t → t k , k ≥ 2). Now using 1-dimensional families of secant lines [q(s), r (s)], where q(s) = z(s) and r (s) = z(ζ s + as e ), ζ d 0 = 1, e ≥ 2 and a ∈ C any constant, it is easy to obtain the limit set of secant lines of this branch as a union of 2-planes. Their explicit description is as follows. Set . This procedure terminates, giving us a finite set of indices with gcd(d 0 , d i 1 , . . . , d i l ) = 1. Now choosing an appropriate root of unity ζ and a suitable value of e in the parameterization above, it is easy to check that the limit set of the branch C 1 is the union of planes v 0 , v i k , k = 1, . . . , l.
We leave the computations to the reader as well as checking that for any parameterization of q(s) and r (s) the limit of the secant line [q(s), r (s)] is inside the union of the planes above.
This completes the case (i). The case (iii) is very similar to (i) and is also left to the reader.
Keeping in mind this model example, let us come back to the proof of Lemma 3.2. At a generic point p ∈ Q we choose a system of linear coordinates with the origin at the point p, such that the projection T p Q → {z 1 = z 2 = z 3 = z 4 = 0} is an isomorphism, so that near p the non-singular subvariety Q admits a parameterization of the form Near p the subvariety B is a union of branches, each of which admits a uniform parameterization satisfying the same properties as the local parameterization of the branch C 1 discussed above (which can be seen, as we mentioned in the beginning of the proof of this lemma, through considering the normalization B → B). Now elementary local computations, similar to those sketched above for the 1dimensional case, show that the limit set of secant lines at the point p is a union of linear space of dimension M − 1, containing T p Q (and, naturally, contained in T p V ). We leave these easy explicit computations to the reader. Proof of Lemma 3.2 is complete. Now for any point p ∈ U Q and some linear subspace ⊂ π P (π −1 B (( p, p))) we have q 2 | ≡ 0, so that the closed set Therefore, the set of triples ( p, p, x) ∈ Sec(B), satisfying the inequality (11), where p ∈ U Q , has the dimension (M − 3) + (M − 2) = 2M − 5, that is, the codimension 2 in Sec(B). This completes the proof of the property (6). Finally, the property (7) is obvious (for instance, follows immediately from the proof of Proposition 2.1, given below). Proposition 3.2 is shown. Q.E.D.
Obviously, Sec(B) aff is the closure of the image of the map σ . Furthermore, it is obvious that for a non-singular point p ∈ B the tangent space T v B aff does not depend on the choice of a non-zero vector v ∈ α −1 ( p) and for that reason we denote it by the symbol T p B aff . It is clear that the embedded tangent space T p B ⊂ P satisfies the equality Let p, q ∈ B be a pair of non-singular points. Obviously, the differential dσ at the point (v 0 , w 0 , (λ 0 , μ 0 )) is so that for a non-singular point r ∈ Sec(B), r ∈ [p, q], we have T r Sec(B) aff = Imdσ = T p B aff + T q B aff (taking into account that v 0 ∈ T p B aff and w 0 ∈ T q B aff ). Set Therefore, Sec(B) = P if and only if dσ is surjective, that is T ( p, q) = C M+2 .
Remark 3.2 At this point we observe that the equality T ( p, q) = C M+2 implies that a general secant line is not a 3-secant. Assume now that Sec(B) = P is a proper irreducible subvariety. Since codim(B ⊂ P) = 3, this implies that codim(T p B aff ∩ T q B aff ) ≤ 5, and the latter holds for any non-singular points p, q ∈ B. Let us show that our assumption leads to a contradiction.
The symbol π p stands for the linear projection P P 2 from the tangent space T p B for a non-singular point p ∈ B. The projection π p is the projectivization of the linear map The differential of the restriction of the latter map onto B aff is not surjective at a point of general position. Indeed, for any smooth point q ∈ B we have: Therefore, π p (B) = P 2 and for that reason π p (B) is either a point or some irreducible curve C ⊂ P 2 . If π p (B) is a point or C is a line, then the subvariety B is contained in a hyperplane, which contradicts the assumption. Therefore, π p (B) = C is a curve of degree d ≥ 2.
Let c ∈ C be a point of general position, the fibre of the projection π p | B . Obviously, B c is a closed subset of pure codimension two in the fibre π −1 p (c) ∼ = P M−1 . For that reason the secant variety Sec(B c ) coincides with its linear span B c (it is sufficient to check this almost obvious fact for a curve in P 3 ).
Therefore, we have three options: Assume that the case (1) takes place. Since Sec(B c ) ⊂ Sec(B), we have On the left we have an irreducible divisor in P, so that by our assumption that Sec(B) = P the equality Sec(B) = π −1 p (C) holds. However, it is obvious, that Sec(B) contains points outside the set π −1 p (C): let c 1 , c 2 ∈ C be a general pair of points, This contradiction excludes the case (1).
The case (3) is impossible, as V does not contain linear subspaces of dimension M − 3.
Therefore, the case (2) takes place. Again we take a general pair of points c 1 , c 2 ∈ C. Let L = [c 1 , c 2 ] ⊂ P 2 be the line through them, H = π −1 p (L) the corresponding hyperplane in P. Set also The linear space P = T p B is of codimension two in H and P 1 ∩ P 2 = P. Furthermore, set these are hyperplanes in P i , i = 1, 2.

Proposition 3.3
The following equality holds: It is clear that since the points c 1 , c 2 are general, Proposition 3.3 implies the equality Sec(B) = P, which contradicts the initial assumption and proves Proposition 3.1. for i = 1, 2. If R 12 = R 1 ∩ R 2 is a line, then we conclude that for a general pair of points (s 1 , s 2 ) ∈ S 1 × S 2 the planes T s 1 S 1 and T s 2 S 2 are disjoint. This implies, that Sec(S 1 ∪ S 2 ) = , so that Proposition 3.3 is shown in this case.
Therefore we assume that R 12 is a plane, that is, By the genericity of the subspace this means that and for that reason 12 = 1 ∩ P = 2 ∩ P. The points c 1 , c 2 are chosen independently of each other, so that we can conclude that there exists (a uniquely determined) hyperplane Q ⊂ P such that for a point of general position c ∈ C we have Let π Q : P P 3 be the projection from the linear subspace Q. By what we proved, π Q ( B c ) is a point and for that reason π Q (B c ) is a point, so that the image π Q (B) is a curve C + (the projection of which from the point π Q (P) is the curve C ⊂ P 2 ). If C + is contained in some plane in P 3 , then B is contained in some hyperplane in P, either, which contradicts our assumption. Thus Now let ξ 1 , ξ 2 ∈ C + be a general pair of points, i = π −1 Q (ξ i ) ⊂ P the corresponding subspaces of codimension 3, B + i = π −1 Q (ξ i ) ∩ B the fibres of the projection π Q | B . We know that B + i ⊂ i are hypersurfaces (possibly reducible) and Since B + i are not cones, we conclude that whence, finally, it follows that Sec(B) = P. Proof of Propositions 3.3 and 3.1 is complete.

