Birational geometry of singular Fano hypersurfaces of index two

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the natural parameter space.

Let V ⊂ P be a hypersurface of degree M. If it is irreducible, reduced, factorial and has at most terminal singularities, then V is a Fano variety of index two: where H is the class of hyperplane section. If P ⊂ P is a linear subspace of codimension 2, then restricting to the hypersurface V the linear projection α P : P P 1 from the subspace P , we define on V a structure of a Fano-Mori fibre space π P : V P 1 .
Let λ: Y → S be a rationally connected fibre space, that is, a surjective morphism of projective varieties, dim S 1, where the fibre of general position λ −1 (s), s ∈ S, and the base S are rationally connected.
Here is the main result of the present paper. holds, (iii) for every V ∈ U and every birational map χ: V → Y onto the total space of the rationally connected fibre space λ: Y → S over a positive-dimensional base S we have S = P 1 and for some isomorphism β: P 1 → S and some subspace P ⊂ P of codimension 2 the equality holds, that is, the following diagram commutes: For a Zariski general smooth hypersurface V ⊂ P the claim (iii) of Theorem 1 was shown for M 16 in [1]. Hypersurfaces with at least one singular point form a divisor in the space F . Thus in the present paper we essentially improve the main result of [1]: we extend it to hypersurfaces with bounded singularities and give an effective estimate for the codimension of the complement to the set U of "correct" hypersurfaces (which grows as 1 2 M 2 when the dimension M grows). Theorem 1 immediately implies the standard set of facts about birational geometry of the variety V ∈ U. Corollary 1. For every hypersurface V ∈ U the following claims are true. (i) On the variety V there are no structures of a rationally connected fibre space (and therefore, of a Fano-Mori fibre space) over a base of dimension 2. In particular, on V there are no structures of a conic bundle and del Pezzo fibrations, and the variety V itself is non-rational.
(ii) Assume that there is a birational map χ: V Y , where Y is a Fano variety of index r 2 with factorial terminal singularities, such that Pic Y = ZH Y , where K Y = −rH Y , and the linear system |H Y | is non-empty and free. Then r = 2 and the map χ is abiregular isomorphism.
(iii) The groups of biregular and birational automorphisms of the variety V are equal: Bir V = Aut V . 0.2. The regularity conditions. Now we give an explicit description of the open set F reg ⊂ F , consisting of hypersurfaces, satisfying the regularity conditions, stated below. We will show that for U = F reg all statements of Theorem 1 are true.
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 hypersurface of degree M. It is defined by an equation f = 0, where f = q 1 + q 2 + · · · + q M is a non-homogeneous polynomial in the variables z * , decomposed into homogeneous components q i of degree i 1. The regularity conditions depend on whether the point o ∈ V is singular or non-singular, that is, whether q 1 ≡ 0 q 1 ≡ 0.
First, we state the regularity conditions for a non-singular point.
(R1.1) For every linear subspace Π ⊂ C M +1 of the standard coordinate space with the coordinates z * , of codimension codim Π = c ∈ {0, 1, 2, 3} and such that q 1 | Π ≡ 0 (that is, Π ⊂ T o V ), the sequence q 1 | Π , q 2 | Π , . . . , q M −c | Π is regular in the local ring O o,Π , that is, the system of equations in the projective space P(Π) ∼ = P M −2 is irreducible, reduced factorial complete intersection of type 2 · 3. Definition 1. We say that the hypersurface V ∈ F is regular, if at every nonsingular point o ∈ V it is regular in the sense of the conditions (R1. [1][2][3], and in every singular point o ∈ V it is regular in the sense of the conditions (R2. [1][2][3].
The set of regular hypersurfaces is denoted by the symbol F reg . Obviously, F reg ⊂ F is a Zariski open subset.
The condition (R2.2) implies that the codimension of the singular set Sing V of a regular hypersurface V is at least 10, so that V is irreducible, reduced and by the well known theorem of Grothendieck [2], factorial. The same condition (R2.2) implies that the singularities of a regular hypersurface V are terminal (see [3] and also [4]; in the latter paper at the end of Subsection 2.1 it is explained that quadratic singularities, the rank of which is bounded from below, are stable with respect to blow ups, which, in its turn, makes it very easy to see the property of being terminal). Therefore, for a regular hypersurface V the claim (i) of Theorem 1 is true. By what was said, Theorem 1 is implied by the following two facts. 0.3. The method of maximal singularities. For an arbitrary subspace P ⊂ P of codimension 2 by the symbol |H − P | we denote the pencil of divisors cut out on V by the pencil of hyperplanes containing P . In the notations of the part (iii) of Theorem 1 let Σ Y be the λ-pull back of some very ample linear system on the base S, and Σ its strict transform on V with respect to χ. The linear system Σ is mobile (that is, has no fixed components) and we may assume that for some n 1 Σ ⊂ |2nH| (replacing, if necessary, the very ample system on the base S by its symmetric square). This whole set of geometric objects: the hypersurface V ∈ F reg , the rationally connected fibre space λ: Y → S, the birational map χ, the linear systems Σ Y on Y and Σ on V , and therefore, the number n 1, is assumed to be fixed. It is well known (see, for instance, [5, Chapter 2, Section 1], and also Subsection 1.1 of the present paper), that the mobile linear system Σ has a maximal singularity: for some exceptional divisor E * over V the Noether-Fano inequality holds: where a(E * ) is the discrepancy of E * with respect to the model V . In a different way this can be expressed as follows: the pair (V, 1 n D) is not canonical for a general divisor D ∈ Σ or, even simpler, the pair (V, 1 n Σ) is not canonical. There are mobile linear systems with a maximal singularity on V . For instance, let E P be the exceptional divisor of the blow up of the subvariety V ∩ P of codimension 2 on V , where P ⊂ P is a linear subspace of codimension 2. Obviously, a(E P ) = 1, so that the "double pencil" |H − P |, that is, the linear system |2H − 2P | ⊂ |2H|, has E P as a maximal singularity, since Theorem 3 essentially means that any linear system Σ with a maximal singularity is composed of a pencil |H − P |. The proof of Theorem 3 consists of two main steps.
Theorem 4. Assume that for a certain linear subspace P ⊂ P of codimension 2 the inequality mult P ∩V Σ > n (2) holds. Then Σ is composed of the pencil |H − P |, that is, every divisor D ∈ Σ is a sum of 2n hyperplane sections from that pencil. Theorem 5. For a linear system Σ with a maximal singularity there is a linear subspace P ⊂ P of codimension 2 satisfying the inequality (2).
Theorem 3 obviously follows from Theorems 4 and 5. Proof of Theorem 5 is most difficult. Proof of Theorem 2 is not hard. 0.4. The structure of the paper. In §1 we show Theorem 4. The following fact is crucial in the proof of Theorem 4: the global log canonical threshold of every hyperplane section of the hypersurface V is equal to 1. The equality lct(F ) = 1 for Fano hypersurfaces F ⊂ P M of degree M, satisfying certain restrictions for the singularities and the regularity conditions at non-singular and singular points, has been recently proven in [4], so that in this paper we just check that every hyperplane section of the hypersurface V satisfies the requirements of [4].
In §2 we prove Theorem 2. For each of the regularity conditions we estimate the codimension of the set of hypersurfaces which do not satisfy that condition at at least one point. After that, by means of the technique of hypertangent divisors we prove certain estimates, bounding the multiplicities of irreducible subvarieties of the hypersurface V at singular points o ∈ Sing V from below. Those estimates will be needed later.
In §3 we start the proof of Theorem 5. Following the traditional scheme of arguments of the method of maximal singularities, we assume that there is no linear subspace P ⊂ P of codimension 2, satisfying the inequality (2). We have to show that under this assumption the linear system Σ has no maximal singularities at all: this contradiction proves Theorem 5. In Subsection 3.1 we prove that the centre B * of the maximal singularity E * is contained in the singular locus Sing V of the hypersurface V . In order to do this, it is sufficient to check that if B ⊂ Sing V , then the maximal singularity E * is excluded by the arguments of [1].
