Canonical and log canonical thresholds of multiple projective spaces

We show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where d⩾4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\geqslant 4$$\end{document}, is equal to 1 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.


Statement of the main results
In [12] 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 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. Then [9] 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 [6]. Let us give precise statements.
Fix a pair of positive integers (d, l) ∈ Z ×2 + in the set described by the following Set M = (d − 1)l. The symbol P stands for the complex projective space P M . Consider the weighted projective space 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 for i = 1, . . . , d). The space 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 [12], 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 M 2 /2 as the dimension M grows.
As in [12], 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 1.1 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 ∼ n H the pair V , 1 n D is canonical. Now [9, 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 is a multiple projective space of the type described above. Claim (ii) of Theorem 1.1 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 claim (i) of Theorem 1.1 has at 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 [9, Theorem 1.1], combined with Theorem 1.1, immediately gives the following result. Theorem 1.2 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 The condition for the cycles of dimension M, described in Theorem 1.2, 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 -pullback of the class of a hyperplane in P on P. The pullback 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 so that for some class R ∈ Pic S the relation holds and for that reason This implies that the condition of Theorem 1.2 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.
there are no other structures of a rationally connected fibre space, apart from projections onto direct fibres V i 1 × · · · × V i k . Property (ii) was shown in [12] for a wider class of multiple projective spaces than the one that is considered in this paper. Of course, Theorem 1.1 implies that 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 1.5.

The regularity conditions
The open subset F reg is 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 of reductions to a hyperplane section, used in the proof of Theorem 1.1, 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 Now let us state the regularity conditions. Let o ∈ V be some point.
with the (non-homogeneous) polynomial a i (z * ) of degree il. Furthermore, the following fact is true [12]: 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 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).
is irreducible and reduced. (R1.4) If ρ 2, then the rank of the quadratic form 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 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.
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 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 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 claim (i) of Theorem 1.1 is true.

The structure of the paper and historical remarks
A proof of Theorem 1.1 (ii) is given in Sects. 2.2 and 2.3. A proof of Theorem 1.1 (iii) in Sect. 2.1 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 Sects. 3 and 4.
The equality of the global (log) canonical threshold to 1 is shown for many families of primitive Fano varieties, starting from the pioneer paper [6] (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 [10]. The double covers were considered in [7]. Fano three-folds, singular and non-singular, were studied in the papers [2][3][4][5] 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 [12], was that the technique of hypertangent divisors does not apply to these varieties in a straightforward way. As it turned out (see [12]), 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.

Proof of the main result
In Sect. 2.1 the proof Theorem 1.1 (iii) is reduced to two intermediate claims, the proofs of which are given in Sect. 3 and 4. In Sects. 2.2, 2.3, we show Theorem 1.1 (ii). First (Sect. 2.2) we give the estimates for the codimension of the sets of polynomials, violating each of the regularity conditions, after that (Sect. 2.3) we explain how to obtain these estimates.

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 Sect. 1.2). Fix a variety V ∈ F reg . Assume that D ∼ n H 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 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 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 = V γ contains one of them. In that case the pair , 1 n D is even non-log canonical.
In any case we obtain a factorial hypersurface ⊂ P = P M of degree dl with at worst quadratic singularities of rank 2ρ +6 8, and an effective divisor D ∼ n H on it (where H is the class of a hyperplane section, so that Pic = ZH ), such that the pair , 1 n D is non-canonical. Now we work only with that pair, forgetting about the original variety V (within the limits of the proof of claim (iii) of Theorem 1.1). Let CS , 1 n D be the union of the centres of all non-canonical singularities of that pair.