Infinitely near case: I-preparatory work
In this section we start the proof of Theorem 1.5, that is, the exclusion of the infinitely near case. Here we carry out preparatory work: we come over to a hyperplane section of the hypersurface V , in order to use the 8n 2 -inequality, list all particular cases that need to be considered and obtain aprioric estimates for the multiplicity of the selfintersection. We use the following tools: the inversion of adjunction, the technique of counting multiplicities and the method of hypertangent divisors.

The method of hypertangent divisors
Let ⊂ |2n H| be a mobile linear system with no maximal subvarieties of codimension two. Fix a maximal singularity E * ⊂ V of the system with the centre B ⊂ V of maximal dimension.

Lemma 4.1 B is a point or a curve on V.
Proof By the 4n 2 -inequality we have where Z = (D 1 • D 2 ) is the self-intersection of the system . Since Z ∼ 4n 2 H 2 , by [19,Proposition 5] it follows that dim B ≤ 1. Q.E.D. for the lemma. The cases dim B = 1 and dim B = 0 are dealt with in word for word the same way, the assumption on the existence of a maximal singularity leads to a contradiction, excluding both cases. We will assume that B = o is a point (if B is a curve, take a general point o ∈ B, and then the proof for the case dim B = 0 applies; note that all local information about the linear system at the point o and its strict transform with respect to the blow up of this point is obtained through restricting the system onto the section of V by a generic linear subspace through o in the dim B = 0 case, so the arguments work in the dim B = 1 case as well). The following fact is true.
and construct a sequence of irreducible subvarieties This contradiction proves our proposition. Q.E.D.
Arguing in a similar way, we obtain the following fact.

Proposition 4.2 For any irreducible subvariety Y ⊂ V of codimension two the following inequality holds:
Proof This contradiction completes the proof of the proposition.
(The somewhat strange writing of the right hand part of the inequality (ii) will become clear below.) Proof (i) Repeating the arguments of the first part of the proof of Proposition 4.2 word for word and taking into account the regularity conditions for the hypersurface V , we obtain the inequality for codim( ⊂ P) = 1, and the inequality for codim( ⊂ P) = 2. It is easy to see that these inequalities are impossible. The contradiction proves the claim (i).
In the case (ii) we repeat the arguments of the second part of the proof of Proposition 4.2 word for word, once again taking into account that by the considerations of dimension we take codim( ⊂ P) = 1 or 2 hypertangent divisors less. Again we get a contradiction, which proves the claim (ii).
Proof of Proposition 4.3 is complete.

The restriction onto a hyperplane section
The next step in the proof of Theorem 1.5 is the restriction of the linear system onto a suitable hyperplane section of the variety V , which allows us to make the estimate for the multiplicity of the self-intersection at the point o twice stronger. If the inequality mult o Z > 8n 2 holds (where Z is the self-intersection of the mobile linear system ), then this step can be skipped, considering below instead of the hyperplane section P o the hypersurface V itself: in that case, the dimension does not drop and all estimates become only stronger, so that the proof given below works without any modifications. Keeping this in mind, assume that mult o Z ≤ 8n 2 . Recall that V + is the blow up of V at the point o with the exceptional divisor E ∼ = P M−1 . The following fact is true.

Proposition 4.4 (The 8n 2 -inequality)
There exists a subspace ⊂ E of codimension 2 (uniquely determined by the system ), satisfying the inequality Proof This is [23, Proposition 4.1]. Now let us consider the linear system |H − |, consisting of hyperplane sections that cut out on E, that is, for a general divisor P ∈ |H − | we have: P ∈ |H | is a hyperplane section, smooth at the point o and P + ⊃ . Obviously, Therefore for a general divisor P ∈ |H − | the effective cycle Z P = (Z • P) of codimension two is well defined and satisfies the inequality Let P = | P be the restriction of the linear system onto P. Obviously, P ⊂ |2n H P |, where H P = H | P is the positive generator of the group Pic P ∼ = Z, whereas the system P is mobile (has no fixed components). The cycle Z P is the self-intersection of the system P : where D 1 , D 2 ∈ P are generic divisors. The variety P is a hypersurface of degree M in P M , which may have isolated singular points, but the point o ∈ P itself is non-singular. Proof The first claim follows from the inversion of adjunction [14]. Let us consider the second one (it is not used in the subsequent proof). If B = , then codim(B ⊂ P) ≥ 3 (otherwise, by the genericity of P, the original system has a maximal subvariety of codimension 2, which is not true by assumption). Therefore, the 4n 2 -inequality holds: Let Q ∈ |H P | be a general (in particular, everywhere non-singular) hyperplane section of P and Z Q = (Z P • Q). Then on Q the cycle Z Q ∼ 4n 2 H Q of codimension two satisfies the inequality mult B∩Q Z Q > 4n 2 and dim B ∩ Q ≤ 1 by Proposition 5 in [19]. Proof is complete.
Note that by the genericity of the hyperplane section P the linear system P satisfies the inequality ν = mult o P (= mult o ) ≤ 3n. Now let 1 ⊂ 2 be a generic pair of linear subspaces of dimensions 5 and 6 in P M = P , containing the point o, and X i = P ∩ i the corresponding sections of the hypersurface P. By the inversion of adjunction the pair (X i , 1 n i ), where i = P | X i , has the point o as an isolated centre of a non log canonical singularity. Let X + i ⊂ P + be the strict transform of X i , so that is the blow up of the point o ∈ X i and E (i) = E ∩ X + i the exceptional divisor of the morphism ϕ i . The pairs and 2 = X + 2 , are not log canonical (recall that ν ≤ 3n) and satisfy the conditions of the connectedness principle with respect to the birational morphisms ϕ 1 and ϕ 2 , respectively (see [14,Section 17.4]). The centre of any non log canonical singularity of the pair i , intersecting E (i) , is contained in E (i) (Proposition 4.5), so that we conclude that the union LC S( ) i of centres of non log canonical singularities of the pair i , intersecting E (i) , is a connected closed subset in E (i) . Recall that E (1) ∼ = P 3 and E (2) ∼ = P 4 . For the pair 1 there are three options: is a union of curves and surfaces, and in this union there is at least one surface.
For the pair 2 there are, respectively, four options, dim LC S( 2 ) ∈ {0, 1, 2, 3}, and if LC S( 2 ) is zero-dimensional, then this set consists of one point. Now looking at the pair we see that LC S( 12 ) is either a line in E (2) ∼ = P 4 , or a connected union of surfaces (every hyperplane section of which is connected), or a union of surfaces and divisors in E (2) . Since the pair 12 is obviously "more effective" than the pair 2 , we have the inclusion There is a non log canonical singularity of the pair * , the centre of which on P + is an irreducible subvariety B ⊂ E P of codimension 2. Case 3. There is a non log canonical singularity of the pair , the centre of which on P + is an irreducible subvariety B ⊂ E P of codimension 1. The six cases listed above correspond to three possible values of the integer dim LC S( 1 ), taking into account the type of the set LC S( 2 ).
The last case is the simplest one. Proof Assume the converse: B ⊂ E P is a prime divisor. We argue as in the proof of Proposition 4.1 in [23] or in [4]: for the self-intersection Z P of the system P , taking into account that the pair is not log canonical at B, we obtain the estimate Therefore, there is an irreducible subvariety Y ⊂ P of codimension two, satisfying the inequality However, this contradicts Proposition 4.3. Proposition 4.6 is shown. Q.E.D.