In Subsection 3.2 for a point o ∈ B of general position (which by what we have already proven is a quadratic singularity of the hypersurface V ) we prove that there is, generally speaking, another singularity E of the linear system Σ, satisfying a Noether-Fano type inequality, which is weaker than (1), but still strong enough for our purposes. We do it by means of the inversion of adjunction [6] similar to the arguments of [1,Subsection 4.2]. Finally, in Subsection 3.3 we recall the improved version of the technique of counting multiplicities for a complete intersection singularity [7].
In §4 we prove certain technical statements about the secant variety of a subvariety of small codimension on a quadratic hypersurface of sufficiently high rank; those technical facts are used in §5 for exclusion of the singularity E.
§5 is the central part of the proof of Theorem 5. Depending on the type of the singularity E (the types are defined in Subsection 3.2), it is excluded by different methods. In accordance with the traditional scheme of the method of maximal singularities (see [5, Chapter 2]), we consider the self-intersection Z = (D 1 • D 2 ) of the mobile linear system Σ and prove that the existence of the singularity E imposes so strong restrictions on the singularities of the self-intersection Z, which can not be satisfied for an effective cycle of codimension 2 on V . Thus we prove that the mobile linear system Σ can not have the singularity E, and therefore can not have the maximal singularity E * , either. This contradiction completes the proof of Theorem 5 (and the main Theorem 1). 0.5. Historical remarks and acknowledgements. The few attempts to study birational geometry of higher-dimensional Fano varieties of index higher than 1 are listed in the introduction to [1], see also the introduction to [4]. Here we note that, starting from the paper [3], the results about birational rigidity of particular classes of Fano varieties become effective in the sense that an explicit effective estimate for the codimension of the subset of non-rigid varieties in the natural parameter space of the given family is produced. Those results (see [9,10]) are very important because they open the way for the study of the problem of birational rigidity for Fano fibre spaces over a base of high (ideally -arbitrary) dimension, the fibres of which belong to a given family of Fano varieties. The first breakthrough in that direction is the paper [4]. In the present paper the result of [1] is extended to singular Fano hypersurfaces of index 2 and becomes effective in the sense described above: we give an explicit effective estimate of the set of hypersurfaces, the birational geometry of which does not satisfy the property (iii) of Theorem 1.
Recently quite a few papers were produced, proving the stable non-rationality of various classes of Fano varieties and Fano-Mori fibre space, see, for instance, [11][12][13][14][15][16][17][18][19] (the list is by no means complete). The importance of those results, obtained by completely different methods (compared to the method of maximal singularities), can not be overestimated. Note, however, that the stable non-rationality is shown for a very general variety in the family. The method of maximal singularities gives birational rigidity (or an explicit exhaustive description of birational geometry like what is done in this paper) for a Zariski general variety, together with an effective estimate for the codimension of the complement in the parameter space.
Note also the recent paper on the birational rigidity of singular Fano three-folds [20], the recently published paper [25] and the papers [22,21] on the groups of birational automorphisms.
The author thanks The Leverhulme Trust for the financial support of the present project (Research Project Grant RPG-2016-279).
The author is also grateful to the members of the Divisions of Algebraic Geometry and Algebra of Steklov Mathematical Institute for the interest to this work and also to the colleagues -algebraic geometers at the University of Liverpool for the general support.

The pencils of hyperplane section and the regularity conditions
In the section we prove Theorem 4. As the first step of the proof, we consider the new model of the hypersurface V , which is obtained by blowing up the subvariety V ∩ P (Subsection 1.1). After that we get the alternative: either the claim of Theorem 4 is satisfied, or the strict transform of the linear system Σ on the new model again has a maximal singularity (Subsection 1.2). Finally, in Subsection 1.3 we show that the results of [4] imply that the second case does not realize, because the global log canonical threshold of every hyperplane section of the hypersurface V is equal to 1. This completes the proof of Theorem 4.
1.1. The structure of a Fano fibre space. Let us prove Theorem 4. Set B = P ∩ V . Let ϕ: V + → V be the blow up of the subvariety B. Denote by the symbol E B the exceptional divisor of this blow up. The variety V + can be seen as the strict transform of the hypersurface V with respect to the blow up ϕ P : P + → P of the linear subspace P , so that E B = V + ∩ E P , where E P = ϕ −1 (P ). The linear projection P P 1 from the subspace P extends to a P M -bundle π P : P + → P 1 . Set π = π P | V + : V + → P 1 . Proposition 1.1. (i) The variety V + and every fibre F t = π −1 (t), t ∈ P 1 , are factorial and have at most terminal singularities. Every fibre F t , t ∈ P 1 , is a Fano variety.
(ii) The equalities 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 and Proof. The fibres of the projection π P are isomorphic to hyperplanes (containing the subspace P ) in P, so that the fibres F t are isomorphic to the corresponding hyperplane sections of the hypersurface V , that is, to hypersurfaces of degree M in P M . The conditions (R1.2) and (R2.2) imply that every hypersurface F t ⊂ P M , t ∈ P 1 , gas at most quadratic singularities of rank at least 11. Therefore, the variety V + also has at most quadratic singularities of rank 11. The claim (i) follows from here. The claim (ii) is checked by obvious computations. Q.E.D. for the proposition. Now let us consider the strict transform Σ + of the linear system Σ on V + . This is a mobile linear system, and for some m ∈ Z + and l ∈ Z we have the inclusion The formulas of part (ii) of Proposition 1.1 imply that m = 2n − mmm B Σ and l = 2(mmm B Σ − n) 2. If m = 0, then the linear system Σ + is composed of the pencil |F |, so that the system Σ is composed of the pencil |H − P |, as Theorem 4 claims. Therefore let us assume that m 1, and show that this assumption leads to a contradiction.

1.2.
Maximal singularities of the system Σ + . The following claim is true. Proposition 1.2. The linear system Σ + has a maximal singularity: for some exceptional divisor E + over V + the Noether-Fano inequality holds, that is, for a general divisor D + ∈ Σ + the pair (V + , 1 m D + ) is not canonical. Proof. This is a particular case of a general well known fact, see [5, Chapter 2, Section 1]. For the convenience of the reader we give a sketch of a proof. Let V → V + be the resolution of singularities of the birational map χ • ϕ: V + Y , and Σ the strict transform of the linear system Σ + on V . Furthermore, let E be the set of all prime exceptional divisors of the resolution V → V + . Recall that Σ is the pull back of the free linear system Σ Y on V . Since divisors of the system Σ Y by assumption are pulled back from the base S, and the general fibre of the projection λ: Y → S is rationally connected, for a general divisor D ∈ Σ the class D + m K is not pseudoeffective ( K = K V for the brevity of writing). However, (for simplicity of notations the pull back of a divisor is denoted by the same symbol as the divisor itself), so that We conclude that in the right hand side for at least one E the corresponding coefficient is negative. Q.E.D. for the proposition. Remark 1.1. In a similar way one proves that the original linear system Σ has a maximal singularity, see Subsection 0.3.
Let R ⊂ V + be the centre of the maximal singularity E + on V + , so that codim(R ⊂ V + ) 2. There are two options: • R covers the base P 1 : π(R) = P 1 , • π(R) is a point on P 1 .
