# Canonical and log canonical thresholds of multiple projective spaces

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## Abstract

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 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.

## Keywords

Maximal singularity (Log) canonical threshold Fano–Mori fibre space Hypertangent divisor## Mathematics Subject Classification

14E05 14E07## 1 Introduction

### 1.1 Statement of the main results

In [12] general *d*-sheeted covers of the complex projective space \({{\mathbb {P}}}={{\mathbb {P}}}^M\) 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 \((d,l)\in {{\mathbb {Z}}}^{\times 2}_+\) in the set described by the following table:

*l*, and a quasi-homogeneous polynomialof degree

*dl*(that is, \(A_i(x_0,\dots ,x_M)\) is a homogeneous polynomial of degree

*il*for \(i=1,\dots ,d\)). The spaceparameterizes all such polynomials. If the hypersurface

*V*is a factorial variety with terminal singularities, see [12], so that

*H*is the class of a “hyperplane section”, that is, of the divisor \(V\cap \{\lambda =0\}\), where \(\lambda (x_0,\dots ,x_M)\) is an arbitrary linear form. Below for all the values of

*d*,

*l*under consideration we will define explicitly a positive integral-valued function \(\varepsilon (d,l)\), which behaves as \(M^2/2\) as the dimension

*M*grows.

As in [12], we identify the polynomial Open image in new window and the corresponding hypersurface \(\{F=0\}\), which makes it possible to write Open image in new window. The following theorem is the main result of the present paper.

### Theorem 1.1

- (i)
every hypersurface Open image in new window has at most quadratic singularities of rank \(\geqslant 8\) and for that reason is a factorial Fano variety of index 1 with terminal singularities,

- (ii)the inequality holds,
- (iii)
for every variety Open image in new window and every divisor \(D\sim nH\) the pair \(\bigl (V,\frac{1}{n}D\bigr )\) 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.

*S*of dimension

*W*| is sufficiently mobile on \({{\mathbb {X}}}\), and the hypersurface

*W*is sufficiently general in that linear system. Set

*W*by claim (i) of Theorem 1.1 has at most quadratic singularities of rank \(\geqslant 8\), and for that reason is a factorial variety with terminal singularities. Therefore, \(\eta :W\rightarrow S\) is a Fano–Mori fibre space, the fibres of which are multiple projective spaces of index 1. Let Open image in new window 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 \((\eta ')^{-1}(s')\), Open image in new window, are rationally connected, and moreover, Open image in new window. Now [9, Theorem 1.1], combined with Theorem 1.1, immediately gives the following result.

### Theorem 1.2

*S*, sweeping out

*S*, and a general curve Open image in new window the class of an algebraic cycleis not effective, that is, it is not rationally equivalent to an effective cycle of dimension

*M*. Then every birational map \(\chi :W\dashrightarrow W'\) onto the total space of the rationally connected fibre space Open image in new window (if such maps exist) is fibre-wise, that is, there is a rational dominant map \(\zeta :S\dashrightarrow S'\), such that the following diagram commutes:

### Corollary 1.3

In the assumptions of Theorem 1.2 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 \(\eta \) and for that reason induces a birational self-map of the base *S*.

*M*, described in Theorem 1.2, is satisfied if the linear system |

*W*| is sufficiently mobile on \({{\mathbb {X}}}\). Let us demonstrate it by an especially visual example, when Open image in new window is the trivial fibre space over

*S*. Let Open image in new window be the only singular point of the weighted projective space \(\overline{\mathbb {P}}\). Consider the projection “from the point \(o^*\)”where Open image in new window. Let \({\overline{H}}\) be the \(\pi _{{\mathbb {P}}}\)-pullback of the class of a hyperplane in \({{\mathbb {P}}}\) on \(\overline{{{\mathbb {P}}}}\). The pullback of the class \({\overline{H}}\) on Open image in new window with respect to the projection onto the first factor we denote for simplicity by the same symbol \({\overline{H}}\). Nowso that for some class \(R\in \mathop {\mathrm{Pic}}\, S\) the relation

*S*, and a general curve Open image in new window, the inequalityholds. Therefore, the following claim is true.