Remark 4.2 Once again, we emphasize that Proposition 4.3 implies the inequality
which we will use in the sequel without special references.

The techniques of counting multiplicities: the aprioric estimates
Following the standard procedure of the method of maximal singularities, let us obtain now bounds from below for the multiplicities of the cycle Z P , improving the 8n 2inequality. We call these estimates aprioric, because they do not make use the additional geometric information available in the cases 1.1-2.3. To exclude those cases, the aprioric estimates are not sufficient and we will need some additional work, which will be carried out in Sects. 4 and 5. mult o Z P + mult Z + P > 12n 2 (16) and mult Z P > 4n 2 . If the case 1.2 takes place, then the following estimate holds: (ii) If either of the cases 2.1 or 2.2 takes place, then the following inequality holds in addition in the case 2.1 the estimate mult Z + P > 4n 2 and in the case 2.2 the estimate mult B * Z + P > 4n 2 hold. (iii) If the case 2.3 takes place, then the following inequality holds: Proof All the inequalities, listed above, belong to one of the two types: the type (16) for a non log canonical singularity of the pair and the type (17) for a singularity of the pair * . The proofs for each of the two types are completely identical, and for this reason we will show only these two inequalities. Let us prove the inequality (16). It is true under a weaker assumption that the pair has a non canonical singularity, the centre of which is the subspace . This is what we will assume. Let be the resolution of the non canonical singularity of the pair , where P 1 = P + , σ 1,0 = ϕ P , E 1 = E P , σ 2,1 is the blow up of the subvariety = B 1 , and in general, B i−1 is the centre of the fixed non canonical singularity of the pair on P i−1 , is the exceptional divisor, finally, i = 1, . . . , K and E K realizes the fixed non canonical singularity. Let be the oriented graph of that resolution, that is, its set of vertices is the set of exceptional divisors E 1 , . . . , E K , and the vertices E i and E j are joined by an oriented edge (an arrow; notation: i → j), if and only if i > j and B i−1 is contained in the strict transform E i−1 j of the exceptional divisor E j on P i−1 , see [17] or [21,Chapter 2], also [24,Chapter 2] for the details. By the symbol p i j we denote the number of paths from the vertex E i to the vertex E j , if i = j; we set p ii = 1. The fact that E K realizes a non canonical singularity of the pair , means that the inequality of the Noether-Fano type holds: where ν i = mult B i−1 i−1 , and δ i = codim B i−1 − 1 is the discrepancy of E i with respect to P i−1 . By linearity of the inequality (19) we may assume that ν K > n (if ν K ≤ n, then E K −1 is a non canonical singularity of the pair and K can be replaced by K − 1). Set The graph breaks into the lower part with the vertices E 1 , . . . , E L and the upper part with the vertices E L+1 , . . . , E K . Now let us the well known trick of removing arrows (see, for instance, [23, §4] or [24,Chapter 2] for the details): let us remove all arrows that go from the vertices of the upper part to the vertex E 1 , if such arrows exist. This operation does not change the numbers p K 2 , . . . , p K K , but, generally speaking, decreases the number of paths from E K to E 1 . Set p i = p K i for i = 2, . . . , K and let p 1 be the number of paths from E K to E 1 in the modified graph. Since ν 1 ≤ 3n, the inequality (19) remains true: in addition, the modification of the graph yields the estimate Set Z i P to be the strict transform of the cycle Z P on P i , i = 1, . . . , L; in particular, Applying the technique of counting multiplicities (see, for example [21,Proposition 2.11] or [24, Chapter 2]), we obtain the inequality whence in the standard way (computing the minimum of the quadratic form p i ν 2 i on the hyperplane, which we obtain, replacing the inequality sign in (20) by the equality sign) we deduce the estimate so that, in particular, p 1 ≤ 0 + 1 . Taking into account that the multiplicities m i do not increase, we obtain the inequality Recall that m 1 = mult o Z P and m 2 = mult Z + P are precisely the multiplicities, which we are interested in, and we prove the inequality m 1 + m 2 > 12n 2 . By linearity in m 1 , m 2 and the last inequality (that is, the inequality (16)) and the inequality (22), it is sufficient to check that the estimate (22) does not hold for m 1 = 8n 2 , m 2 = 4n 2 and for m 1 = 12n 2 , m 2 = 0. Since p 1 ≤ 0 + 1 , it is sufficient to consider the first case. Setting in (22) m 1 = 8n 2 and m 2 = 4n 2 , cancelling n 2 and moving everything to the right hand side, we obtain the inequality We obtained a contradiction, which proves the inequality (16). Now let us show the inequality (17). The arguments are completely similar to those above, with the only difference that the coefficient at p 1 in the Noether-Fano inequality is 4, the elementary discrepancies can take four, not three values, that is, δ i ∈ {1, 2, 3, 4}, so that there are, generally speaking, four groups of vertices of the graph and we must set and the inequality p 1 ≤ 0 + 1 + 2 holds. The technique of counting multiplicities gives the following estimate, which is similar to the inequality (22): Since m 1 ≤ 12n 2 , to prove the inequality (17) (which in the notations of the resolution of singularities takes the form of the inequality m 2 > 4n 2 ), it is sufficient to check that the inequality (23) can not be true for m 1 = 12n 2 and m 2 = 4n 2 . Substituting these values into (23), cancelling n 2 and moving everything to the right hand side, we get the inequality where in the brackets we have a quadratic form in s, t 0 , t 1 , t 2 , t 3 with nonnegative coefficients. We obtained a contradiction, proving the inequality (17). The remaining inequalities of Proposition 4.7 are shown word for word in the same way as the inequality (16) or (17), depending on the type of the inequality.
Proof of Proposition 4.7 is complete.
The further work, completing the proof of Theorem 1.5, is organized in the following way: we exclude the cases 1.1-2.3, inspecting all geometric possibilities.