Assume that the first option takes place. Restricting the linear system Σ + onto the fibre F t of general position, we obtain a mobile linear system Σ t ⊂ |mH t |, where H t is the class of a hyperplane section of F t ⊂ P M , and moreover, the pair (F t , 1 m Σ t ) is not canonical (that is, Σ t has a maximal singularity). In [3] it was shown that this is impossible (under the weaker assumptions about the singularities of the hypersurface F t ⊂ P M and for weaker, than in the present paper, regularity conditions at every point). Therefore we may assume that the second option takes place: R ⊂ F t for some t ∈ P 1 . Somewhat abusing the notations, we write F instead of F t . Since the linear system Σ + is mobile, it can be restricted onto F and by inversion of adjunction [6,Section 17.4] obtain an effective divisor D F ∼ mH F , such that the pair (F, 1 m D F ) is not log canonical. However, this contradicts to the following fact. Proposition 1.3. For every divisor ∆ ∈ |mH F | the pair (F, 1 m ∆) is log canonical.
Proof is given below in Subsection 1.3. The contradiction obtained above shows that the case m 1 is impossible. Proof of Theorem 4 is complete. Q.E.D.
1.3. The global log canonical threshold of a fiber. The claim of Proposition 1.3 is shown in [4,Theorem 1.4] under the assumption that the hypersurface F ⊂ P M satisfies certain regularity conditions at every point o ∈ F (the conditions of the same type that the conditions (Rα.β) of the present paper). Therefore, in order to prove Proposition 1.3, it is sufficient to compare the conditions used in [4] with the conditions in Subsection 0.2 of the present paper and make sure that the latter are not weaker. In order to make the reading more convenient, we reproduce the regularity conditions from [4] below. To avoid any misunderstanding, the condition which in [4] has number (Rα.β) (for instance, (R2.1)), will be denoted by (R * α.β).
So let F ⊂ P M be a hypersurface of degree M, o ∈ F an arbitrary point, (u 1 , . . . , u M ) a system of affine coordinates with the origin at the point o and w = q * 1 + q * 2 + q * 3 + · · · + q * M the affine equation of the hypersurface F with respect to that system of coordinates, decomposed into homogeneous components. Here is the list of conditions, which should be satisfied for the hypersurface F in [4]. Let us first consider a non-singular This condition is satisfied because it is a particular case of the condition (R1.1) (for c = 1).
(R * 1. 2) The quadratic form q * 2 | {q * 1 =0} is of rank 6 and the linear span of every irreducible component of the closed set is the hyperplane {q * 1 = 0}. The first part of this condition follows from (R1.2) (the rank of the quadratic form q * 2 | {q * 1 =0} turns out to be at least 9), and the second part follows from (R1.3), since by the condition (R1.3) the closed set (3) is irreducible, reduced and of codimension 2 in the hyperplane {q * 1 = 0}, that is, forms an irreducible and reduced complete intersection of type 2 · 3.
(R * 1.3) For every hyperplane P ⊂ P M , P ∋ o, P = T o F , the algebraic cycle of the scheme-theoretic intersection This condition holds in our case: a section of F by two hyperplanes is a section of V by three hyperplanes, and therefore it has at most quadratic singularities of rank 7 and for that reason, (P • {q * 1 = 0} • F ) is a factorial hypersurface in the projective space P ∩ {q * 1 = 0}. The restriction of the quadratic form q * 2 (that is, the restriction of q 2 ) onto this projective space is of rank 7. Therefore, the condition (R * 1.3) holds. Now let us consider a singular point o ∈ F . q * 1 ≡ 0, so that the equation w starts with q * 2 .
If the singularity o ∈ F comes from a singularity of the original hypersurface V , then the condition (R * 2.1) follows from the condition (R2.1). If the singularity o ∈ F comes from a non-singular point of the hypersurface V (that is, F is a section of V by the hyperplane which is tangent to V at this point), then (R * 2.1) follows from (R1.1). In any case the condition (R * 2.1) holds.
(R * 2. 2) The quadratic form q * 2 is of rank at least 8. In our case this rank is at least 11. (R * 2.3) Considering (u 1 , . . . , u M ) as homogeneous coordinates (u 1 : · · · : u M ) on P M −1 , and the quadric hypersurface {q * 2 = 0} ⊂ P M −1 , let us construct the divisor . This divisor should not be a sum of three (not necessarily distinct) hyperplane sections of this quadric, taken from one linear pencil.
This condition follows from (R1.3), if the point o ∈ F comes from a non-singular point of the hypersurface V , and from (R2.3), if o ∈ F comes from a singular point of V .
Thus we have checked that every hyperplane section of V satisfies the regularity conditions of the paper [4]. Therefore, the global log canonical threshold of every fibre F of the fibre space π: V + → P 1 is equal to 1. The proof of Proposition 1.3 is complete.

Regular hypersurfaces
In this section we prove Theorem 2 and its immediate geometric implications. In Subsection 2.1 we consider all regularity conditions but the last one (R2.3). In Subsection 2.2 we estimate the codimension for the violation of the condition (R2.3), using a technical fact which is shown in 2.3. This completes the proof of Theorem 2. In Subsection 2.4 we prove geometric facts which follow from the regularity conditions and will be used in the proof of Theorem 5 in Sections 3-5. For that purpose in Subsection 2.4 we briefly recall the technique of hypertangent divisors.

Violations of the regularity conditions. Let us prove Theorem 2.
We need to estimate the number of independent conditions which are imposed on the hypersurface V (that is, on the coefficients of the polynomial f , defining this hypersurface) by violation of each of the six regularity conditions. Let us define the following polynomials of one real variable: . Let F α.β ⊂ F be the closed subset of hypersurfaces that do not satisfy the condition (Rα.β) at at least one point, where α ∈ {1, 2} and β ∈ {1, 2, 3}. The following claim is true.
Proposition 2.1. The following inequality holds: It is easy to see that for M 16 the minimum of the values γ α.β (M) is attained for α = 1, β = 2, which implies Theorem 2.
Proof of Proposition 2.1. Let us consider each of the conditions (Rα.β) separately.
The case (α, β) = (2, 2). This is an elementary exercise. The only case which is non-trivial and was not considered in the previous papers, is the case (α, β) = (2, 3). For that case, we give a complete detailed proof. We may (and will) assume the condition (R2.2) to be satisfied.
For the convenience of our arguments set • are singular at that point, • do not satisfy the condition (R2.3) at that point for the subspace Π.
Now we reduce the global statement (4) to the corresponding local statement. Proposition 2.2. The following inequality holds: Taking into account that the point o runs through P, and Π varies in the 2(M −1)dimensional Grassmanian, and that the point o lies on V and is a singular point of that hypersurface, by an elementary dimension count we check that Proposition 2.2 implies the inequality (4) for (α, β) = (2, 3). Q.E.D. for Proposition 2.1.

Violation of the condition (R2.3). Let us prove Proposition 2.2.
The symbol P k,Π stands for the space of homogeneous polynomials of degree k on P(Π) ∼ = P M −2 . The restrictions of the polynomials q 2 and q 3 onto P(Π) are denoted by the symbolsq 2 andq 3 , respectively, and the set of their common zeros {q 2 =q 3 = 0} ⊂ P(Π) by the symbol Z(q 2 ,q 3 ). It is obvious that the codimension of the set F 2.3 (o, Π), which we need to estimate, is equal to the codimension of the set B ⊂ P 2,Π × P 3,Π of pairs (q 2 ,q 3 ), such that the set Z(q 2 ,q 3 ) is not an irreducible reduced factorial complete intersection of type 2 · 3 in P(Π).