### Theorem 1.4

Assume that the class \(R+K_S\in \mathrm{Pic}\, S\) is pseudo-effective and for every point \(s\in S\) we have Open image in new window. Then in the notations of Theorem 1.2 every birational map \(\chi :W \dashrightarrow W'\) is fibre-wise. In particular, every birational self-map \(\chi \in \mathrm{Bir}\, W\) induces a birational self-map of the base *S*.

Another standard application of Theorem 1.1 is given by the theorem on birational geometry of Fano direct products [6, Theorem 1]. Recall that the following statement is true.

### Theorem 1.5

- (i)for every effective divisorthe pair \(\bigl (V_i,\frac{1}{n}D_i\bigr )\) is log canonical,$$\begin{aligned} D_i\sim {}-nK_{V_i} \end{aligned}$$
- (ii)for every mobile linear systemand a general divisor \(D_i\in \Sigma _i\) the pair \(\bigl (V_i,\frac{1}{n}D_i\bigr )\) is canonical.$$\begin{aligned} \Sigma _i\subset |{-}nK_{V_i}| \end{aligned}$$

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 Open image in new window. Therefore, every variety considered in the present paper can be taken as a factor of the direct product in Theorem 1.5.

### 1.2 The regularity conditions

The open subset Open image in new window is given by explicit local regularity conditions, which we will now describe. To begin with, let us introduce an auxiliary integral-valued parameter \(\rho \in \{1,2,3,4\}\), 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 \(\rho =4\), if \(d=4\) and \(21\leqslant l\leqslant 25\) and \(\rho =1\), if \(d\geqslant 18\) and \(l\geqslant 2\). For the remaining possible pairs (*d*, *l*) the value \(\rho \geqslant 2\) is given by the following table:

If the pair (*d*, *l*) is not in the table, then \(\rho =1\) (for instance, for \(d=14\), \(l\geqslant 4\)).

One more table gives the function \(\varepsilon (d,l)\), bounding from below the codimension of the complement to the set Open image in new window. Write this function as a function of the dimension Open image in new window, for each of the possible values of the parameter \(\rho \) defined above.

Now let us state the regularity conditions.

*l*the equation \(\xi =\gamma (x_*)\) defines a hypersurface \(R_{\gamma }\subset \overline{{\mathbb {P}}}\) that does not contain the point Open image in new window, and moreover the projection

*o*, decomposed into homogeneous components (with respect to an arbitrary system of linear coordinates on

*P*).

- (R1.1)For any linear formthe sequence of homogeneous polynomials$$\begin{aligned} \lambda \not \in \langle q_1\rangle \end{aligned}$$is regular in the local ring Open image in new window.$$\begin{aligned} q_1|_{\{\lambda =0\}},\quad q_2|_{\{\lambda =0\}},\quad \dots ,\quad q_{M-\rho -2}|_{\{\lambda =0\}} \end{aligned}$$
- (R1.2)The linear span of any irreducible component of the closed set is the hyperplane \(\{q_1=0\}\).
- (R1.3)For any linear form \(\lambda \not \in \langle q_1\rangle \) the setis irreducible and reduced.$$\begin{aligned} \overline{P\cap V_{\gamma }\cap \{q_1=q_2=0\}\cap \{\lambda =0\}} \end{aligned}$$
- (R1.4)If \(\rho \geqslant 2\), then the rank of the quadratic formis at least \(8+2(\rho -2)\).$$\begin{aligned} q_2|_{\{q_1=0\}} \end{aligned}$$

*regular*, if for a general polynomial \(\gamma (x_*)\) and any subspace \(P\not \subset T_oV_{\gamma }\) conditions (R1.1–3) are satisfied.

- (R2.1)
The point \(o\in V_{\gamma }\) is a quadratic singularity of rank \(\geqslant 2\rho +6\).

