Canonical and log canonical thresholds of multiple projective spaces

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano-Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

Introduction 0.1. Statement of the main results. In [1] general d-sheeted covers of the complex projective space P = P M which are Fano varieties of index 1 with at most quadratic singularities, the rank of which is bounded from below, were shown to be birationally superrigid. In this paper we will prove that for almost all values of the discrete parameters defining these varieties a general multiple projective space of index 1 satisfies a much stronger property: its global canonical (and the more so, log canonical) threshold is equal to 1. Now [2] immediately implies the birational rigidity type results for fibre spaces, the fibres of which are multiple projective spaces, and new classes of Fano direct products [3]. Let us give precise statements.
Fix a pair of positive integers (d, l) ∈ Z ×2 + in the set described by the following table: with homogeneous coordinates x 0 , . . . , x M , ξ, where x i are of weight 1 and ξ is of weight l, and a quasi-homogeneous polynomial of degree dl (that is, A i (x 0 , . . . , x M ) is a homogeneous polynomial of degree il i = 1, . . . , d). The space H 0 (P, O P (il)) parameterizes all such polynomials. If the hypersurface V = {F = 0} ⊂ P has at most quadratic singularities of rank 7 (and we will consider hypersurfaces with stronger restrictions for the rank), then V is a factorial variety with terminal singularities, see [1], so that where H is the class of a "hyperplane section", that is, of the divisor V ∩ {λ = 0}, where λ(x 0 , . . . , x M ) is an arbitrary linear form. Below for all the values of d, l under consideration we will define explicitly a positive integral-valued function ε(d, l), which behaves as 1 2 M 2 as the dimension M grows. As in [1], we identify the polynomial F ∈ F and the corresponding hypersurface {F = 0}, which makes it possible to write V ∈ F . The following theorem is the main result of the present paper.
Theorem 0.1. There is a Zariski open subset F reg ⊂ F such that: (i) every hypersurface V ∈ F reg has at most quadratic singularities of rank 8 and for that reason is a factorial Fano variety of index 1 with terminal singularities, (ii) the inequality codim((F \F reg ) ⊂ F ) ε(d, l) holds, (iii) for every variety V ∈ F reg and every divisor D ∼ nH the pair (V, 1 n D) is canonical. Now [2, Theorem 1.1] makes it possible to describe the birational geometry of Fano-Mori fibre spaces, the fibres of which are multiple projective spaces of index 1.
Let η: X → S be a locally trivial fibre space, the base of which is a non-singular projective rationally connected variety S of dimension dim S < ε(d, l), and the fibre is the weighted projective space P. Consider an irreducible hypersurface W ⊂ X, such that for every point s ∈ S the intersection η −1 (s) ∩ W ∈ F is a multiple projective space of the type described above. The claim (ii) of Theorem 0.1 implies that we may assume that W s = η −1 (s) ∩ W ∈ F reg for every point of the base s ∈ S, if the linear system |W | is sufficiently mobile on X, and the hypersurface W is sufficiently general in that linear system. Set The variety W by the claim (i) of Theorem 0.1 has ay most quadratic singularities of rank 8, and for that reason is a factorial variety with terminal singularities. Therefore, η: W → S is a Fano-Mori fibre space, the fibres of which are multiple projective spaces of index 1. Let η ′ : W ′ → S ′ be an arbitrary rationally connected fibre space, that is, a morphism of projective algebraic varieties, where the base S ′ and the fibre of general position (η ′ ) −1 (s ′ ), s ′ ∈ S ′ , are rationally connected, and moreover, dim W ′ = dim W . Now [2, Theorem 1.1], combined with Theorem 0.1, immediately gives the following result.
Theorem 0.2. Assume that the Fano-Mori fibre space η: W → S satisfies the following condition: for every mobile family C of curves on the base S, sweeping out S, and a general curve C ∈ C the class of an algebraic cycle is not effective, that is, it is not rationally equivalent to an effective cycle of dimension M. Then every birational map χ: W W ′ onto the total space of the rationally connected fibre space W ′ /S ′ (if such maps exist) is fibre-wise, that is, there is a rational dominant map ζ: S S ′ , such that the following diagram commutes: Corollary 0.1. In the assumptions of Theorem 0.2 on the variety W there are no structures of a rationally connected fibre space (and, the more so, of a Fano-Mori fibre space), the fibre of which is of dimension less than M. In particular, the variety W is non-rational and every birational self-map of the variety W commutes with the projection η and for that reason induces a birational self-map of the base S.