We note at once that the quadratic formq 2 is by the condition (R2.2) of rank at least 9, so that the quadric {q 2 = 0} ⊂ P(Π) is for sure factorial. It is easy to compute that for a fixed formq 2 of rank 9 the set of cubic polynomialsq 3 ∈ P 3,Π , such that the divisor {q 3 | {q 2 =0} = 0} on the quadric {q 2 = 0} is non-reduced or reducible has codimension 1 6 M(M + 1)(M − 4) in P 3,Π . Since this is much higher than γ 2.3 (M), we may (and will) assume that the set Z(q 2 ,q 3 ) is irreducible, reduced and of codimension 2 in P(Π). It remains to consider the condition for this set to be factorial. Let p ∈ Z(q 2 ,q 3 ) be an arbitrary point and (u 1 , . . . , u M −2 ) a system of affine coordinates on P(Π) with the origin at the point p. Let P(Π) + → P(Π) be the blow up of the point p with the exceptional divisor E p ∼ = P M −3 , equipped with the natural homogeneous coordinates (u 1 : · · · : u M −2 ). The affine polynomials in the (non-homogeneous) variables u * , corresponding to to the homogeneous polynomials q 2 ,q 3 , we denote, somewhat abusing the notations, by the same symbolsq 2 ,q 3 . We getq whereq i,j are homogeneous of degree j. We say that the point p is a correct biquadratic point of the set Z(q 2 ,q 3 ), ifq 2,1 ≡q 3,1 ≡ 0, and the closed set Let X ⊂ P 2,Π × P 3,Π be the set of pairs such that Z(q 2 ,q 3 ) is irreducible, reduced, of codimension 2 in P(Π), and its every point p ∈ Z(q 2 ,q 3 ) • either is non-singular, • or is a quadratic singularity of rank 5, • or is a correct bi-quadratic point.
By Grothendieck's theorem [2] for the pair (q 2 ,q 3 ) ∈ X the complete intersection Z(q 2 ,q 3 ) is factorial, so that B ∩ X = ∅ and in order to prove Proposition 2.2, it is sufficient to show that the codimension of the complement to the set X in P 2,Π ×P 3,Π is at least γ * 2.3 (M). Recall that rkq 2 9, and the complete intersection Z(q 2 ,q 3 ) is irreducible and reduced. Fix a point p ∈ Z(q 2 ,q 3 ). Assume first that at least one of the linear forms q 2,1 ,q 3,1 is not identically zero, but these forms are linearly dependent. If, moreover, q 2,1 ≡ 0, thenq 3,1 ≡ 0 and the point p is a quadratic singularity of rank which is what we need. Ifq 2,1 ≡ 0, then there is a unique constant λ, such that q 3,1 = λq 2,1 . In that case p is a quadratic singularity of rank If this rank 4, then for a fixed polynomialq 2 we get 1 2 (M − 7)(M − 6) independent conditions for the polynomialq 3 . Taking into account that the constant λ varies in a 1-dimensional family, there is the dependenceq 3,1 = λq 2,1 , the point p varies in P M −2 and the polynomialsq 2 ,q 3 vanish at that point, we get precisely the codimension γ * 2.3 (M) for the violation of the condition about the rank of quadratic points. It remains to consider the caseq 2.1 ≡q 3.1 ≡ 0 and estimate the codimension for the violation of the condition about bi-quadratic points. For this purpose we state and solve the following general problem. Let P 2,N +1 be the space of quadratic forms on P N , where N 8. Let Y ⊂ P ×2 2,N +1 be the set of pairs (g 1 , g 2 ), such that the closed set of common zeros is an irreducible reduced complete intersection of codimension 2, and moreover, The following fact is true.
Let us complete the proof of Proposition 2.2. Setting in Proposition 2.3 N = M − 3, we get that violation of the condition about bi-quadratic points at the fixed point p gives the codimension 1 2 (M − 7)(M − 6) − 2. Now the standard dimension count (taking into account thatq 2,1 ≡q 3,1 ≡ 0, and also the conditions p ∈ Z(q 2 ,q 3 ) and the variation of the point p) completes the proof of Proposition 2.2.

Complete intersections of two quadrics. Let us prove Proposition 2.3.
Since the set of quadratic forms of rank 4 has codimension 1 2 (N − 3)(N − 2) in P 2,N +1 , we may assume that rk g i 5 for i = 1, 2, so that the quadric {g 1 = 0} is factorial. If the condition of irreducibility and reducedness of the set Z(g 1 , g 2 ) is violated, this imposes on g 2 a lot more conditions than the required 1 Thus only the condition about the singularities of the set Z(g 1 , g @ ) needs to be considered. We argue as in [9, Section 3.3], somewhat improving the estimate obtained in that paper. The key observation (used in [9]) is that if p ∈ Sing Z(g 1 , g 2 ), then for some λ 1 , λ 2 (where (λ 1 , λ 2 ) = (0, 0)) the point p is a singular point of the quadric {λ 1 g 1 + λ 2 g 2 = 0}. In order to obtain a somewhat more precise, than in [9], estimate for the codimension of the set of "incorrect" pairs, we have to consider several cases. For a quadratic polynomial g ∈ P 2,N +1 the symbol C(g, k) stands for the cone with the vertex g, the base of which is the set of quadratic forms of rank k. The vertex can lie on the base: C(g, k) is the closure Case 1: rk g 1 = 5. We have 1 2 (N − 4)(N − 3) independent conditions for g 1 . If for a fixed quadric g 1 we have, into the bargain, g 2 ∈ C(g 1 , 6), this gives in addition 1 2 (N − 5)(N − 4) − 2 independent conditions for g 2 , and we get the total (N − 4) 2 − 2 independent conditions for the pair (g 1 , g 2 ), which is much higher than what we need. Therefore we may assume that g 2 ∈ C(g 1 , 6).
This implies that in the pencil {λ 1 g 1 + λ 2 g 2 = 0} all quadrics, apart from g 1 , are of rank 7. Therefore, the codimension of the set (λ 1 :λ 2 ) =(1:0) in P N is at least 6. On the other hand, Sing{g 1 = 0} is a (N − 5)-dimensional subspace in P N , so that the condition gives for g 2 the codimension 1 2 (N −4)(N −3) (for a fixed g 1 ), and for the pair (g 1 , g 2 ) the codimension (N − 4)(N − 3). Removing this set of high codimension, we may assume that is of dimension at most N − 6. We get finally that the codimension of the set Sing Z(g 1 , g 2 ) is at least 5 in P N in the case under consideration (for the pairs (g 1 , g 2 ) ∈ P ×2 2,N +1 , lying outside a closed subset of high codimension). Case 2: rk g 1 = 6. Here we argue word for word as in Case 1. The only difference is that we get somewhat fewer ( 1 2 (N − 5)(N − 4)) independent conditions for g 1 . Together with the conditions for the form g 2 we get the total codimension of the set of pairs (g 1 , g 2 ) to be much higher than we need. In this case we do not need to exclude the option g 2 | Sing{g 1 =0} ≡ 0.
Case 3: rk g 1 7. This is the case of general position for g 1 , so that no conditions are imposed on g 1 . For a fixed form g 1 the condition conditions. This is the estimate that we need. If g 2 ∈ C(g 1 , 5), then a general quadric in the pencil {λ 1 g 1 + λ 2 g 2 = 0} is of rank 7 and at most finitely many quadrics are of rank 6. This implies that codim(Sing(g 1 , g 2 ) ⊂ P N ) 6, which is what we need. Proof of Proposition 2.3 is complete. Q.E.D.   (5) is trivial. Assume that Y ∋ o. In that case for a general hypertangent divisor T a+2 ∈ Λ a+2 we get: Y ⊂ |T a+2 | (the vertical lines mean the support of the divisor), so that the effective cycle of the scheme-theoretic intersection (Y • T a+2 ) of codimension (a + 1) on V Π is well defined. Since Λ j ⊂ |jH Π | (where H Π is the class of a hyperplane section of V Π ) and mult o Λ j (j + 1), we get whence the inequality (5) Proof. Let us apply Proposition 2.4 to the effective cycle of the intersection (Y • T 2 ) of codimension 2 on V Π . Q.E.D.