- (R2.2)The sequence of homogeneous polynomialsis regular in the local ring Open image in new window.$$\begin{aligned} q_2,\;\; q_3,\;\;,\dots ,\;\; q_{M-\rho -4} \end{aligned}$$

*regular*, if for a general polynomial \(\gamma (x_*)\) and any subspace \(P\subset {{\mathbb {A}}}^M\) of codimension \(\rho +2\) conditions (R2.1–2) hold.

*V*is

*regular*, if it is regular at every point \(o\in V\), singular or non-singular. Setto 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 Open image in new window has at worst quadratic singularities of rank \(\geqslant 8\), so that claim (i) of Theorem 1.1 is true.

### 1.3 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 \({{\mathbb {P}}}^N\), which are applied to the hypersurface \(V_{\gamma }\subset {{\mathbb {P}}}\), 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.

## 2 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.

### 2.1 Exclusion of maximal singularities

Fix the parameters *d*, *l*. Recall that the integer \(\rho \in \{1,2,3,4\}\) depends on *d*, *l* (see the table in Sect. 1.2). Fix a variety Open image in new window. Assume that \(D\sim nH\) is an effective divisor on *V* such that the pair \(\bigl (V,\frac{1}{n}D\bigr )\) is not canonical. Our aim is to get a contradiction. This would prove claim (iii).

*dl*with at worst quadratic singularities of rank \(\geqslant 2\rho +6\geqslant 8\), and an effective divisor \(D_{\Gamma }\sim nH_{\Gamma }\) on it (where \(H_{\Gamma }\) is the class of a hyperplane section, so that \(\mathrm{Pic}\,\Gamma ={{\mathbb {Z}}}H_{\Gamma }\)), such that the pair \(\bigl (\Gamma ,\frac{1}{n}D_{\Gamma }\bigr )\) 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). Letbe the union of the centres of all non-canonical singularities of that pair.

### Proposition 2.1

The closed set Open image in new window is contained in the singular locus \(\mathrm{Sing}\,\Gamma \) of the hypersurface \(\Gamma \).

The proof makes the contents of Sect. 3.

*o*. Let

*P*be the section of the hypersurface \(\Gamma \) by that subspace. Obviously, \(P\subset {{\mathbb {P}}}^5\) is a hypersurface of degree

*dl*with a unique singular point, a non-degenerate quadratic point

*o*. Denoting \(D_{\Gamma |_P}\) by the symbol \(D_P\), we get \(D_P\sim nH_P\), 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 \(\bigl (P,\frac{1}{n}D_P\bigr )\), and moreover,This implies that

### Proposition 2.2

- (i)
\(\Gamma _0=\Gamma \) and \(\Gamma _{i+1}\) is a hyperplane section of the hypersurface \(\Gamma _i\subset {{\mathbb {P}}}^{M-i}\), containing the point

*o*, - (ii)on the variety \(\Gamma _{\rho }\) there is a prime divisor Open image in new window, where \(H^*\) is the class of a hyperplane section of the hypersurface \(\Gamma _{\rho }\), satisfying the inequality

The proof makes the contents of Sect. 4.

*d*,

*l*). Thus we obtained a contradiction, which completes the proof of claim (iii) of Theorem 1.1.

### 2.2 Estimating the codimension of the set Open image in new window

*i*.

*j*) at at least one point. Here

*i*.

*j*we set, respectively,We omit the symbols

*d*,

*l*in order to simplify the formulas, however \(\varepsilon _{i.j}=\varepsilon _{i.j}(d,l)\) are functions of these parameters. The following claim is true.

### Proposition 2.3

- (i)
\(\varepsilon _{1.1}\geqslant (M^2-(4\rho +5)M+(3\rho ^2+3\rho ))/2\),

- (ii)
\(\varepsilon _{1.2}\geqslant (M^2-(4\rho +11)M+(3\rho ^2-15\rho +32))/2\),

- (iii)
\(\varepsilon _{1.3}\geqslant (M^2-(4\rho +13)M+(3\rho ^2+11\rho +42))/2\),

- (iv)
\(\varepsilon _{1.4}\geqslant (M^2-(4\rho +9)M+(4\rho ^2+14\rho +16))/2\),

- (v)
\(\varepsilon _{2.1}\geqslant (M^2-(6\rho +7)M+(4\rho ^2+14\rho +12))/2\),

- (vi)
\(\varepsilon _{2.2}\geqslant (M^2-(4\rho +1)M+(3\rho ^2-\rho ))/2\).