Infinitely near case: II-exclusion of the linear case
In this section we prove that the cases 1.1 and 1.2 do not realize: it is sufficient to exclude the first one, which immediately implies that the second one is impossible.

Decomposition of an effective cycle
Let us forget for a moment about the proof of Theorem 1.5 and consider one very simple construction which will be used below many times. Let X be an arbitrary algebraic variety, Y ⊂ X an irreducible subvariety and Z an effective cycle of codimension two on X . Assume first that codim(Y ⊂ X ) ≤ 2, that is, Y is a prime Weil divisor on X or an irreducible subvariety of codimension two.

Definition 5.1 We say that the presentation
is a Y -decomposition of the cycle Z , if both cycles Z 0 , Z 1 are effective and an irreducible component of the cycle Z is contained in Z 0 (respectively, in Z 1 ) if and only if it is contained in Y (respectively, not contained in Y ).

Definition 5.2
We say that the presentation is a Y -decomposition of the cycle Z , if both cycles Z 0 , Z 1 are effective and an irreducible component of the cycle Z is contained in Z 0 (respectively, in Z 1 ) if and only if it does not contain Y (respectively, does contain Y ).
Note that the definitions are not symmetric. Let us come back to the proof of Theorem 1.5.

Restriction onto a hyperplane section
The main result of this section is the following Proof Assume the converse: the case 1.1 takes place. Our purpose is to get a contradiction. We will do it in several steps, since the case under consideration is the hardest of the six ones. We use both inequalities of Proposition 4.7 for the case 1.1 without special comments. First of all, let us repeat the operation of restricting onto a hyperplane section that was used in Sect. 3.
Let R ⊂ P be a general hyperplane section, such that: • o ∈ R, the variety R is non-singular at that point, • the hyperplane E R = R + ∩ E P in E P contains the subspace .
Let us restrict the system P onto R and obtain a mobile linear system R on the hypersurface R ⊂ R ∼ = P M−1 with the self-intersection Z R = Z P | R , satisfying the estimates and mult Z R > 4n 2 .
The advantage of this situation is that the subspaces ⊂ ⊂ E R are of codimension 3 and 2, respectively. Let We obtain the equality and the inequality Furthermore, set λ 1 = 1 n 2 mult Z + 1 , where Z + 1 is the strict transform of the cycle Z 1 on R + , so that the following inequality holds: Proposition 4.3 implies that the multiplicities μ i can be estimated in terms of the degrees d i in the following way: for M ≥ 18 the inequality holds, for M ≤ 17 a weaker estimate is true: Since none of the components of the cycle Z 1 is contained in the tangent section holds. Finally, it is obvious that the following estimate holds: The system of six equations and inequalities (24-30) (it is six, because depending on whether M ≥ 18 or M ≤ 17, we choose the inequality (27) or (28)) forms the first system of relations for the five parameters introduced above.
Since q 2 | ≡ 0, the components of the cycle Z R , the strict transforms of which contain the linear subspace , can not be contained in T R . For that reason, mult o Z 1 ≥ mult Z + 1 > 4n 2 , so that the following inequality holds:

Additional estimates for the cycle Z 1
Now let us consider the cycle Z 1 , the most important part of the self-intersection Z R , since it contains the linear subspace . First of all, none of the components of the cycle Z 1 is contained in the tangent section T R = R ∩ T o R and for that reason (Z 1 • T R ) is an effective cycle of codimension 2 on the hypersurface T R . The latter has a quadratic singularity at the point o, so that is its tangent cone at that point. Its projectivization will be denoted by the symbol Q R . Obviously, By the condition (R2) the intersection [Q R ∩ ] is an irreducible quadric; it is subvariety of codimension two on Q R . Now let us compute the multiplicity mult o (Z 1 • T R ). By the rules of the intersection theory (see [8] or [24, Chapter 2]), write where Z Q is an effective divisor on the quadric Q R (outside Q R the effective cycles Setting μ 2 = 1 n 2 mult o (Z 1 • T R ), and deg Z Q = 2δn 2 , we obtain the equality Obviously, deg(Z 1 • T R ) = deg Z 1 . Furthermore, set Since the following inequality is obviously true: by Corollary 5.1, which is shown below, we get the estimate Now we have to take into account the input of the infinitely near subvariety [Q R ∩ ].