The condition for the cycles of dimension M, described in Theorem 0.2, is satisfied if the linear system |W | is sufficiently mobile on X. Let us demonstrate it by an especially visual example, when X = P × S is the trivial fibre space over S. Let o * = (0 : . . . : 0 : 1) = (0 M +1 : 1) ∈ P be the only singular point of the weighted projective space P. Consider the projection "from the point o * " where π P ((x 0 : . . . : x M : ξ)) = (x 0 : . . . : x M ). Let H be the π P -pull back of the class of a hyperplane in P on P. The pull back of the class H on X = P × S with respect to the projection onto the first factor we denote for simplicity by the same symbol H. Now Pic X = ZH ⊕ η * Pic S, so that for some class R ∈ Pic S the relation W ∼ dlH + η * R holds and for that reason This implies that the condition of Theorem 0.2 holds if for any mobile family of curves C, sweeping out S, and a general curve C ∈ C, the inequality holds. Therefore, the following claim is true. Theorem 0.3. Assume that the class R + K S ∈ Pic S is pseudo-effective and for every point s ∈ S we have η −1 (s) = W s ∈ F reg . Then in the notations of Theorem 0.2 every birational map χ: W W ′ is fibre-wise. In particular, every birational self-map χ ∈ Bir W induces a birational self-map of the base S.
Another standard application of Theorem 0.1 is given by the theorem on birational geometry of Fano direct products [3,Theorem 1]. Recall that the following statement is true.
Theorem 0.4. Assume that primitive Fano varieties V 1 , . . . , V N satisfy the following properties: (i) for every effective divisor there are no other structures of a rationally connected fibre space, apart from projections on to direct fibres The property (ii) was shown in [1] for a wider class of multiple projective spaces than the one that is considered in this paper. Of course, Theorem 0.1 implies that the conditions (i) and (ii) are satisfied for every variety V ∈ F reg . Therefore, every variety considered in the present paper can be taken as a factor of the direct product in Theorem 0.4. 0.2. The regularity conditions. The open subset F reg are given by explicit local regularity conditions, which we will now describe. To begin with, let us introduce an auxiliary integral-valued parameter ρ ∈ {1, 2, 3, 4}, depending on (d, l). Its meaning, the number reductions to a hyperplane section, used in the proof of Theorem 0.1, will become clear later. Set ρ = 4, if d = 4 and 21 l 25 and ρ = 1, if d 18 and l 2. For the remaining possible pairs (d, l) the value ρ 2 is given by the following If the pair (d, l) is not in the table, then ρ = 1 (for instance, for d = 14, l 4). One more table gives the function ε(d, l), bounding from below the codimension of the complement to the set F reg . Write this function as a function of the dimension M = (d − 1)l, for each of the possible values of the parameter ρ defined above.
Now let us state the regularity conditions. Let o ∈ V be some point. The coordinate system (x 0 : x 1 : · · · : x M : ξ) can be chosen in such a way that o = (1 : 0 : · · · : 0 : 0) (see [1, §1]). The corresponding affine coordinates are where the (non-homogeneous) polynomial a i (z * ) is of degree il. Furthermore, the following fact is true ([1, ]): for any homogeneous polynomial γ(x 0 , . . . , x M ) of degree l the equation ξ = γ(x * ) defines a hypersurface R γ ⊂ P that does not contain the point o * = (0 M +1 : 1), and moreover the projection π P | Rγ : R γ → P is an isomorphism. In this way, the hypersurface V ∩ R γ in R γ identifies naturally with a hypersurface in P = P M , and its intersection with the affine chart {x 0 = 0} identifies with a hypersurface in the affine space A M z 1 ,...,z M . The regularity conditions, given below, are assumed to be satisfied for the hypersurface V γ = V ∩ R γ for a general polynomial γ(x * ).
Assume that the point o ∈ V is non-singular, so that o ∈ V γ is non-singular, too. Let P ⊂ A M be an arbitrary linear subspace of codimension (ρ − 1), that is, o ∈ P , that is not contained in the tangent hyperplane T o V γ . Let f P = q 1 + q 2 + · · · + q dl be the affine equation of the hypersurface P ∩ V γ , which is non-singular at the point o, decomposed into homogeneous components (with respect to an arbitrary system of linear coordinates on P ).
(R1.1) For any linear form λ ∈ q 1 the sequence of homogeneous polynomials is at least 8 + 2(ρ − 2). We say that a non-singular point o ∈ V is regular, if for a general polynomial γ(x * ) and any subspace P ⊂ T o V γ the conditions (R1.1-3) are satisfied.
Assume now that the point o ∈ V is singular, so that the hypersurface V γ is also singular at that point.
(R2.1) The point o ∈ V γ is a quadratic singularity of rank 2ρ + 6. Let P ⊂ A M be an arbitrary linear subspace of codimension ρ + 2, that is, o ∈ P , and f P = q 2 + q 3 + · · · + q dl is the affine equation of the hypersurface P ∩ V γ , decomposed into homogeneous components (in particular, q 2 is a quadratic form of rank 2). (R2. 2) The sequence of homogeneous polynomials is regular in the local ring O o,P . We say that a singular point o ∈ V is regular, if for a general polynomial γ(x * ) and any subspace P ⊂ A M of codimension ρ + 2 the conditions (R2.1,2) hold.