### Proof

The regularity conditions must be satisfied for *any* point *o*, *any* linear subspace *P* of the required codimension, and *any* linear form \(\lambda \) (the polynomial \(\gamma (x_*\)) is assumed to be general and does not influence the estimating of the codimension of the sets Open image in new window). Therefore, the problem of getting a lower bound for the numbers \(\varepsilon _{i.j}\) reduces obviously to a similar problem for varieties Open image in new window violating condition (R *i*.*j*) at a *fixed* point *o*, for a *fixed* linear subspace and a *fixed* linear form \(\lambda \). The solution of the latter problem comes from the claims of Propositions 2.4 and 2.6, shown below. More precisely, the estimates for conditions (R1.4) and (R2.1) follow from Proposition 2.4 (i), for condition (R1.2) from Proposition 2.4 (ii), for condition (R1.3) from Proposition 2.4 (iii). The estimates for conditions (R1.1) and (R2.2) follow from Proposition 2.6. \(\square \)

Now, in order to prove claim (ii) of Theorem 1.1, it is sufficient to check that the function \(\varepsilon (d,l)\) is the minimum of the right-hand sides in inequalities (i)–(vi) of Proposition 2.3. This work is elementary and we do not give it here.

### 2.3 Quadratic forms and regular sequences

*N*variables \(u_1,\dots ,u_N\). For \(i\leqslant j\) we writeand Open image in new window. The number of variables

*N*is fixed, so we omit the symbol

*N*and write Open image in new window and so on. Letbe the closed subset of quadratic forms of rank \(\leqslant r\). Letbe the closed subset of pairs \((w_2,w_3)\), such that the closed set \(\{w_2=w_3=0\}\subset {{\mathbb {P}}}^{N-1}\) has at least one degenerate component (that is, a component, the linear span of which is of dimension \(\leqslant N-2\)). Let \(Q\subset {\mathbb P}^{N-1}\) be a factorial quadric. For \(m\geqslant 4\) letbe 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 2.4

- (i)The following equality holds:
- (ii)The following inequality holds:
- (iii)The following inequality holds:

### Proof

*m*with positive coefficients. This implies that for \(0<s<t\leqslant m/2\) the inequality

*Q*, which are cut out on

*Q*by hypersurfaces of degree \(1\leqslant a\leqslant m/2\) and \(m-a\). For that reason,By what was said above, the right-hand side of that inequality is \(h_Q(1)+h_Q(m-1)\), so thatElementary computations show that the right-hand side of the last equality is

### Remark 2.5

*P*and the linear forms \(q_1\) and \(\lambda \), consider the quadric

*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 \(P\cap V_{\gamma }\) 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 \(q_3,\dots ,q_{dl}\) are arbitrary. In inequality (iii) of Proposition 2.4 the codimension of the “bad” set Open image in new window is considered with respect to the whole space Open image in new window, 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 Open image in new window in Open image in new window is very high.

### Proposition 2.6

### Proof

See [8, Chapter 3, Section 1].\(\square \)

## 3 Exclusion of maximal singularities at smooth points

In this section we consider factorial hypersurfaces \(X\subset {{\mathbb {P}}}^N\), satisfying certain additional conditions. We show that the centre of every non-canonical singularity of the pair \(\bigl (X,\frac{1}{n} D_X\bigr )\), where \(D_X\sim nH_X\) is cut out on *X* by a hypersurface of degree \(n\geqslant 1\), is contained in the singular locus \(\mathrm{Sing} \,X\). 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 (\(\leqslant 3\)) 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 \(\bigl (X,\frac{1}{n} D_X\bigr )\), the centre of which is not contained in \(\mathrm{Sing} \,X\). 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.