Proposition 5.2
The following estimate holds: Proof Let H R be the class of a hyperplane section of the hypersurface R ⊂ R ∼ = P M−1 . Consider the pencil |H R − | of hyperplane sections of R, defined by the condition: for S ∈ |H R − | we have S + ⊃ . The base set R of the pencil |H R − | is an irreducible subvariety of codimension two in T R , of codimension 3 in R; more precisely, R ⊂ R ∼ = P M−3 is a hypersurface of degree M, where the linear span R is determined by the condition Now let Y be an arbitrary subvariety of codimension 2 in R. For a general divisor S ∈ |H R − | we have Y ⊂ S, so that (Y • S) is an effective cycle of codimension 3 on R. By construction, . However, S is a section of the singular hypersurface V ∩T o V by a linear subspace of codimension 3, so that, applying the regularity condition (R1) and arguing in word for word the same way as in the proof of Proposition 4.3, by means of the technique of hypertangent divisors, applied to the cycle (Y • S), we obtain the estimate (recall that on S the cycle (Y • S) has codimension 2, so that this cycle can be considered as an effective cycle of codimension 3 on a section of the hypersurface V by a linear subspace of codimension 3, which by the condition (R1) satisfies the regularity condition).
The inequality (35) follows from (36) in a trivial way. Proof of Proposition 5.2 is complete.

On the multiplicities of subvarieties on a quadric
Let us put off the proof of Theorem 1.5 and show the fact about multiplicities of subvarieties on a quadric hypersurface that was used in Sect. 4.3. Let Q ⊂ P N be an irreducible quadric, dim Sing Q = s Q , and Y ⊂ X ⊂ Q irreducible subvarieties.

Proposition 5.3 Assume that the inequality
holds. Then the following estimate is true: Proof Assume the converse: By the assumption on the dimensions Y ⊂ Sing Q. Take an arbitrary point p ∈ Y \Sing Q.

Lemma 5.1 The variety X is contained in the tangent hyperplane T p Q.
Proof Assume the converse: X ⊂ T p Q. Then the effective cycle (X • (T p Q ∩ Q)) is well defined. Its degree is equal to deg X and its multiplicity at the point p is at least 2mult p X > deg X , which is impossible. Q.E.D. for the lemma. Therefore, the following inclusion takes place Proof of the proposition will be complete, if we show that the dimension of the right hand side of the inclusion is strictly smaller than dim X . Note that Sing Q ⊂ P N is a linear subspace and for any non-singular point p ∈ Q we have Sing Q ⊂ T p Q. Consider the section Q * of the quadric Q by a general linear subspace of codimension s Q + 1 (in particular, not meeting Sing Q). The quadric Q * is non-singular. Let Y * ⊂ X * be the corresponding sections of the varieties Y and X .
Obviously, Y * contains at least linearly independent points, so that the linear space p∈Y * T p Q * has the dimension not higher than the number However, by assumption the opposite inequality holds. This contradiction completes the proof of Proposition 5.3.

Exclusion of the linear case
Let us complete the proof of Proposition 5.1. The six linear equations and inequalities (32-35) form the second system of relations for now 5 + 3 = 8 parameters d * , μ * , λ * and δ. Joining the first and second systems of relations, adding to them the inequality (31), we obtain 11 linear equations and inequalities for 8 nonnegative real parameters. replacing the strict inequalities everywhere by the non-strict ones, we obtain 11+8=19 linear equations and non-strict linear inequalities, defining some (obviously, compact) convex subset ⊂ R 8 .

Corollary 5.2 The case 1.2 does not take place.
Proof Since a non log canonical singularity of the pair * is automatically a non log canonical singularity of the pair , the case 1.2 is a version of the case 1.1 (for one can take any hyperplane in ). Q.E.D. for the corollary.

Infinitely near case: III-exclusion of the non-linear case
In this section we exclude the case 2, which completes the exclusion of the infinitely near case (and so the proof of Theorem 1.5).

The case 2.1, B is not contained in a quadric
Let us consider first the 2.1 and assume that the subvariety B is not contained in any quadric hypersurface in E P . (Note, that in the case 2.1 the subvariety B is certainly not contained in the quadric Q P , since B ⊃ and ⊂ Q P .) Proposition 6.1 The following inequality holds: Proof Recall that the multiplicity mult o P is denoted by the letter ν and set ν B = mult B + P . In terms of the resolution of the maximal singularity of the system P we have ν = ν 1 and ν B = ν 2 . Assume that the opposite inequality holds: 5ν B > 2ν. Let us show that this assumption leads to a contradiction with the mobility of the linear system + P : the divisor E P can not be a fixed component of + P , that is, We will obtain a contradiction, thus proving the claim of our proposition.
Since the subvariety B is not contained in any quadric hypersurface, its degree deg B (as a subvariety of the projective space E P ) is at least 5.
Indeed, d B = deg B ≥ 3. Furthermore, B is not a cone over a curve: otherwise, B contains a linear subspace of codimension 3 in E P , which is excluded by the proof of Proposition 5.1. Now, projecting from a point of general position p ∈ B, we exclude the option d B = 3. If d B = 4 and B has at least one singular point of multiplicity 2 or 3, then B is contained in a hyperplane or an irreducible quadric, contrary to the assumption. Since a non-singular projective subvariety of codimension 2 and degree 4 in P k , k ≥ 4, is a complete intersection of two quadrics (this is a well known fact; see also [12]), then B is a complete intersection of two quadrics, either, if B is non-singular or is a cone over a subvariety of degree 4 and dimension ≥2. We have inspected all options. Therefore, d B ≥ 5.
Let Proof Assume the converse: there are three distinct points p 1 , p 2 , p 3 ∈ B , lying on the line L. Since ν B > n and ν ≤ 3n, we obtain, that L ⊂ Bs + P . As we noted above, this is impossible. Q.E.D. for the lemma. Now let us consider any 5 distinct points p 1 , . . . , p 5 ∈ B and the unique conic C ⊂ , containing those points. As the plane is generic, the conic C is not contained in the base locus Bs + P , therefore, the restriction C = + P | C is well defined. It is a linear series of degree 2ν with 5 base points of multiplicity ν B . Since 5ν B > 2ν, we have C ⊂ Bs + P , which is impossible. Proof of Proposition 6.1 is complete.
Now we can apply the technique of counting multiplicities and estimate the multiplicity of the self-intersection Z P at the point o and its strict transform Z + P along the subvariety B.
Set μ = mult o Z P , μ B = mult B Z + P .