Finally, we say that the variety V is regular, if it is regular at every point o ∈ V , singular or non-singular. Set F reg ⊂ F to be the Zariski open subset of regular hypersurfaces (that it is non-empty, follows from the estimate for the codimension of the complement). Obviously, every hypersurface V ∈ F reg has at worst quadratic singularities of rank 8, so that the claim (i) of Theorem 0.1 is true.
0.3. The structure of the paper, historical remarks and acknowledgements. A proof of the claim (ii) of Theorem 0.1 is given in Subsections 1.2 and 1.3. A proof of the claim (iii) of Theorem 0.1 in Subsection 1.1 is reduced to two facts about hypersurfaces in the projective space P N , which are applied to the hypersurface V γ ⊂ P, both in the singular and non-singular cases. Proofs of those two facts are given, respectively, in §2 and §3.
The equality of the global (log) canonical threshold to one is shown for many families of primitive Fano varieties, starting from the pioneer paper [3] (for a general variety in the family). For Fano complete intersections in the projective space the best progress in that direction (in the sense of covering the largest class of families) was made in [4]. The double covers were considered in [5]. Fano three-folds, singular and non-singular, were studied in the papers [6,7,8,9] and many others. However, the non-cyclic covers of index 1 in the arbitrary dimension were never studied up to now: the reason, as it was explained in [1], was that the technique of hypertangent divisors does not apply to these varieties in a straightforward way. As it turned out (see [1]), the technique of hypertangent divisors should be applied to a certain subvariety, which identifies naturally with a hypersurface (of general type) in the projective space. This approach is used in the present paper, too.
The author thanks The Leverhulme Trust for the support of the present work (Research Project Grant RPG-2016-279).
The author is also grateful to the colleagues in the Divisions of Algebraic Geometry and Algebra at Steklov Institute of Mathematics for the interest to his work, and to the colleagues-algebraic geometers at the University of Liverpool for the general support.

Proof of the main result
In Subsection 1.1 the proof of part (iii) of Theorem 0.1 is reduced to two intermediate claims, the proofs of which are given in §2 and §3. In Subsections 1.2,3 we show part (ii) of Theorem 0.1. First (Subsection 1.2) we give the estimates for the codimension of the sets of polynomials, violating each of the regularity conditions, after that (Subsection 1.3) we explain how to obtain these estimates.
1.1. Exclusion of maximal singularities. Fix the parameters d, l. Recall that the integer ρ ∈ {1, 2, 3, 4} depends on d, l (see the table in Subsection 0.2). Fix a variety V ∈ F reg . Assume that D ∼ nH is an effective divisor on V , such that the pair (V, 1 n D) is not canonical. Our aim is to get a contradiction. This would prove the claim (iii).
If there is a non-canonical singularity of the pair (V, 1 n D), the centre of which is of positive dimension, then the pair where Γ = V γ for some polynomial γ(x * ) of general position (see Subsection 0.2) and D Γ = D| Γ , is again non-canonical. If the centres of all non-canonical singularities of the pair (V, 1 n D) are points, let us take a polynomial γ(x * ) of general position such that the hypersurface Γ = V γ contains one of them. In that case the pair (Γ, 1 n D Γ ) is even non log canonical.
In any case we obtain a factorial hypersurface Γ ⊂ P = P M of degree dl with at worst quadratic singularities of rank 2ρ+6 8, and an effective divisor D Γ ∼ nH Γ on it (where H Γ is the class of a hyperplane section, so that Pic Γ = ZH Γ ), such that the pair (Γ, 1 n D Γ ) is non-canonical. Now we work only with that pair, forgetting about the original variety V (within the limits of the proof of the claim (iii) of Theorem 0.1). Let be the union of the centres of all non-canonical singularities of that pair. Proposition 1.1. The closed set CS(Γ, 1 n D) is contained in the singular locus Sing Γ of the hypersurface Γ.
Proof makes the contents of §2. Therefore, Let us define a sequence of rational numbers α k , k ∈ Z + , in the following way: (We can simply write α k = 2 − 1 2 k , but for us it is important how α k+1 and α k are related.) In order to exclude the maximal (non-canonical) singularities, we will need only the four values: Let o ∈ Γ be a point of general position on the irreducible component of maximal dimension of the closed set CS(Γ, 1 n D Γ ). Consider a general 5-dimensional subspace in P, containing the point o. Let P be the section of the hypersurface Γ by that subspace. Obviously, P ⊂ P 5 is a hypersurface of degree dl with a unique singular point, a non-degenerate quadratic point o. Denoting D Γ| P by the symbol D P , we get D P ∼ nH P , where H P is the class of a hyperplane section. By the inversion of adjunction, the point o is the centre of a non log canonical singularity of the pair (P, 1 n D P ), and moreover, LCS P, This implies that mult o D P > 2n and therefore mult o D Γ > 2n = 2α 0 n.