### 3.1 Regular hypersurfaces

*X*is factorial and \(\mathrm{Pic}\,X={\mathbb {Z}}H_X\), where \(H_X\) is the class of a hyperplane section. Let \(o\in X\) be a non-singular point and

*o*, and the hypersurface

*X*in this coordinate system is given by the equation \(h=0\), where

*i*. We assume that the inequality

*X*at the point

*o*.

- (N1)For any linear formthe sequence of homogeneous polynomials$$\begin{aligned} \lambda (z_*)\not \in \langle h_1\rangle \end{aligned}$$is regular (in the local ring Open image in new window).$$\begin{aligned} h_1|_{\{\lambda =0\}},\quad h_2|_{\{\lambda =0\}},\quad ,\dots ,\quad h_{N-3}|_{\{\lambda =0\}} \end{aligned}$$
- (N2)The linear span of every irreducible component of the closed setis the hyperplane \(\{h_1=0\}\).$$\begin{aligned} h_1=h_2=h_3=0 \end{aligned}$$
- (N3)For any linear form \(\lambda \not \in \langle h_1\rangle \) the setis irreducible and reduced.$$\begin{aligned} \overline{X\cap \{h_1=h_2=0\}\cap \{\lambda =0\}} \end{aligned}$$

### Proposition 3.1

Assume that the hypersurface *X* satisfies conditions (N1–3) at every non-singular point \(o\in X\). Then for every pair \(\bigl (X,\frac{1}{n}D_X\bigr )\), where \(D_X\sim nH_X\) is an effective divisor, the union of the centres of all non-canonical singularities Open image in new window of that pair is contained in the closed set \(\mathrm{Sing}\, X\).

### Proof

*Y*be an irreducible component of the set Open image in new window, which is not contained in \(\mathrm{Sing}\, X\), the dimension of which is maximal among all such components.

### Lemma 3.2

### Proof

*Y*is the centre of some non-canonical singularity of the pair \(\bigl (X,\frac{1}{n}D_X\bigr )\) and \(Y\not \subset \mathrm{Sing}\, X\), we get the inequality \(\mathrm{mult}_YD_X>n\). Since the codimension of the set \(\mathrm{Sing}\, X\) is at least 5, we can take a curve \(C\subset X\) such thatObviously, \(\mathrm{mult}_C D_X>n\). 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. \(\blacksquare \)

### 3.2 Restriction onto a hyperplane section

*o*. The hypersurface \(P\subset {{\mathbb {P}}}^4\) is non-singular, so that \(\mathrm{Pic}\, P={{\mathbb {Z}}}H_P\) 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 \(D_P\sim nH_P\). By inversion of adjunction, the pair \(\bigl (P,\frac{1}{n}D_P\bigr )\) is not log canonical; moreover, by construction,Let \(\varphi _P:P^+\!\rightarrow P\) be the blow-up of the point

*o*, \(E_P=\varphi ^{-1}_P(o)\cong {{\mathbb {P}}}^2\) the exceptional divisor, \(D^+_P\) the strict transform of the divisor \(D_P\) on \(P^+\).

### Lemma 3.3

### Proof

This follows from [6, Proposition 9]. \(\blacksquare \)

*P*of the blow-p \(\varphi _X:X^+\!\rightarrow X\) of the point

*o*with the exceptional divisor \(E_X\cong {\mathbb {P}}^{N-2}\). Lemma 3.3 implies that there is a hyperplane \(\Theta \subset E_X\), satisfying the inequality

*R*of the hypersurface

*X*, such that \(R\ni o\) and \(R^+\cap E_X=\Theta \) (where \(R^+\!\subset X^+\) is the strict transform). Let \(R\in |H_X-\Theta |\) be a general element of the pencil. Set \(D_R=D_X|_R\).