Proposition 6.2
The following inequality holds: Proof As in the proof of Proposition 4.7, fix a maximal singularity, the centre of which on P + is a subvariety B and take its resolution. We use the standard notations, associated with the resolution. The graph is assumed to be modified, so that the inequality We have the inequality of Noether-Fano type where 1 = K i=L+1 p i , and, besides, we know that ν 1 ≤ 3n and 5ν 2 ≤ 2ν 1 ; the multiplicities ν i do not increase, By the technique of counting multiplicities, taking into account the inequalities we obtain the estimate For ν 1 = ν fixed, the minimum of the right hand side of the latter inequality on the hyperplane is attained at ν 2 = · · · = ν K = θ , where the value θ is computed from the equation Therefore, the inequality holds. On the other hand, the equality (41) can be re-written in the following way: Recall that ν and θ are connected by the inequality 5θ ≤ 2ν. As a result, we obtain that the sum μ + μ B is strictly higher than the minimum of the function x + y on the interval, cut out by the inequalities The more so, this minimum is strictly higher than the number It is easy to check that the minimum of the function (43) on the triangle is attained for ν = 3n, θ = 6 5 n and is equal to 81 5 n 2 . This completes the proof of Proposition 6.2.
The inequality (39) is so strong that it makes it possible to easily complete the exclusion of the case 2.1 (under the assumption that B is not contained in any quadric hypersurface in E P ). Indeed, since d B ≥ 5, we have the inequality It is easy to check that it is incompatible with the inequalities (39) and μ ≤ 12n 2 . This excludes the case under consideration (that is, the case 2.1 under the assumption that B is not contained in any quadric in E P ).

Case 2.1, B is contained in a quadric, but not in a hyperplane
Now let us consider the case 2.1 under the assumption that B is contained in some quadric in E P , but B = E P , that is, B is not contained in any hyperplane in E P .

Proposition 6.3
The following inequality holds: Proof Again we write ν B = mult B + P and ν = mult o P . Since B is not contained in a hyperplane, Sec(B) = E P . Let L be a general secant line of the variety B. Since the system + P has no fixed components, for a general divisor D ∈ P we have L ⊂ D + . Therefore, as we claimed. The proposition is shown.

Corollary 6.1
The following estimate is true: ν B ≤ 3 2 n. The following claim is an analog of Proposition 6.2 in the situation under consideration Proposition 6.4 The following inequality holds: Proof is completely similar to the proof of Proposition 6.2 given above: we argue in word for word the same way and, recalling that μ > 8n 2 , we get that the value μ + μ B is strictly higher than the minimum of the function max(ν 2 , 8n 2 ) + nθ 2 θ − n on the triangle This minimum is attained for ν = 2 √ 2n, θ = √ 2n and is equal to (10 + 2 √ 2)n 2 , which is what we need. Q.E.D.
The estimate (44) is essentially weaker than (39), however, this is compensated by the additional geometric information about the subvariety B: we know that B ⊂ Q * , where Q * = Q P is some irreducible quadric, and moreover by assumption B is not a hyperplane section of the quadric Q * .

Lemma 6.2 The degree of the subvariety B is at least 4.
Proof We must exclude the option d B = deg B = 3. Assume that this is the case. Then the rank of the quadratic form, defining Q * , is equal to 3 or 4, so that B is swept out by a one-dimensional family of linear subspaces of codimension 3 in E P . Proposition 5.1 excludes this situation. Q.E.D. for the lemma.

Corollary 6.2
The following inequality holds: Now we exclude the case under consideration in the same way as we used to exclude the case 1.1, with some simplifications. Let Z P = Z 0 + Z 1 be the T P -decomposition of the cycle Z P . Since B ⊂ Q P = T + P ∩ E P , we have mult B Z + 1 = μ B . Now, introducing the normalized parameters d i , μ i , i = 0, 1, and λ 1 , we obtain for them the system of the following inequalities: (24), (25), instead of (27) and (28) we have the estimate instead of (29) we have the estimate finally, instead of (26) we have the estimate and instead of (30) the stronger estimate Using MAPLE, it is easy to check that this system of linear equations and inequalities has no solutions already for M ≥ 5. This completes the exclusion of the case 2.1 under the assumption that B = E P .

The case 2.1, B is contained in a hyperplane
Assume that B is contained in some hyperplane ⊂ E P . By Proposition 5.1, B is a hypersurface of degree d B ≥ 2 in . Consider the linear system |H P − |, that is, the pencil of hyperplane sections, the base set of which is the intersection of the tangent section T P with the hyperplane in P that has as the tangent cone. Let Z P = Z 0 + Z 1 be such a decomposition of the cycle Z P , that Z + P = Z + 0 + Z + 1 is the B-decomposition of the effective cycle Z + P .
Let R ∈ |H P − | be a general divisor. For the effective cycle (Z 1 • R) we have: However, in the case under consideration (Z 1 • R) is an effective cycle of codimension two on the hyperplane section R, which itself satisfies the regularity conditions, and for that reason the inequality holds; the right hand side for M ≥ 18 does not exceed 3/M. Taking into account that mult o Z 0 ≤ 3 M deg Z 0 , we obtain a contradiction with the aprioric inequality (18). This excludes the case under consideration for M ≥ 18.

Remark 6.2
In the argument given above we used the fact that B ⊂ Q P : it is for that reason that the scheme-theoretic intersection (Z 1 • R) is well defined. However, if B ⊂ Q P , then B = ∩ Q P . Set = Bs|H P − | (see above). Obviously, where a ∈ Z + and Z * does not contain as a component, we repeat the previous argument and come to a contradiction for M ≥ 18.
If we use all the information available, we can exclude the case under consideration for smaller values of M as well. Namely, write Z P = Z 0 + Z 1 , where Z + P = Z + 0 + Z + 1 is the -decomposition of the effective cycle Z + P . The cycle (Z 1 • R) is well defined for a general divisor R ∈ |H P − |. Furthermore, write where the strict transform of this equality on P + is the -decomposition of the cycle (Z 1 • R) + . The cycle Z 10 satisfies the estimate but for Z 11 a much stronger inequality holds: since none of the components of the cycle Z 11 is contained in the tangent section T R = R ∩ T o R, so that we can form the effective cycle (Z 11 • T R ) and then apply to this cycle of codimension two on T R the technique of hypertangent divisors. Finally, setting we get the following system of linear equations and inequalities: (24)-(27), and also μ 10 + μ 11 = μ 1 + 2λ 1 + δ 1 , ξ 1 > 4, Applying MAPLE we see that this system is incompatible (even when we replace all strict inequalities by the non-strict ones) already for M ≥ 11. This completes the exclusion of the case 2.1.