3 Maximal singularities of the system Σ In this section we begin to study the maximal singularity E * . First (Subsection 3.1) we show that the centre B * of this singularity is contained in Sing V . In order to do this, we check that the arguments of [1] exclude a maximal singularity, the centre of which is not contained in Sing V . After that (Subsection 3.2), using inversion of adjunction, we derive from the existence of the singularity E * the existence of a, generally speaking, another singularity E of the linear system Σ, the centre of which is a point o ∈ Sing V , and moreover, E has some good properties (which may not be satisfied for E * ). Then we classify the types of the singularity E. Finally, in Subsection 3.3 we recall the technique of counting multiplicities which makes use of combinatorial invariants of the oriented graph associated with the singularity E.
3.1. The linear system Σ at non-singular points of the hypersurface V . Starting from this moment, we assume that the mobile linear system Σ ⊂ |2H| satisfies the inequality mult P ∩V Σ n for every linear subspace P ⊂ P of codimension 2. On the other hand, the system Σ has the maximal singularity E * (see Subsection 0.3). Theorem 5 will be shown if we derive a contradiction from this fact. We will do it in several steps, excluding the possible types of the maximal singularity E * one after another. The first step is given by the following statement. Proposition 3.1. The centre B * of the maximal singularity E * on V is contained in the singular locus Sing V .
Proof. Let us assume the converse: B * ⊂ Sing V and show that the arguments of [1] exclude this option. Let us consider separately the three cases: The maximal singularity is excluded in each of these three cases in a different way. Note that the inequality mult B * Σ > n holds.
Assume that Case 1 takes place. In this case B * is a maximal subvariety of the system Σ and, arguing in a word for word the same way as in [1, Section 3.1], we conclude that B * = P (since every hyperplane section of the hypersurface V is a factorial variety). Furthermore, in the notations of [1, Section 3.1] we conclude that Sec(B * ) = π P (Sec(B * )) = P, that is, the claim of [1, Proposition 3.1] is true in our case, either.
Indeed, the proof of that claim, given in [1, Section 3.3], makes use of only one fact, that B * is contained in a non-singular hypersurface, which does not contain cones over a positive-dimensional base (that is, cones of dimension 2). In our case the inequality codim(Sing V ⊂ V ) 12 holds, which follows from the regularity condition (R2.2), so that for a general linear subspace Π ⊂ P of dimension 12 the hypersurface V Π = V ∩ Π is non-singular and B * ∩Π is an irreducible subvariety of codimension 2 on V Π . Furthermore, V Π does not contain cones of dimension 2, because V does not contain any (by the conditions (R1.1) and (R2.1), there are at most finitely many lines on V through every point of V ). Therefore, we have Sec(B * ∩ Π) = Π, which implies that Sec(B * ) = P, as we claimed. Now the arguments of [1, Sections 3.1,3.2], excluding a maximal subvariety of codimension 2 -in our case it is denoted by B * -work word for word. The curves C ± , R do not touch the set Sing V for a general point x ∈ P and a general curve Γ, see [1, Section 3.1] (p. 739-740). Therefore, Case 1 can not take place.
Assume that Case 2 takes place. Here we argue as in the proof of [1, Lemma 4.1]. Let Z = (D 1 • D 2 be the self-intersection of the linear system Σ, where D 1 , D 2 ∈ Σ are general divisors. The 4n 2 -inequality holds: Again let Π be a general linear subspace of dimension 12. Then V Π = V ∩ Π ⊂ Π ∼ = P 12 is a non-singular hypersurface and B ∩Π an irreducible subvariety, dim B * ∩Π 2, and the effective cycle Z Π = (Z •V Π ) of codimension 2 on V Π satisfies the inequality mult B * ∩Π Z Π > 4n 2 .
is a mobile linear system (H R means the class of a hyperplane section of the hypersurface V R ). Therefore, for a general linear subspace R 1 , o ∈ R 1 ⊂ R, where 5 dim R 1 10, by inversion of adjunction the pair is not log canonical at the point o. This implies the 8n 2 -inequality (see [5, Chapter 2, Section 4]) and the existence of a hyperplane section P ∋ o, such that • the linear system Σ P = Σ| P ⊂ |2nH P | is mobile, (H P is the class of a hyperplane section of the variety P ), • its self-intersection Z P = (Z • P ) satisfies the inequality mult o Z P > 8n 2 , ]. This condition is stronger than the regularity condition (R1.2) used in this paper. However, that condition (R2) is used in [1, Section 5.4] only once -in the proof of Corollary 5.1, and it is easy to see from the proof that the condition (R2) is unnecessarily strong: the inequality which is equivalent to the condition (R1.2) of the present paper, is sufficient. Thus all arguments of the paper [1], excluding the maximal singularity in Sections 4-6, work without modifications in our Case 3 and exclude the maximal singularity, the centre of which is not contained in Sing V . This completes the proof of Proposition 3.1.
3.2. The linear system Σ at singular points of the hypersurface V . Let us fix a maximal singularity E * , the centre B * of which has the maximal dimension among all centres of maximal singularities of the linear system Σ. We have B * ⊂ Sing V . Let o ∈ B * be a point of general position. For a general 13-dimensional linear subspace Π ⊂ P, where o ∈ Π, the pair where V Π = V ∩ Π and Σ Π is the restriction of Σ onto V Π , has the point o as an isolated centre of a non canonical singularity, that is, this pair is canonical outside the point o in a neighborhood of that point. By inversion of adjunction for a general proper subspace Π 1 ⊂ Π, containing the point o, the pair is not log canonical at the point o, but canonical outside that point in a neighborhood of that point. Let ϕ P : P + → P be the blow up of the point o with the exceptional divisor E P ∼ = P M , and ϕ: V + → V the restriction of that blow up onto the hypersurface V . The exceptional divisor Q = V + ∩ E P of the blow up ϕ is by the condition (R2.2) a quadric hypersurface of rank at least 13, embedded in E P . The symbol H Q stands for the class of a hyperplane section of Q. Every irreducible subvariety R ⊂ Q of codimension 5 is numerically equivalent to the class dQ (R)H codim(R⊂Q) Q for some dQ (R) ∈ Z + ; by linearity, the integer-valued function dQ (·) extends for all equidimensional cycles of codimension 5.
Furthermore, let R ⊂ V be an irreducible subvariety of codimension 5. We get the numerical equivalence for some d (R) ∈ Z + ; again, this integer-valued function extends by linearity to all equidimensional cycles of this codimension. Let R + ⊂ V + be the strict transform of R on V + . Obviously, again extends to equidimensional cycles. For simplicity of notations we will often omit the pull back symbol: for instance, we write R instead of ϕ * R. The obvious equalities hold: For a general divisor D ∈ Σ set ν = m(D), that is, D + ∼ D − νQ. Proposition 2.5 implies the inequality ν 8 3 n. Consider the self-intersection Z = (D 1 •D 2 ) of the mobile system Σ. By construction, d (Z) = 4n 2 . The singularity o ∈ V satisfies the assumptions of the main theorem of [7], therefore the inequality m(Z) > 4n 2 holds (which, unfortunately, is insufficient for the exclusion of the maximal singularity). Let Π ⊂ P be a general 6-dimensional subspace, containing the point o, V Π = V ∩ Π and Σ Π = Σ| V Π . The symbol H Π stands for the class of a hyperplane section of the hypersurface V Π . Obviously, V Π has a unique singular point -the nondegenerate quadratic point o. Let V + Π be the strict transform of V Π on V + , that is, ϕ Π : V + Π → V Π is the blow up of the point o with the exceptional divisor Q Π = V + Π ∩ E P , which is a non-singular 4-dimensional quadric in Π + ∩ E P ∼ = P 5 . We get Since for the discrepancy of the exceptional divisor Q Π we have the equality a(Q Π , V Π ) = 3 and as we mentioned above, ν 3n and the pair (V Π , 1 n Σ Π ) is not log canonical at the point o, we obtain the following fact: the pair is not log canonical, and moreover, the centre of every non log canonical singularity of the pair Π , intersecting the exceptional quadric Q Π , is contained in it.  [24]), and taking into account the fact that the point o is a double point of the hypersurface V , we obtain the following inequality for the self-intersection Z: Therefore, since d (Z) = 4n 2 , the inequality holds, which contradicts the estimate (5) for a = 2, c = 1 (see Remark 2.1 (i)). This contradiction proves Proposition 3.3. Q.E.D. So the codimension α can take at most two values: 2 and 3. In order to exclude these options as well, let us consider one more characteristic of the subvariety B. For a pair of distinct points p, q ∈ B the symbol [p, q] denotes the line in E P , joining the points p and q, provided that this line is contained in Q. If this line is not contained in Q, then set [p, q] = ∅. Now set: We say that the case α.β takes place, where α ∈ {2, 3} and 0 β α, if codim(B ⊂ Q) = α and codim(Sec(B ⊂ Q) ⊂ Q) = β. In the description of the cases 3.β given above, the only not quite obvious statement is the description of the case 3.2, which is based on an analog of Proposition 3.4 for a subvariety of codimension 3. This analog is stated and proven in Section 4 (the proof is very simple).