There is a sequence of irreducible varieties Γ i , i = 0, 1, . . . , ρ, such that: (i) Γ 0 = Γ and Γ i+1 is a hyperplane section of the hypersurface Γ i ⊂ P M −i , containing the point o, (ii) on the variety Γ ρ there is a prime divisor D * ∼ n * H * , where H * is the class of a hyperplane section of the hypersurface Γ ρ , satisfying the inequality Proof makes the contents of §3. Note that by the condition (R2.1) all hypersurfaces Γ 1 , . . . , Γ ρ are factorial, so that Pic Γ ρ = ZH * . Furthermore, ρ 1, so that with this construction, is ensured by the condition (R2.2). Note that the first step of this construction is possible because the hypertangent divisor D 2 is irreducible, D 2 ∼ 2H * and the equality mult o D 2 = 6 = 3 · 2 holds, so that Y 1 = D 2 . The hypertangent divisor D 3 does not take part in the construction.
For the irreducible surface which is impossible (the last inequality checks directly for each of the possible values of ρ and the corresponding values of d, l). Thus we obtained a contradiction, which completes the proof of the claim (iii) of Theorem 0.1. For these values of i.j we set, respectively, We omit the symbols d, l in order to simplify the formulas, however ε i.j = ε i.j (d, l) are functions of these parameters. The following claim is true. Proposition 1.3. The following inequalities hold: Proof. The regularity conditions must be satisfied for any point o, any linear subspace P of the required codimension and any linear form λ (the polynomial γ(x * ) is assumed to be general and does not influence the estimating of the codimension of the sets F i.j ). Therefore, the problem of getting a lower bound for the numbers ε i.j reduces obviously to a similar problem for varieties V ∈ F violating the condition   be the closed subset of quadratic forms of rank r. Let X 2,3 ⊂ P [2,3] be the closed subset of pairs (w 2 , w 3 ), such that the closed set {w 2 = w 3 = 0} ⊂ P N −1 has at least one degenerate component (that is, a component, the linear span of which is of dimension N − 2). Let Q ⊂ P N −1 be a factorial quadric. For m 4 let X m,Q ⊂ P m be the closed subset of polynomials w m , such that the divisor {w m | Q = 0} on Q is reducible or non-reduced. The following claim is true. Proposition 1.4. (i) The following equality holds: codim(X 2, r ⊂ P 2 ) = N − r + 1 2 .
(iii) The following inequality holds: Proof. The claim (i) is well known. Let us show the inequality (ii). Taking into account the part (i), we may assume that the quadratic form w 2 is of rank 5, so that the quadric {w 2 = 0} is factorial. If the closed set w 2 = w 3 = 0 has a degenerate component, then the divisor {w 3 | {w 2 =0} = 0} on the quadric {w 2 = 0} is either reducible, or non-reduced, so that in any case it is a sum of a hyperplane section and a section of the quadric {w 2 = 0} by some quadratic hypersurface. Calculating the dimensions of the corresponding linear systems, we get that for a fixed quadratic form w 2 of rank 5 the closed set of polynomials w 3 ∈ P 3 , such that the divisor {w 3 | {w 2 =0} = 0} is reducible or non-reduced, is of codimension It is easy to see that this expression is higher than the right hand side of the inequality (ii). This proves the claim (ii). Let us show the inequality (iii). Recall that the quadric Q ⊂ P N −1 is assumed to be factorial (that is, the rank of the corresponding quadratic form is at least 5).