The case 2.2, B is not contained in Q P
Now assume that the case 2.2 takes place, where B ⊂ Q P . Now, if B is not contained in a quadric, we obtain a contradiction, arguing as in Sect. 5.1. If B is contained in a quadric, but not contained in a hyperplane, then we obtain a contradiction, arguing as in Sect. 5.2. Therefore we assume that B ⊂ , where ⊂ E P is some hyperplane. Now, if M ≥ 18 or if B * ⊂ Q P , then we obtain a contradiction in word for word the same way as in Sect. 5.3. Therefore we assume that M ≤ 17 and B * ⊂ Q P is a subvariety of codimension 2. Let us consider the pencil of hyperplane sections |H P − |. Its base set = Bs|H P − | is a hyperplane section of the tangent section T P . Write where a ∈ Z + and Z * does not contain as a component.
and mult B * Z + * = mult B * Z + P − a > 4n 2 − a. Now let R ∈ |H P − | be a general divisor. By construction, R does not contain irreducible components of the cycle Z * and for that reason the effective cycle (Z * • R) of codimension 2 on R is well defined. Let Z R = (Z * • R) = Z 0 + Z 1 be the T R -decomposition of the cycle Z R .
Let Q R = Q P ∩ = T + R ∩ E P be the (projectivized) tangent cone to T R , a quadric in E R = R + ∩ E P = . The subvariety B * is a prime divisor on Q R and for that reason is cut out on Q R by a hypersurface in E R of degree δ * ≥ 1, so that 2δ * = d * = deg B * . Proposition 6.5 The equality δ * = 1 holds, that is, B * is a hyperplane section of Q R .
Proof Assume the converse: δ * ≥ 2. In that case d * ≥ 4. Therefore, the inequality holds. To compute the left hand part, write where N is an effective divisor on , not containing B as a component and β ≥ μ B . By the intersection theory, where d N = deg N . On the other hand, by assumption B * is not contained in a hyperplane , that is, B * = and for that reason the inequalities hold. Therefore, we have the estimate Besides, we remember that the inequalities hold, and also the inequalities μ > 8n 2 and μ + μ B > 12n 2 . Using MAPLE, it is easy to check (replacing, as usual, strict inequalities by non-strict ones), that the system of linear equations and inequalities, obtained above, has no solutions for M ≥ 13. Proof of Proposition 6.5 is complete. Therefore, B * = ∩ Q R , where ⊂ = E R is a hyperplane. Instead of the inequality (45) we have a weaker estimate and it is no longer sufficient to obtain a contradiction. Let us consider the linear system |H R − | on R and set * = Bs|H R − | to be its base set (a divisor on the tangent section). By the regularity conditions we have mult B * ( * ) + = 1. Write down Z 0 = c * + Z , where c ∈ Z + and Z does not contain * as a component. For a general divisor D ∈ |H R − | the effective cycle (D • Z ) of codimension two on T R is well defined and satisfies the inequalities Adding the corresponding normalized inequalities to the previous ones and using MAPLE, we see that for M ≥ 13 the case under consideration is impossible. This completes the exclusion of the case 2.2 under the assumption that B ⊂ Q P .

The case 2.2, B is contained in Q P
Assume that B ⊂ Q P . Note, first of all, that B is not contained in a hyperplane (that is, it is not a hyperplane section of Q P ): such an option is excluded by word for word the same arguments as those that were used in the case B ⊂ Q P , B ⊂ , where ⊂ E P is a hyperplane. In particular, d B ≥ 4 and the estimate (44) holds.

Proposition 6.6
The subvariety B * ⊂ Q P of codimension two is contained in a hyperplane ⊂ E P .
Proof Assume the converse. Let ⊂ Q P be a general linear subspace of maximal dimension, B * = B * ∩ ⊂ an irreducible subvariety of codimension two. For the linear span B * there are three options: 1. B * = , 2. B * is a hyperplane in , 3. B * = B * is a subspace of codimension two in .
Note at once, that the third option does not realize: (3) implies that deg B * = 2 and then B * is contained in a hyperplane, contrary to our assumption.
Furthermore, Sec(B * ) = B * . Set It follows from what was said that either W is an irreducible divisor on Q P , or W = Q P . However, in the first case W ∩ is an irreducible hypersurface in (for a general ) and for that reason W ∩ = B * is a hyperplane in , and then W is a hyperplane section of the quadric Q P , where B * ⊂ W , contrary to our assumption. Therefore, W = Q P . From here we get the following fact.

Lemma 6.3 For any effective divisor Y on the quadric Q P the inequality
holds (the degree deg Y is understood as the degree of an effective cycle of codimension 2 on E P ).
Proof Denote by the symbol H Q the class of a hyperplane section of the quadric Q P , so that Y ∼ γ H Q for some γ ≥ 1, where deg Y = 2γ . Let be a general linear subspace of maximal dimension on Q P and L ⊂ a general secant line of the variety B * . Since the lines L sweep out Q P , we may assume that L ⊂ |Y |. Let x, y ∈ B * be general points, where L = [x, y]. We have when the claim of the lemma follows. Q.E.D. Now let Z P = Z 0 + Z 1 be, as usual, the T P -decomposition of the cycle Z P . Setting λ i = 1 n 2 mult B Z i , i = 0, 1, we obtain the following system of linear equations and inequalities: (24,25,27), and also the estimate instead of (26), and also the estimates Now set ξ i = 1 n 2 mult B * Z + i , i = 0, 1. By the lemma shown above, the estimate μ 0 ≥ 4ξ 0 holds, besides, μ 1 ≥ ξ 1 and, as we know, ξ 0 + ξ 1 > 4. The inequality (47) can be sharpened. Write down where N is an effective divisor on the quadric Q P . Obviously, so that, applying Lemma 6.3 once again, we get the inequality Using MAPLE, we check that the system of linear equations and inequalities, obtained above, is incompatible. Q.E.D. for Proposition 6.6.