Remark 3.1. Let us sum up what has been done so far. Assuming the existence of a maximal singularity E * of the linear system Σ (and the non-existence of maximal subvarieties of the form P ∩ V , where P ⊂ P is a linear subspace of codimension 2), we proved the existence of another singularity E of the linear system Σ, the centre of which is a double point o ∈ V . This singularity E "looks like a maximal one" in the sense that it satisfies the Noether-Fano type inequality (in the brackets we could have added +1, since the pair is not log canonical, but we do not need this). That inequality is weaker than the standard Noether-Fano inequality, but this is compensated by a high dimension of the centre B of the singularity E on V + . We will show below that in each of the cases α.β listed above, the existence of the singularity E leads to a contradiction. Thus, the existence of the original maximal singularity E * leads to a contradiction, either. This will complete the proof of Theorem 5.

Resolution of the singularity E.
Let us consider the standard procedure of resolving the singularity E. Let ϕ i,i−1 : V i → V i−1 be the blow up of the centre The exceptional divisor of the blow up ϕ i,i−1 is denoted by the symbol E i , so that E 1 = Q. The sequence of these blow ups terminates: for some i = K the exceptional divisor E K is the centre of E on V K , and there is nothing more to blow up. The varieties V i have, generally speaking, uncontrollable singularities, however, at the general point of the subvariety B i the variety V i is non-singular for i 1, and this is the only property that we need for all computations. For i > j the composition of blow ups In particular, we always have (i + 1) → i. By the symbol p ij we denote the number of paths in the just constructed oriented graph from the vertex i to the vertex j, if i = j; by definition, p ii = 1. For i < j, obviously, p ij = 0. In order to simplify the notations, we write p i instead of p Ki . For i = 1, . . . , K we define the numbers ν i ∈ Z + (the "elementary multiplicities"), taking a general divisor D ∈ Σ and writing down where the upper index j denotes, as usual, the operation of taking the strict transform on V j (see [5]). Now . Now the Noether-Fano type inequality (7) can be explicitly re-written in the form of the estimate Since from the definition of the numbers p i = p Ki we have the obvious equality a > i: we conclude that if ν K δ K n, then for some K 1 < K the estimate holds. For this reason we may (and will) assume that ν K > δ K n. Furthermore, we break the set of vertices {1, . . . , K} of the constructed oriented graph into the lower part {1, . . . , L} and the upper part (which can be empty) {L + 1, . . . , K}, setting i L, if and only if δ i 2. Finally, we use the well known procedure of erasing arrows in the constructed graph: let us remove all arrows i → 1, going from the vertices of the upper part (i L + 1). Recall that by Proposition 3.3 we have L 2, so that at least the vertex 2 lies in the lower part. The procedure of erasing arrows may decrease p 1 , but does not change the numbers p i for i 2, therefore the inequality (8) can only get stronger. For this reason we assume that in the oriented graph there are no arrows from the vertices of the upper part to the vertex 1. Set and Σ 0 = p 1 + Σ + 0 . The procedure of erasing arrows gives the following fact. Proposition 3.5. The following inequality holds: Proof. Indeed, every path from the vertex K to the vertex 1 is of the form where a L. Therefore, where Z 1 is an effective divisor on the quadric Q = E 1 , we obtain the equality whence, arguing as in [7] and in [5,Chapter 2], we obtain by means of the technique of counting multiplicities the following estimate: where m 1 m 2 · · · m L and ν 1 ν 2 · · · ν K . (Note that the first inequalities in both expressions, that is, m 1 m 2 and ν 1 ν 2 , are non-trivial, although for the quadric Q their proof is very simple; for the general case see [7,Proposition 2]). If there is no additional information about the multiplicities ν i , then we estimate the minimum of the quadratic form in the right hand side of the inequality (9) on the hyperplane, the equation of which is obtained from the inequality (8) by replacing the > sign by =, and get the estimate Replacing the left hand side of this inequality by (which can only make the inequality sharper), we get finally: Estimating the left hand side from above by the expression (Σ 0 + Σ 1 )p 1 , we get the 4n 2 -inequality m 1 > 4n 2 , mentioned above (as it was done in [7]). However, we can say more. Proposition 3.6. The following inequality holds: Proof. For a fixed value of the linear form p 1 m 1 + (Σ + 0 + Σ 1 )m 2 (in the variables m 1 m 2 0) the minimum of the expression m 1 + m 2 is attained for m 1 = m 2 (recall that p 1 Σ + 0 + Σ 1 ). In that case m 1 = m 2 > 4n 2 , which proves Proposition 3.6. Q.E.D.
For certain types of the singularity E there is some additional information about the multiplicities m i and ν i , which makes it possible to make the estimates for m 1 and m 2 sharper. In order to obtain such information, we need some facts about the secant variety Sec(B ⊂ Q). Thses questions are dealt with in the next section.

Subvarieties of the quadric Q
The aim of this section is to prove the classification of options given in Subsection 3.2. First we consider the problem of irreducibility of the intersection of an irreducible subvariety X ⊂ Q of a small codimension with a general linear subspace P ⊂ Q of maximal dimension (Subsection 4.1). On this basis it is easy to prove our classification, that is, the description of the cases 2.1, 2.2 and 3.2, 3.3 (Subsection 4.2). In Subsection 4.3 we discuss non-degenerate subvarieties X ⊂ P N of codimension 3, the secant variety Sec(X) of which are strictly smaller than P N . 4.1. Irreducibility of the intersection with a linear subspace. Let L be a closed algebraic set, parameterizing linear subspaces of maximal dimension on the quadric Q, that is, of dimension M − 1 2 rk Q . Depending on whether the rank of the quadric is odd or even, the set L can be irreducible variety or a union of two irreducible varieties. For a subspace P ∈ L by the symbol π P we denote the projection P M from the subspace P . For a subspace Λ ⊂ P M , such that Λ ⊃ P and dim Λ = dim P +1 (that is, a fibre of the projection π P ) we have Q∩Λ = P ∪Q(P, Λ), where Q(P, Λ) ∈ L. If P ∈ L is a subspace of general position and Λ a general fibre of the projection π P , then Q(P, Λ) is also a subspace of general position.
Proposition 4.1. Let X ⊂ Q be an irreducible subvariety of codimension 3 and P ∈ L a linear subspace of general position. Then X ∩ P is an irreducible subvariety of codimension codim(X ⊂ Q) in the projective space P .
Proof. The equality is obvious, we only need to show the irreducibility. Assume that where |I| 2. Then for any general P ∈ L we have a similar picture with the same value of |I|. For a fibre Λ of the projection π P the intersection is a hyperplane in P , and moreover, it is easy to check that varying Λ, we obtain the complete family of hyperplanes in P , containing the vertex space of the quadric Q. For that reason for a general space Λ ⊃ P the closed space is irreducible for all i ∈ I, and we may assume that all sets X i (P ) ∩ P Λ , i ∈ I, are distinct, so that the components X i (P ) are identified by the intersections with the hyperplane P Λ . However, our construction is symmetric with respect to the subspaces P and Q(P, Λ), so that for a general fibre Λ of the projection π P there is a bijective correspondence between irreducible components of the intersection X ∩P and the components of the intersection X ∩Q(P, Λ), which makes it possible to write down X ∩ Q(P, Λ) = i∈I X i (Q(P, Λ)).