is a polynomial in m with positive coefficients. This implies that for 0 < s < t 1 2 m the inequality holds, which can be re-written as If the divisor {w m | Q = 0} is not irreducible and reduced, then it is a sum of two effective divisors on Q, which are cut out on Q by hypersurfaces of degree 1 a By what was said above, the right hand side of that inequality is h Elementary computations show that the right hand side of the last equality is (m + (N − 4)) . . . It is slightly less obvious, how to obtain the estimate for ε 1.3 , starting from the claim (iii) of Proposition 1.4, for that reason we will explain briefly, how to do it. Fixing the linear subspace P and the linear forms q 1 and λ, consider the quadric The codimension of the set of quadratic forms, for which this quadric is of rank 4 and so not factorial, is given by the claim (i) of Proposition 1.4. It is from here that we get the estimate for ε 1.3 in Proposition 1.3. It remains to show that the violation of the condition (R1.3) under the assumption that the quadric (1) is factorial, gives at least the same (in fact, much higher) codimension. It is to the factorial quadric (1) that we apply the estimate (iii) of Proposition 1.4. There is, however, a delicate point here. The hypersurface P ∩ V γ is given by a polynomial that has at the point o the linear part q 1 and the quadratic part q 2 , which both vanish when restricted onto the quadric (1). The other homogeneous components q 3 , . . . , q dl are arbitrary. In the inequality (iii) of Proposition 1.4 the codimension of the "bad" set X m,Q is considered with respect to the whole space P m , whereas in order to prove the inequality (iii) of Proposition 1.3, we need the codimension with respect to the space of homogeneous polynomials of degree dl, the non-homogeneous presentation of which at the fixed point o has zero linear and quadratic components. However, this does not make any influence on the final result, because the codimension of the set X m,Q in P m is very high. Now let for 2 k N − 2 X [2,k] ⊂ P [2,k] be the set of non-regular tuples (h 2 , . . . , h k ) of length k − 1 N − 3, where h i ∈ P i = P i,N , that is, the system of equations

Exclusion of maximal singularities at smooth points
In this section we consider factorial hypersurfaces X ⊂ P N , satisfying certain additional conditions. We show that the centre of every non-canonical singularity of the pair (X, 1 n D X ), where D X ∼ nH X is cut out on X by a hypersurface of degree n 1, is contained in the singular locus Sing X. In Subsection 2.1 we list the conditions that are satisfied by the hypersurface X, state the main result and exclude non-canonical singularities with the centre of a small ( 3) codimension on X. In Subsections 2.2 and 2.3, following (with minor modification) the arguments of Subsection 2.1 in [3], we exclude non-canonical singularities of the pair (X, 1 n D X ), the centre of which is not contained in Sing X. In subsection 2.3 we use, for this purpose, the standard technique of hypertangent divisors. As a first application, we obtain a proof of Proposition 1.1.

Regular hypersurfaces.
Let X ⊂ P N , where N 8, be a hypersurface, satisfying the condition codim(Sing X ⊂ X) 5.
In particular, X is factorial and Pic X = ZH X , where H X is the class of a hyperplane section. Let o ∈ X be a non-singular point and is irreducible and reduced. Proposition 2.1. Assume that the hypersurface X satisfied the conditions (N1-3) at every non-singular point o ∈ X. Then for every pair (X, 1 n D X ), where D X ∼ nH X is an effective divisor, the union of the centres of all non-canonical singularities CS (X, 1 n D X ) of that pair is contained in the closed set Sing X. Proof. Assume the converse: for some effective divisor D X ∼ nH X CS X, 1 n D X ⊂ Sing X.
Let Y be an irreducible component of the set CS (X, 1 n D X ), which is not contained in Sing X, the dimension of which is maximal among all such components.
Lemma 2.1. The following inequality holds: Proof. Assume the converse: codim(Y ⊂ X) 3. Since Y is the centre of some non canonical singularity of the pair (X, 1 n D X ) and Y ⊂ Sing X, we get the inequality mult Y D X > n. Since the codimension of the set Sing X is at least 5, we can take a curve C ⊂ X, such that C ⊂ X\ Sing X.
Obviously, mult C D X > n. Now repeating the arguments in the proof of Lemma 2.1 in [10, Chapter 2] word for word, we get a contradiction which completes the proof of Lemma 2.1.

Restriction onto a hyperplane section.
Let o ∈ Y be a point of general position, o ∈ Sing X. Consider the section P ⊂ X by a general linear subspace of dimension 4, containing the point o. The hypersurface P ⊂ P 4 is non-singular, so that Pic P = ZH P by the Lefschetz theorem, where H P is the class of a hyperplane section of the variety P . Set D P = D X | P , so that D P ∼ nH P . By inversion of adjunction, the pair (P, 1 n D P ) is not log canonical; moreover, by construction, Let ϕ P : P + → P be the blow up of the point o, E P = ϕ −1 P (o) ∼ = P 2 the exceptional divisor, D + P the strict transform of the divisor D P on P + . Lemma 2.2. There is a line L ⊂ E P , satisfying the inequality Proof. This follows from [3,Proposition 9]. Q.E.D. The blow up ϕ P can be viewed as the restriction onto the subvariety P of the blow up ϕ X : X + → X of the point o with the exceptional divisor E X ∼ = P N −2 . Lemma 2.2 implies that there is a hyperplane Θ ⊂ E X , satisfying the inequality The rest of the proof of Proposition 2.1 repeats the proof of part (i) of Theorem 2 in [3, . 2.1] almost word for word. For the convenience of the reader we briefly reproduce those arguments. By the symbol |H X − Θ| we denote the pencil of hyperplane sections R of the hypersurface X, such that R ∋ o and R + ∩ E X = Θ (where R + ⊂ X + is the strict transform). Let R ∈ |H X − Θ| be a general element of the pencil. Set D R = D X | R . Lemma 2.3. The following inequality holds: Proof. This is Lemma 3 in [3] (our claim follows directly from the inequality (3) and the choice of the section R). Q.E.D. for the lemma.