Proposition 2.1
The closed set CS , 1 n D is contained in the singular locus Sing of the hypersurface .
The proof makes the contents of Sect. 3. 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 ∈ be a point of general position on the irreducible component of maximal dimension of the closed set CS , 1 n D . Consider a general 5-dimensional subspace in P, containing the point o. Let P be the section of the hypersurface by that subspace. Obviously, P ⊂ P 5 is a hypersurface of degree dl with a unique singular point, a nondegenerate quadratic point o. Denoting D | P by the symbol D P , we get D P ∼ n H 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, This implies that and therefore

Proposition 2.2
There is a sequence of irreducible varieties i , i = 0, 1, . . . , ρ, such that: (i) 0 = and i+1 is a hyperplane section of the hypersurface i ⊂ P M−i , containing the point o, (ii) on the variety ρ there is a prime divisor D * ∼ n * H * , where H * is the class of a hyperplane section of the hypersurface ρ , satisfying the inequality The proof makes the contents of Sect. 4. Note that by condition (R2.1) all hypersurfaces 1 , . . . , ρ are factorial, so that Pic ρ = ZH * . Furthermore, ρ 1, so that with the maximal value of the ratio of the multiplicity mult o to the degree. That it is possible to go through with this construction, is ensured by 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 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 claim (iii) of Theorem 1.1. For these values of i. j we set, respectively,

Estimating the codimension of the set F \ F reg
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 2.3
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 condition

Quadratic forms and regular sequences
By the symbol P i,N we denote the linear space of homogeneous polynomials of degree i ∈ Z + in N variables u 1 , . . . , u N . For i j we write and P i,N = i k=0 P k,N . The number of variables N is fixed, so we omit the symbol N and write P k , P [i, j] and so on. Let X 2, r ⊂ P 2 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.
(iii) The following inequality holds: Proof. Claim (i) is well known. Let us show inequality (ii). Taking into account 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 inequality (ii). This proves claim (ii). Let us show 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). Set h Q (m) = h 0 (Q, O Q (m)) for m 1. It is easy to check that is a polynomial in m with positive coefficients. This implies that for 0 < s < t m/2 the inequality holds, which can be re-written as By what was said above, the right-hand side of that inequality is Elementary computations show that the right-hand side of the last equality is which is certainly higher than
The codimension of the set of quadratic forms, for which this quadric is of rank 4 and so not factorial, is given by Proposition 2.4 (i). It is from here that we get the estimate for ε 1.3 in Proposition 2.3. It remains to show that the violation of 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 estimate (iii) of Proposition 2.4. 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 inequality (iii) of Proposition 2.4 the codimension of the "bad" set X m,Q is considered with respect to the whole space P m , whereas in order to prove inequality (iii) of Proposition 2.3, 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 h 2 = · · · = h k = 0 defines in P N −1 a closed subset of codimension k − 2.

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 ∼ n H X is cut out on X by a hypersurface of degree n 1, is contained in the singular locus Sing X . In Sect. 3.1 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 Sects. 3.2 and 3.3, following (with minor modification) the arguments of [6, Subsection 2.1], we exclude non-canonical singularities of the pair X , 1 n D X , the centre of which is not contained in Sing X . In Sect. 3.3 we use, for this purpose, the standard technique of hypertangent divisors. As a first application, we obtain a proof of Proposition 2.1.

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 z 1 , . . . , z N a system of affine coordinates on A N ⊂ P N with the origin at the point o, and the hypersurface X in this coordinate system is given by the equation h = 0, where h = h 1 + h 2 + · · · + h degX and the polynomials h i are homogeneous of degree i. We assume that the inequality holds. Now let us state the regularity conditions for the hypersurface X at the point o.
is irreducible and reduced.

Proposition 3.1 Assume that the hypersurface X satisfies conditions (N1-3) at every non-singular point o ∈ X . Then for every pair X , 1 n D X , where D X ∼ n H X is an effective divisor, the union of the centres of all non-canonical singularities
n D X of that pair is contained in the closed set Sing X.
Proof Assume the converse: for some effective divisor D X ∼ n H 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 3.2 The following inequality holds:
Proof Assume the converse: codim(Y ⊂ X ) 3. Since Y is the centre of some noncanonical 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 Obviously, mult C D X > n. Now repeating the arguments in the proof of [8, Chapter 2, Lemma 2.1] word for word, we get a contradiction which completes the proof.