But then for each i ∈ I X i = Λ∈U X i (Q(P, Λ)) (the union is taken over a non-empty Zariski open subset U ⊂ P M −1−dim P ) is an irreducible component of the original set X, which contradicts the assumption about its irreducibility. Q.E.D. for the proposition. Taking into account the previous case, we can claim that the last inequality is an equality. According to Proposition 3.4, in this case contains a hyperplane in P , and for that reason by Proposition 4.1 is a hyperplane in P . Then dQ (Sec(B ⊂ Q)) = 1, so that deg Sec(B ⊂ Q) = 2. Therefore, Sec(B ⊂ Q) is a hyperplane section of the quadric Q, which is itself a factorial quadric. Since dQ (B) 2 (if dQ (B) = 1, then we are in the case 2.2), we obtain precisely the description of the case 2.1. The description of the case 2.0 requires no proof.
Assume now that codim(B ⊂ Q) = 3. If Sec(B ⊂ Q) has codimension 3, then we argue as above in the case 2.2 and obtain the description of the case 3.3. If Sec(B ⊂ Q) has codimension 2, then we argue as above in the case 2.1, using the following simple fact. Proposition 4.2. Let X ⊂ P N be an irreducible subvariety of dimension dim X 2, and moreover, dim Sec(X) = dim X + 1. Then Sec(X) is a linear subspace and X is a hypersurface of degree d X 2 in that subspace.
Proof. For a point p ∈ X of general position consider the cone C(p, X) with the vertex p and the base X (the closure of the union of all lines [p, q], where q ∈ X \{p}). This is an irreducible subvariety of some degree d 1 and dimension dim X + 1, so that Sec(X) = C(p, X). Therefore, for all points p ∈ X, that is, for every point p ∈ X the subvariety Sec(X) is a cone with the vertex at the point p. Since X is a subvariety of codimension 1 in Sec(X), it is easy to see that d = 1. Q.E.D. for the proposition.
The proposition proven above implies the description of the case 3.2. The cases 3.1 and 3.0 require no proof: in those two cases we just note the codimension of the set Sec(B ⊂ Q). This completes the proof of the list of options given in Subsection 3.2.

A remark on the secant variety.
In [1,Section 3] it was shown that if B ⊂ V is a subvariety of codimension 2 on a general smooth hypersurface V ⊂ P of degree M and B is not contained in a hyperplane (that is, B = P), then Sec(B) = P. Since in that case codim(B ⊂ P) = 3, it is natural to ask: what is an analog of Proposition 3.4 for subvarieties of codimension 3? Let X ⊂ P N be an irreducible subvariety of codimension 3. If dim X N − 1, then applying Proposition 3.4 to the projective space X , we get a complete classification of options. Assume therefore that X = P N .
Example 4.1. Let Γ ⊂ P 4 be a non-degenerate curve. Obviously, Sec(Γ) is a hypersurface in P 4 . Considering the cone over Γ ⊂ P 4 ⊂ P N with a vertex subspace of dimension N − 5, we obtain an irreducible subvariety X ⊂ P N , for which Sec(X) is a hypersurface in P N . We say that a subvariety X, obtained in this way, is a cone over a non-degenerate curve in P 4 . The following fact takes place. Proposition 4.3. Let X ⊂ P N be a non-degenerate irreducible subvariety of codimension 3 and assume that Then X is a cone over a non-degenerate curve in P 4 .
We do not give a proof here, because we do not use this fact. The proof follows the same scheme of arguments as in [1,Section 3]. A close look at that proof shows that the condition B ⊂ V was used only to claim that for every point p ∈ B there are at most finitely many lines on B passing through this point.
The proof given in [1,Section 3] can be improved to a proof of Proposition 4.3.

Exclusion of the maximal singularity
In this section we prove Theorem 5. First we exclude the case 3.3 (Subsection 5.1), then the case 3.2 (Subsection 5.2). These are the most difficult cases, requiring considerable efforts; the remaining five cases are excluded in Subsection 5.3. This completes the proof of Theorem 5.
Assume that the case 3.3 takes place. Our aim is to obtain a contradiction. By Proposition 3.6 the inequality holds. This inequality is linear in Z. For that reason we may assume that Z is an irreducible subvariety of codimension 2. By Proposition 2.4 we have m(Z) 2 d (Z), so that Z + contains B and thus Z ⊂ T 2 (by the regularity condition (R2.1)).
Set Π ⊂ P to be the uniquely determined subspace of codimension 3, containing the point o and "cutting out B on Q", that is, The corresponding section V ∩ Π is denoted by the symbol ∆ (in Subsection 2.4 it was denoted by the symbol V Π ). So ∆ + ∩Q = B. Consider the linear system |H −∆| of hyperplane sections of the hypersurface V , containing ∆. Let R 1 ∈ |H − ∆| be a general divisor. Since B ⊂ T + 2 , we have ∆ ⊂ T 2 , which implies that none of the irreducible components of the effective cycle (Z • R 1 ) of codimension 3 is contained in T 2 .
Since Bs |H − ∆| = ∆, and Bs |H − ∆| + = ∆ + (in the scheme-theoretic sense), the following equalities hold: where a ∈ Z + and the effective cycle Z ♯ does not contain ∆ as a component. Since d (∆) = 1, m(∆) = 1 and mult B ∆ + = 1, we obtain the inequality Let us consider one more general divisor R 2 ∈ |H − ∆|. Obviously, none of the components of the cycle Z ♯ is contained in R 2 , so that the effective cycle (Z ♯ • R 2 ) is well defined. It has codimension 4 in V , 3 in R 1 and 2 in R 1 ∩ R 2 . The following inequality holds: Proof. Assume the converse: Y ⊂ T 2 . By construction, Y is an irreducible component of the intersection of the divisor R 2 with one of the irreducible components of the cycle Z ♯ , which, as we know, is not contained in T 2 . Therefore, Y is an irreducible component of the effective cycle (Z ♯ • T 2 ), which is contained in R 2 . The cycle (Z ♯ • T 2 ) is of codimension 3 in R 1 . Since that cycle does not depend on R 2 , and R 2 ∈ |H − ∆| by assumption is a general divisor of this linear system, we conclude that Y ⊂ ∆.
So Y is a prime divisor on ∆. By the condition (R2.1) the divisor (T 2 • ∆) on ∆ is irreducible and reduced, so that we obtain the equality Y = (T 2 • ∆). But this is impossible: by the condition (R2.1) we have m(T 2 • ∆) = 3. Since d (T 2 • ∆) = 2, we get a contradiction with the inequality (11). Q.E.D. for the lemma.

Exclusion of the case 3.2.
Assume that the case 3.2 takes place. Again our aim is to obtain a contradiction. Now by the symbol ∆ we denote the section of the hypersurface V by the uniquely determined subspace Π ⊂ P of codimension 2, such that Π + ∩ Q = Sec(B ⊂ Q).
(Recall that Sec(B ⊂ Q) is the section of the quadric Q ⊂ E P = P M by the linear subspace B ⊂ P M of codimension 2.) Therefore, We continue to assume that the self-intersection Z is an irreducible subvariety of codimension 2. Since d (∆) = 1 and m(∆) = mult B ∆ + = 1, we have Z = ∆.