Consider the tangent hyperplane T o R ⊂ P N −1 to the hypersurface R at the point o. The intersection T R = R ∩ T o R is a hyperplane section of R. Therefore, T R ∼ H R is a prime divisor on R. By the condition (N1) the equality mult o T R = 2 holds. Therefore, if D R = aT R + D ♯ R , where a ∈ Z + and the effective divisor D ♯ R ∼ (n − a)H R does not contain T R as a component, then the inequality holds. In order not to make the notations too complicated, we assume that a = 0, that is, D R ∼ nH R does not contain T R as a component. Moreover, by the linearity of the inequality (4) in D R , we may assume that D R is a prime divisor.

Hypertangent divisors.
Getting back to the coordinates z 1 , . . . , z N , write down . . , deg X and consider the second hypertangent system where s 0 ∈ C and s 1 runs through the space of linear forms in z * . By the condition (N3) the base set Bs Λ R 2 is irreducible and reduced, and by the condition (N1) it is of codimension 2 on R. Therefore, a general divisor D 2 ∈ Λ R 2 does not contain the prime divisor D R as a component, so that we get a well defined effective cycle By the linearity of the equivalent inequality in Y 2 we may replace the cycle Y 2 by its suitable irreducible component and assume Y 2 to be an irreducible subvariety of codimension 2. Lemma 2.4. The subvariety Y 2 is not contained in the tangent divisor T R .
Proof. The base set of the hypertangent system Λ R 2 is It is irreducible, reduced and therefore deg S R = 2 deg X.
By the condition (N1) the equality holds. Therefore, Y 2 = S R . However, a certain polynomial vanishes on Y 2 , where s 0 = 0, since the divisor D 2 ∈ Λ R 2 is chosen to be general. If we had h 1 | Y 2 ≡ 0, then we would have got h 2 | Y 2 ≡ 0. Since h 2 = h 1 + h 2 , this would have implied that h 2 | Y 2 ≡ 0 and Y 2 ⊂ Bs Λ R 2 = S R , which is not true. Q.E.D. for the lemma. By the lemma that we have just shown, the effective cycle The cycle Y 3 can be assumed to be an irreducible subvariety of codimension 3 on R for the same reason as Y 2 . Now applying the technique of hypertangent divisors in the usual way [10, Chapter 3], we intersect Y 3 with general hypertangent divisors using the condition (N1), and obtain an irreducible curve C ⊂ R, satisfying by (2) the inequality which is impossible. This proves Proposition 2.1. Q.E.D.
Proof of Proposition 1.1. It is sufficient to check that the hypersurface Γ satisfies all the assumptions that were made about the hypersurface X. Indeed, Γ has at most quadratic singularities of rank 8, so that codim (Sing X ⊂ X) 7.

Reduction to a hyperplane section
In this section we consider hypersurfaces X ⊂ P N with at most quadratic singularities, the rank of which is bounded from below, which also satisfy some additional conditions. For a non-canonical pair (X, 1 n D X ), where D X ∼ nH X does not contain hyperplane sections of the hypersurface X, we construct a special hyperplane section ∆, such that the pair (∆, 1 n D ∆ ), where D ∆ = D X | ∆ , is again non-canonical and, into the bargain, somewhat "better" than the original pair: the multiplicity of the divisor D ∆ at some point o ∈ ∆ is higher than the multiplicity of the original divisor D X at this point.
3.1. Hypersurfaces with singularities. Take N 8 and let X ⊂ P N be a hypersurface, satisfying the following conditions: (S1) every point o ∈ X is either non-singular, or a quadratic singularity of rank 7, (S2) for every effective divisor D ∼ nH X , where H X ∈ Pic X is the class of a hyperplane section and n 1, the union CS(X, 1 n D X ) os the centres of all non log canonical singularities of the pair (X, 1 n D X ) is contained in Sing X, (S3) for every effective divisor Y on the section of X by a linear subspace of codimension 1 or 2 in P N and every point o ∈ Y , singular on X, the following inequality holds: The condition (S1), Grothendieck's theorem on parafactoriality [11] and the Lefschetz theorem imply that X is a factorial variety and Cl X = Pic X = ZH X , since codim (Sing X ⊂ X) 6. As every hyperplane section of the hypersurface X is a hypersurface in P N −1 , the singular locus of which has codimension at least 4, it is also factorial.