Exclusion of the case 2.2
Now let us assume that the hyperplane ⊃ B * is the only hyperplane in E P with that property, that is, B * is not the intersection of Q P with a linear subspace ⊂ E P of codimension two. In particular, d * = deg B * ≥ 4. Let R ∈ |H P − | be a general divisor of the pencil. Write down Z P = a + Z * , where = Bs|H P − |, a ∈ Z + and Z * does not contain as a component. To simplify the formulas, we will assume that a = 0 and Z * = Z P : if a ≥ 1, then the system of linear equations and inequalities, obtained below, remains incompatible, which is easy to check.
So Z P = Z 0 + Z 1 is the T P -decomposition of the cycle Z P and is not an irreducible component of the cycle Z 0 . Setting, as usual, and d i = 1 Mn 2 deg Z i , i = 0, 1, we obtain the standard set of linear equations and inequalities: (24,25,46), and also the inequalities (27), (48) with d B = 4 and the estimate Set ξ i = 1 n 2 mult B * Z + i , i = 0, 1. In our case ξ 0 + ξ 1 > 4.
Proof Z + 0 is an effective divisor on T + P , and its projectivized tangent cone Z + 0 ∩ E P is an effective divisor on the quadric Q P . Let ⊂ Q P be a general linear subspace.
Let p ∈ B * ∩ and q ∈ B ∩ be points of general position. The lines L = [pq] sweep out and for that reason we may assume that L ⊂ Z + 0 . Therefore, for the intersection numbers on Q P we have: 1 2 μ 0 n 2 = (L · (Z + 0 ∩ E P )) Q P ≥ (L · (Z + 0 ∩ E P )) p +(L · (Z + 0 ∩ E P )) q ≥ (ξ 0 +λ 0 )n 2 , which is what we claimed. Q.E.D. for the lemma. Now write down where N 1 is an effective divisor on , and N 0 = c( ∩ Q P ), c ∈ Z + . Since B * = , the inequality deg N i ≥ 2mult B * N i holds (for i = 0 it is the equality, since obviously mult B * N 0 = 1). Setting we obtain inequalities where n i = 1 n 2 deg N i . Setting α i = 1 n 2 mult o (Z i • R), i = 0, 1, we obtain the set of standard estimates (the last is true by the inequality deg B * ≥ 4, as B * = ). Besides, one more important inequality holds.

Lemma 6.5
The following estimate holds: Proof Once again, let ⊂ Q P be a general linear subspace of the maximal dimension and L a general secant line of the variety B * ∩ . The lines L sweep out the hyperplane section ∩ Q P and for that reason it is sufficient to show the inequality Since n 0 ≥ 2β, the inequality (50) implies the claim of our lemma.
Consider a general line L * ⊂ , intersecting L, and let S o be a generic twodimensional germ of an isolated quadratic singularity at the point o, S ⊂ T P , such that S + ∩ E P = L + L * , and S + is a non-singular surface. Obviously, whereas the effective 1-cycle Z + 0 | S + has the line L as a component of the multiplicity βn 2 . Taking this component out, we obtain that the effective 1-cycle does not have L as a component, and its multiplicity at two distinct points p, q ∈ L is at least Computing the intersection (C · L), we obtain the inequality which is what we need. Q.E.D. for the lemma. Finally, adding the inequality of Lemma 6.5 to the previous estimates, we obtain an incompatible system of linear equations and inequalities (checked using MAPLE), which completes the exclusion of the case under consideration.
Therefore, the only remaining possibility is when B * = ∩ Q P , where ⊂ E P is a linear subspace of codimension two. The claim of Lemma 6.4 is valid. Let R ∈ |H P − | be a general divisor, Z R = (Z P • R) an effective cycle of codimension two on R, Z P = Z 0 + Z 1 is, as usual, the T P -decomposition of the cycle Z P . We get the standard set of linear equalities and inequalities for that decomposition: (24,25,27,46,48) and (47) with d B = 4. Now let us consider the cycle Z R more carefully. Set where = Bs|H P − | is a hyperplane section of the hypersurface T R ; note that none of the components of the cycle Z 1 is not contained in T R . The support of the cycle Z 2 is contained in T R . For the subvariety we obviously have: deg = M, mult o = 2 and mult B * + = 1. Obviously, Setting μ i = 1 n 2 mult o Z i for i = 0, 1, we obtain the inequality Furthermore, the following equalities hold. The cycle Z 0 is an effective divisor on T R , which does not contain as a component. Let R * ∈ |H P − | be another general divisor. Obviously, R * ∩ T R = R ∩ R * ∩ T P = , so that none of the components of the cycle Z 0 is not contained in R * and therefore the cycle Z * 0 = (Z 0 • R * ) of codimension two on T R is well defined. The cycle Z * 0 is an effective divisor on . Setting μ * 0 = mult o Z * 0 , we obtain the inequality By the regularity conditions on the hypersurface R the inequality holds. But it is not hard to obtain a stronger estimate. By the regularity conditions and the Lefschetz theorem we have: is an irreducible reduced divisor on , which has the degree 2M and the multiplicity precisely 6 at the point o. Let Y be an irreducible component of the cycle Z * 0 . Then either the inequality (mult o /deg)Y ≤ (3/M) holds, or Y is not contained in the hypertangent divisor T o (T P ), so that the estimate is true. From this it follows, that the inequality holds. Using MAPLE, it is easy to check that the system of linear equations and inequalities for μ * , d * , c * , μ * and μ * 0 , obtained above, has no solutions. The case 2.2 is completely excluded.

Exclusion of the case 2.3
Assume that the case 2.3 takes place. We have μ B = mult B Z + P > 4n 2 , so that we get the following sequence of inequalities: where d B = deg B ≥ 2 (the case of a linear subspace was excluded by Proposition 5.1), whence we conclude that d B = 2, that is, B is a quadric in some hyperplane ⊂ E P .
If B ⊂ Q P , then we argue as in Sect. 5.3: we write down Z P = Z 0 + Z 1 and intersect Z 1 with a general divisor R ∈ |H P − |. Since μ B > 4n 2 , we obtain the linear inequalities which hold for M ≥ 6. Putting together and recalling that mult o Z P > 8n 2 , we obtain a contradiction, excluding the possibility B ⊂ Q P . So let us assume that B = ∩ Q P is a hyperplane section of the quadric Q P . Consider = Bs|H P − |, which is a hyperplane section of the variety T P . Write down where a ∈ Z + and Z * does not contain as a component. Now, arguing in word for word the same way as above in the case B ⊂ Q P , where Z P is replaced by Z * , we obtain a contradiction. The case 2.3 is excluded. Proof of Theorem 1.5 is complete.
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