Consider the pencil of hyperplane sections |H −∆|, containing ∆, and take a general divisor R ∈ |H − ∆|. By construction, Z ⊂ R, so that the effective cycle Z R = (Z • R) of codimension 2 on the hyperplane section R is well defined. Setting for the convenience of notations ∆ Q = ∆ + ∩ Q, let us write down where a ∈ Z + . Such a writing is possible, because ∆ Q = Bs |H − ∆| + Q . Now we get m(Z R ) = m(Z) + a and a + mult B Z + R mult B Z + , since obviously mult B ∆ Q = 1. From there we obtain the inequality By the linearity of this inequality in Z R we may assume that Z R = Y is an irreducible subvariety of codimension 2 (with respect to R). We know that B is a prime divisor on the quadric ∆ Q ⊂ B = P M −2 , cut out on this quadric by a hypersurface of degree dQ (B) 2. The effective cycle ( this implies the inequality m(Y ) dQ (B) · mult B Y + . Now we have to consider two cases: (1) Y ⊂ ∆ is a prime divisor, From the regularity conditions we deduce that Y = (T 2 • ∆). Indeed, the quadratic form q 2 | Π is of rank 9, the variety ∆ is factorial, so that the divisor (T 2 • ∆) is irreducible and reduced. By the condition (R2.1) this divisor has multiplicity precisely 6 at the point o and for that reason the equality holds, from which we get that Y = (T 2 • ∆)). So we obtain the following well defined effective cycle (Y • (T 2 • ∆)) of codimension 2 on ∆, satisfying the inequality which can be re-written in the form This contradicts the inequality (5) Recall that for Z R = Y the inequality (12) holds. It is easy to check that the minimum of the function s + 2t of the real variables s, t on the set {s + t 2, s 2t} ⊂ R 2 + is attained at the point ( 4 3 , 2 3 ) and equal to 8 3 . Therefore, ). The last inequality can be re-written in the form of the estimate which contradicts the inequality (5) for c = a = 2. We obtained a contradiction which proves the lemma. Q.E.D. By Lemmas 5.2 and 5.3, is the case 3.2 takes place, then on ∆ there is a prime divisor Y , satisfying the inequality m(Y ) + mult B Y + > 2 d (Y ), and moreover, dQ (B) = 2 and m(Y ) > 2 mult B Y + , so that m(Y ) > 4 3 d (Y ). In order to exclude that last option, we use the condition (R2.3). Denote the subvariety that is cut out on the quadric ∆ Q by the equation q 3 | ∆ Q = 0, by the symbol G. It is easy to see that G belongs to the family of varieties, which are irreducible, reduced and factorial by the condition (R2.3). Therefore, G is a factorial complete intersection of type 2 · 3 in P M −2 . For that reason the kernel of the surjective restriction map is one-dimensional and is generated by the quadratic form q 2 | P(Π) . The irreducible subvariety B is cut out on the quadric ∆ Q by a quadratic equation β = 0, where β ∈ q 2 | P(Π) . Therefore, the equation β|| G = 0 defines an effective divisor on G.
Proof. By the factoriality of the complete intersection G, reducibility or nonreducedness of this divisor means that it is a sum of two hyperplane sections. Therefore, if the divisor {β| G = 0} were reducible or non-reduced, for some linear forms l 1 , l 2 on P M −2 P(Π) we would have had the equality where λ ∈ C is some constant. But then the divisor B ⊂ ∆ Q , given by the equation β = 0, would have been reducible or non-reduced. Q.E.D. for the lemma.
Therefore, B∩G = {β| G = 0} is an irreducible reduced subvariety of codimension 4 on Q, and moreover B ∩ G ∼ 6H 4 Q , that is, dQ (B ∩ G) = 6. Now let us come back to the prime divisor Y on ∆. If Y + does not contain B, then the inequality m(Y ) > 2 d (Y ) holds, which is excluded by the proof of Lemma 5.2 (where we proved that m(Y ) 3 2 d (Y )). So Y + must contain B.
Besides, as we pointed out above, m(Y ) > 4 3 d (Y ), which implies the inequality which will be used later. Consider again the irreducible reduced subvariety B ∩ G. It can be viewed as a complete intersection of type 2 · 2 · 3 in P(Π) = P M −2 . Therefore B ∩ G is cut out in the scheme-theoretic sense by cubic hypersurfaces. Let Obviously, d (Y * • W ) = 3 d (Y * ). Taking into account that the minimum of the function s + 2t of real variables s, t on the set {s + 3t 3, s 2} ⊂ R 2 + is attained at the point (2, 1 3 ) and is equal to 8 3 , we get by the inequalities (13) and (14): holds. This contradicts the inequality (5) for c = 2, a = 3 when M 16. We obtained a contradiction which completes the exclusion of the case 3.2.

Exclusion of all remaining cases.
Assume that one of the following four cases takes place: 3.1, 3.0, 2.1, 2.0.
Let us consider the self-intersection Z of the mobile system Σ. Proof. In the notations of §4 let P ∈ L be a general subspace of maximal dimension on the quadric Q, Z P = (Z + • P ) an effective cycle of codimension 2. Assume the converse: m(Z) < 2 mult B Z + .
Then Z P is an effective cycle of degree m(Z) (recall that m(Z) = dQ (Z + •Q)), which satisfies the inequality deg Z P < 2 mult B∩P Z P .
Let p, q ∈ B ∩ P be a pair of distinct points and [p, q] ⊂ P the line through them. Furthermore, let Θ ⊃ [p, q] be a two-dimensional plane of general position in P , containing the line [p, q]. By the symbol |Z P | we denote the support of the cycle Z P . If the intersection Θ ∩ |Z P | were zero-dimensional, we would have got the following chain of equalities and inequalities, where all intersection numbers are meant to be in the projective space P : deg Z P = (Z P · Θ) = x∈Θ∩|Z P | (Z P · Θ) x (Z P · Θ) p + (Z P · Θ) q mult p Z P + mult q Z P 2 mult B∩P Z P > deg Z P , which is impossible. Therefore, the set Θ ∩ |Z P | is positive-dimensional. Since the plane Θ is arbitrary, we conclude that [p, q] ⊂ |Z P |. Therefore, the support |Z Q | of the cycle Z Q = (Z + • Q) contains the secant subvariety Sec(B ⊂ Q). However, codim(|Z Q | ⊂ Q) = 2, whereas by assumption we are in one of the four cases when codim(Sec(B ⊂ Q) ⊂ Q) 1.
This contradiction proves the lemma. Q.E.D. Now let us use the technique of counting multiplicities (Subsection 3.3). In the notations of Subsection 3.3 Lemma 5.5 claims that m 1 2m 2 .
It is easy to bring the last inequality to the following form: The contradiction proves the lemma. Q.E.D. Thus in each of the four cases under consideration the self-intersection Z satisfies the inequality This contradicts Proposition 2.4 (see Remark 2.1 (i)). The cases 3.1, 3.0, 2.1 and 2.0 are excluded. The only remaining case is the case 2.2. Assume that this case takes place. Recall that B in this case is a section of the quadric Q by a linear subspace of codimension 2 in E P ∼ = P M . Let ∆ be the section of the hypersurface V by the uniquely determined subspace of codimension 2 in P, such that ∆ + ∩ Q = B. Then m(∆) = d (∆) = mult B ∆ + = 1. Write down where a ∈ Z + and the cycle Z 1 does not contain ∆ as a component. Since d (Z) = 4n 2 , from Proposition 3.6 (and the equalities for ∆, written above) we obtain: m(Z 1 ) + mult B Z + 1 > 2 d (Z 1 ).
By the linearity of this inequality we may assume that Z 1 = Y is an irreducible subvariety of codimension 2, and moreover, Y = ∆. Consider a general hyperplane section R ⊃ ∆. Since Y ⊂ R, the effective cycle (Y • R) of codimension 2 on R is well defined. This cycle satisfies the inequality which is equivalent to the estimate contradicting the inequality (5) for c = 1, a = 2. This contradiction excludes the case 2.2 and completes the proof of Theorem 5.