Assume, furthermore, that D X ∼ nH X is an effective divisor, such that we have CS(X, 1 n D X ) = ∅, and moreover, there is a point o ∈ CS(X, 1 n D X ) ⊂ Sing X (see the condition (S2)), which is a quadratic singularity of rank 8. Let ϕ: X + → X be its blow up with the exceptional divisor E = ϕ −1 (o), which by our assumption is a quadric of rank 8. For the strict transform D + X ⊂ X + we can write where by the condition (S3) we have α < 2, since mult o D X < 4n. Remark 3.1. As we will see below, under our assumptions the inequality α > 1 holds. Since for every hyperplane section ∆ ∋ o of the hypersurface X and its strict transform ∆ + ⊂ X + we have the pair (X, ∆) is canonical, so that we may assume that the effective divisor D X does not contain hyperplane sections of the hypersurface X as components (if there are such components, they can be removed with all assumptions being kept). For that reason, for any hyperplane section ∆ ∋ o the effective cycle (∆ • D X ) of codimension 2 on X is well defined. We will understand this cycle as an effective divisor on the hypersurface ∆ ⊂ P N −1 and denote it by the symbol D ∆ . and D + ∆ ∼ n(H ∆ − α ∆ E ∆ ), and moreover the following inequality holds: (Here H ∆ is the class of a hyperplane section of the hypersurface ∆ ⊂ P N −1 , and E ∆ = ∆ + ∩ E is the exceptional divisor of the blow up ϕ ∆ : ∆ + → ∆, where ∆ + is the strict transform of ∆ on X + and D + ∆ is the strict transform of the divisor D ∆ on ∆ + .) Proof. Obviously, D ∆ ∼ nH ∆ . We have for every hyperplane section ∆, so that we only need to show the existence of the hyperplane section ∆ for which the inequality (6)

Preliminary constructions.
Consider the section P of the hypersurface X by a general 5-dimensional linear subspace, containing the point o. Obviously, P ⊂ P 5 is a factorial hypersurface, o ∈ P is an isolated quadratic singularity of the maximal rank. Let P + ⊂ X + be the strict transform of the hypersurface P , so that E P = P + ∩E is a non-singular three-dimensional quadric. Set D P = (D • P ) = D| P . Obviously, by the inversion of adjunction the pair (P, 1 n D P ) has the point o as an isolated centre of a non log canonical singularity. Since a(E P ) = 2 and D + P ∼ nH P − αnE P (where H P is the class of a hyperplane section of the hypersurface P ⊂ P 5 ), and moreover α < 2, we conclude that the pair (P + , 1 n D + P ) is not log canonical and the union LCS E (P + , 1 n D + P ) of the centres of all non log canonical singularities of that pair, intersecting the exceptional divisor E P , is a connected closed subset of the quadric E P . Let S P be an irreducible component of maximal dimension of that set. Since S P is the centre of certain non log canonical singularity of the pair (P + , 1 n D + P ), the inequality mult S P D + P > n holds. Furthermore, codim(S P ⊂ E P ) ∈ {1, 2, 3} (and if S P is a point, then we have LCS E (P + , 1 n D + P ) = S P by the connectedness of that set). Coming back to the original pair (X, 1 n D X ), we see that the pair (X + , 1 n D + X ) has a non log canonical singularity, the centre of which is an irreducible subvariety S ⊂ E, such that S∩E P = S P ; in particular, codim(S ⊂ E) = codim(S P ⊂ E P ) ∈ {1, 2, 3}, and if the last codimension is equal to 3, then S ∩ E P is a point and for that reason S ⊂ E is a linear subspace of codimension 3. However, on a quadric of rank 8 there can be no linear subspaces of codimension 3, so that codim(S ⊂ E) ∈ {1, 2}.
Proof. Assume that this case takes place. Then S ⊂ E is a prime divisor, which is cut out on E by a hypersurface of degree d S 1, that is, S ∼ d S H E , where H E is the class of a hyperplane section of the quadric E. We have (D + X • E) ∼ αnH E , so that 2 > α d S , and for that reason S ∼ H E is a hyperplane section of the quadric E. Let ∆ ∈ |H| be the uniquely determined hyperplane section of the hypersurface X, such that ∆ ∋ o and (∆ + • E) = ∆ + ∩ E = S. For the effective divisor D ∆ the inequality holds. Taking into account that deg(∆ • D X ) = n deg X, we get a contradiction with the condition (S3), which by assumption is satisfied for the hypersurface X. Q.E.D. for the proposition.
3.3. The case of codimension 2. We proved above that S ⊂ E is a subvariety of codimension 2. Following [12, Section 3], for distinct points p = q on the quadric E we denote by the symbol [p, q] the line joining these two points, provided that it is contained in E, and the empty set, otherwise, and set (where the line above means the closure).
Lemma 3.1. One of the following two options takes place: (1) Sec (S ⊂ E) is a hyperplane section of the quadric E, on which S is cut out by a hypersurface of degree d S 2, (2) S = Sec (S ⊂ E) is the section of the quadric E by a linear subspace of codimension 2.