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 ∼ n H 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 3.3 There is a line L ⊂ E P , satisfying the inequality
Proof This follows from [6, Proposition 9].
The blow-up ϕ P can be viewed as the restriction onto the subvariety P of the blow-p ϕ X : X + → X of the point o with the exceptional divisor E X ∼ = P N −2 . Lemma 3.3 implies that there is a hyperplane ⊂ E X , satisfying the inequality The rest of the proof of Proposition 3.1 repeats the proof of [6, 2.1, Theorem 2 (i)] almost word for word. For the convenience of the reader we briefly reproduce those arguments. By the symbol |H X − | we denote the pencil of hyperplane sections R of the hypersurface X , such that R o and R + ∩ E X = (where R + ⊂ X + is the strict transform). Let R ∈ |H X − | be a general element of the pencil. Set D R = D X | R .

Lemma 3.4 The following inequality holds:
Proof This is [6, Lemma 3] (our claim follows directly from inequality (3) and the choice of the section R).
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 condition (N1) the equality mult o T R = 2 holds. Therefore, if 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 ∼ n H R does not contain T R as a component. Moreover, by the linearity of 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 condition (N3) the base set Bs R 2 is irreducible and reduced, and by condition (N1) it is of codimension 2 on R. Therefore, a general divisor D 2 ∈ R 2 does not contain the prime divisor D R as a component, so that we get a well-defined effective cycle of codimension 2 on R, satisfying the inequality 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 3.5 The subvariety Y 2 is not contained in the tangent divisor T R .
Proof The base set of the hypertangent system R 2 is It is irreducible, reduced and therefore deg S R = 2 deg X .
By 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 ∈ R 2 is chosen to be general. If we had By the lemma that we have just shown, the effective cycle of codimension 3 on R is well defined. It satisfies the inequality 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 [8, Chapter 3], we intersect Y 3 with general hypertangent divisors using condition (N1), and obtain an irreducible curve C ⊂ R, satisfying by (2) the inequality which is impossible. This completes the proof of Proposition 3.1.

Proof of Proposition 2.1
It is sufficient to check that the hypersurface satisfies all the assumptions that were made about the hypersurface X . Indeed, 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 ∼ n H 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.

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 ∼ n H X , where H X ∈ Pic X is the class of a hyperplane section and n 1, the union CS X , 1 n D X of 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: Condition (S1), Grothendieck's theorem on parafactoriality [1], 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 ∼ n H 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 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 condition (S3) we have α < 2, since mult o D X < 4n.

Remark 4.1
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 .

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 3-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 ∼ n H P − αn E 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, 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}.

Proposition 4.3 The case codim(S ⊂ E) = 1 is impossible.
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 holds. Taking into account that deg( • D X ) = ndeg X , we get a contradiction with condition (S3), which by assumption is satisfied for the hypersurface X .

The case of codimension 2
We proved above that S ⊂ E is a subvariety of codimension 2. Following [11,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). Proof Assume the converse: case (b) 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 + a P, where a ∈ Z + and G is an effective divisor on , not containing P as a component. Obviously, G ∈ |m H |, 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 following inequality holds: Since deg(G • P) = m deg X , we obtain a contradiction with condition (S3), which is satisfied for the hypersurface X .

The hyperplane section 1
We have shown that case (a) takes place. Set = S = Sec(S ⊂ E). This is a hyperplane section of the quadric E, where ⊂ D + X . Set We know that μ > n and μ αn < 2n (the second inequality holds, because for a general linear subspace ⊂ E of maximal dimension the divisor (D + X • ) = D + X ∩ on is a hypersurface of degree αn, containing every point of the set S ∩ with multiplicity μ).