Proof repeats the proof of Lemma 4.1 in [2], and we do not give it here. (The key point in the arguments is that due to the inequality α < 2 every line L = [p, q] ⊂ E, joining some point p, q ∈ S and lying on E, is contained in D + X , because mult S D + X > n.) Proposition 3.3. The option (2) does not take place. Proof. Assume the converse: the case (2) takes place. Let P ⊂ X be the section of the hypersurface X by the linear subspace of codimension 2 in P N , that is uniquely determined by the conditions P ∋ o and P + ∩ E = S.
The symbol |H − P | stands for the pencil of hyperplane sections of the hypersurface X, containing P . For a general divisor ∆ ∈ |H − P | we have the equality Write down (∆ • D X ) = G + aP , where a ∈ Z + and G is an effective divisor on ∆, not containing P as a component. Obviously, G ∈ |mH ∆ |, where m = n − a and H ∆ is the class of a hyperplane section of ∆ ⊂ P N −1 . The symbols G + and ∆ + stand for the strict transforms of G and ∆ on X + , respectively. Now where E ∆ = ∆ + ∩ E is a hyperplane section of the quadric E and, besides, By construction, the effective cycle (G•P ) of codimension 2 on ∆ is well defined. One can consider it as an effective divisor on the hypersurface P ⊂ P N −2 . The Proposition 3.4. For some irreducible divisor S 1 ⊂ E S , such that the projection σ S | S 1 is birational, the inequality holds, where D ∆ and Λ are the strict transforms, respectively, of D + ∆ and Λ on ∆. Proof. This is a well known fact, see [3,Proposition 9]. (Note that the subvariety S is, generally speaking, singular, however ∆ + is non-singular at the general point of S and ∆ is non-singular at the general point of S 1 .) 3.5. End of the proof. Set µ S = mult S D + ∆ and β = mult S 1 D ∆ . One of the two cases takes place: -the case of general position S 1 = E S ∩ Λ, so that S 1 ⊂ Λ, -the special case S 1 = E S ∩ Λ. Let us consider them separately. In the case of general position the inequality (8) takes the form µ S + β + a > 2n, since mult S 1 Λ = 0. Furthermore, µ S β, so that the more so 2µ S + a > 2n.
The inequality (6) in the case of general position is now proven.
Proof of Proposition 1.2. Let us check that the operation of reduction, described in Subsection 3.1, can be ρ times applied to the hypersurface Γ ⊂ P M . Consider the hypersurface Γ i ⊂ P M −i , where i ∈ {0, . . . , ρ − 1}. Let us show, in the first place, that Γ i satisfies the condition (S1). Let p ∈ Γ i be an arbitrary singularity. If i = 0, then by the condition (R2.1), the point p is a quadratic singularity of rank 8. If i 1, then there are two options: either p ∈ Γ is a non-singular point, or p ∈ Γ is a singularity (recall that Γ i is a section of the hypersurface Γ by a linear subspace of codimension i in P M ). In the second case by the condition (R2.1) the point p is a quadratic singularity of Γ of rank 2ρ + 6 2i + 8, since ρ i + 1. Since a hyperplane section of a quadric of rank r 3 is a quadric of rank r − 2, we conclude that p ∈ Γ i is a quadratic singularity of rank 8, so that the condition (S1) is satisfied at that point (for the hypersurface Γ i ).
In the first case the point p is non-singular on Γ, so that Γ i is a section of Γ by a linear subspace of codimension i, which is contained in the tangent hyperplane T p Γ. By the condition (R1.4) the point p ∈ Γ i is a quadratic singularity of rank 8 + 2(i + 1) − 4 − 2(i − 1) = 8 (one should take into account that the cutting subspace is of codimension i − 1 in T p Γ). Therefore, the condition (S1) is satisfied in any case.
Let us show that the hypersurface Γ i satisfies the condition (S2) as well. In order to do it, we must check that for Γ i all assumptions of Subsection 2.1 are satisfied. By what was said above, the codimension of the set Sing Γ i with respect to Γ i it at least 7 -this is higher than we need. The inequality (2) takes the form of the estimate which is easy to check. Finally, the conditions (N1), (N2) and (N3) follow from the conditions (R1.1), (R1.2) and (R1.3), respectively. By Proposition 2.1 we conclude that the hypersurface Γ i satisfies the condition (S2). Finally, let us consider the condition (S3). Obviously, it is sufficient to check that the inequality (5) holds for any prime divisor Y on the section of the hypersurface Γ i by a linear subspace P * of codimension 2 in P M −i . Assume the converse: In some affine coordinates with the origin at the point o on the subspace P * = P M −i−2 the equation of the hypersurface P * ∩ Γ i has the form 0 = q * 2 + q * 3 + · · · + q * dl , where by the condition (R2.2) the sequence of homogeneous polynomials q * 2 , q * 3 , . . . , q *