Lemma 4.6
The following inequality holds: Proof This is [9,Lemma 4.2]. The claim of the lemma is a local fact and for that reason the proof given in [9, 4.3] does not require any modifications and works word for word. Now let us consider the uniquely determined hyperplane section of the hypersurface X ⊂ P N such that o and + ∩ E = , where + ⊂ X + is the strict transform of . Write down Obviously, mult o D = 2(αn + a), so that α = α + a n (recall that D + ∼ n(H − α E ), where E = ). Since the subvariety S is cut out on the quadric by a hypersurface of degree d S 2, we obtain the inequality Since S ⊂ LCS X + , 1 n D + X , we get Consider the blow-up σ S : → + of the subvariety S ⊂ + of codimension 2 and denote its exceptional divisor σ −1 S (S) by the symbol E S . Proposition 4.7 For some irreducible divisor S 1 ⊂ E S , such that the projection σ S | S 1 is birational, the inequality holds, where D and are the strict transforms, respectively, of D + and on .
Proof This is a well-known fact, see [6,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 .)

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 ∩ , so that S 1 ⊂ , • the special case S 1 = E S ∩ .
Let us consider them separately. In the case of general position inequality (8) takes the form μ S + β + a > 2n, since mult S 1 = 0. Furthermore, μ S β, so that the more so 2μ S + a > 2n.
Inequality (6) in the case of general position is now proven. Let us consider the special case. Here mult S 1 = 1, so that inequality (8) takes the form Besides, the effective cycle (D + • ), considered as an effective divisor on , is cut out on the quadric by a hypersurface of degree αn + a, and contains the divisor S ∼ d S H (where H is the class of a hyperplane section of ) with multiplicity μ S + β, so that 2(μ S + β) αn + a, whence we get αn + 5a > 4n and for that reason 5 α n = 5 (αn + a) > 4 (α + 1) n, that is, α > 4 5 α + 4 5 > 3 5 α + 1 (since α > 1). This inequality is stronger than (6), which completes the proof in the special case.
The proof of Proposition 4.2 is now complete.

Proof of Proposition 2.2
Let us check that the operation of reduction, described in Sect. 4.1, can be ρ times applied to the hypersurface ⊂ P M . Consider the hypersurface i ⊂ P M−i , where i ∈ {0, . . . , ρ − 1}. Let us show, in the first place, that i satisfies condition (S1). Let p ∈ i be an arbitrary singularity. If i = 0, then by condition (R2.1), the point p is a quadratic singularity of rank 8. If i 1, then there are two options: either p ∈ is a non-singular point, or p ∈ is a singularity (recall that i is a section of the hypersurface by a linear subspace of codimension i in P M ). In the second case by condition (R2.1) the point p is a quadratic singularity of 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 ∈ i is a quadratic singularity of rank 8, so that condition (S1) is satisfied at that point (for the hypersurface i ).
In the first case the point p is non-singular on , so that i is a section of by a linear subspace of codimension i, which is contained in the tangent hyperplane T p . By condition (R1.4) the point p ∈ 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 ). Therefore, condition (S1) is satisfied in any case.
Let us show that the hypersurface i satisfies condition (S2) as well. In order to do it, we must check that for i all assumptions of Sect. 3.1 are satisfied. By what was said above, the codimension of the set Sing i with respect to i it at least 7 -this is higher than we need. Inequality (2) takes the form of the estimate which is easy to check. Finally, conditions (N1), (N2) and (N3) follow from conditions (R1.1), (R1.2) and (R1.3), respectively. By Proposition 3.1 we conclude that the hypersurface i satisfies condition (S2).
where codim(Y * j ⊂ (P * ∩ i )) = j and the last subvariety Y * M−ρ−6 (the dimension of which is ρ − i + 3 4) satisfies the inequality It is easy to check that the right-hand side of the last inequality for the values of d, l and ρ under consideration is higher than 1, which gives a contradiction with assumption (9) and proves that the hypersurface i satisfies condition (S3). Note that all singular points of i are quadratic singularities of rank 8, so that the additional assumption about the point o made in Sect. 4.1 is satisfied. Now applying Proposition 4.2, we complete the proof of Proposition 2.2.