### Lemma 3.4

### Proof

This is [6, Lemma 3] (our claim follows directly from inequality (3) and the choice of the section *R*). \(\blacksquare \)

*R*at the point

*o*. The intersection \(T_R=R\cap T_oR\) is a hyperplane section of

*R*. Therefore, \(T_R\sim H_R\) is a prime divisor on

*R*. By condition (N1) the equality \(\mathrm{mult}_oT_R=2\) holds. Therefore, if

### 3.3 Hypertangent divisors

*R*. Therefore, a general divisor Open image in new window does not contain the prime divisor \(D_R\) as a component, so that we get a well-defined effective cycleof codimension 2 on

*R*, satisfying the inequalityBy the linearity of the equivalent inequality

### Lemma 3.5

The subvariety \(Y_2\) is not contained in the tangent divisor \(T_R\).

### Proof

*R*is well defined. It satisfies the inequalityThe cycle \(Y_3\) can be assumed to be an irreducible subvariety of codimension 3 on

*R*for the same reason as \(Y_2\).

This completes the proof of Proposition 3.1. \(\square \)

### Proof of Proposition 2.1

*X*. Indeed, \(\Gamma \) has at most quadratic singularities of rank \(\geqslant 8\), so thatThat inequality (2) is true for \(\Gamma \), one checks by elementary computations. Condition (N1) follows from (R1.1), condition (N2) from (R1.2), condition (N3) from (R1.3). Therefore, we can apply Proposition 3.1. \(\square \)

## 4 Reduction to a hyperplane section

In this section we consider hypersurfaces \(X\subset {{\mathbb {P}}}^N\) with at most quadratic singularities, the rank of which is bounded from below, which also satisfy some additional conditions. For a non-canonical pair \(\bigl (X,\frac{1}{n} D_X\bigr )\), where \(D_X\sim nH_X\) does not contain hyperplane sections of the hypersurface *X*, we construct a special hyperplane section \(\Delta \), such that the pair \(\bigl (\Delta ,\frac{1}{n} D_{\Delta }\bigr )\), where \(D_{\Delta }=D_X|_{\Delta }\), is again non-canonical and, into the bargain, somewhat “better” than the original pair: the multiplicity of the divisor \(D_{\Delta }\) at some point \(o\in \Delta \) is higher than the multiplicity of the original divisor \(D_X\) at this point.

### 4.1 Hypersurfaces with singularities

- (S1)
every point \(o\in X\) is either non-singular, or a quadratic singularity of rank \(\geqslant 7\),

- (S2)
for every effective divisor \(D\sim nH_X\), where \(H_X\in \mathrm{Pic}\, X\) is the class of a hyperplane section and \(n\geqslant 1\), the union Open image in new window of the centres of all non-log canonical singularities of the pair \(\bigl (X,\frac{1}{n}D_X\bigr )\) is contained in \(\mathrm{Sing}\, X\),

- (S3)for every effective divisor
*Y*on the section of*X*by a linear subspace of codimension 1 or 2 in \({{\mathbb {P}}}^N\) and every point \(o\in Y\), singular on*X*, the following inequality holds:

*X*is a factorial variety and \(\mathrm{Cl}\, X=\mathrm{Pic}\, X={{\mathbb {Z}}} H_X\), since Open image in new window. As every hyperplane section of the hypersurface

*X*is a hypersurface in \({{\mathbb {P}}}^{N-1}\), the singular locus of which has codimension at least 4, it is also factorial.

### Remark 4.1

*X*and its strict transform Open image in new window we have

*X*as components (if there are such components, they can be removed with all assumptions being kept). For that reason, for any hyperplane section \(\Delta \ni o\) the effective cycle Open image in new window of codimension 2 on

*X*is well defined. We will understand this cycle as an effective divisor on the hypersurface \(\Delta \subset {{\mathbb {P}}}^{N-1}\) and denote it by the symbol \(D_{\Delta }\).

### Proposition 4.2

*X*such thatand \(D^+_{\Delta }\sim n(H_{\Delta }-\alpha _{\Delta }E_{\Delta })\), and, moreover, the following inequality holds:

### Proof

*every*hyperplane section \(\Delta \), so that we only need to show the existence of the hyperplane section \(\Delta \) for which inequality (6) is satisfied. This fact is obtained by the arguments, repeating the proof of [9, Theorem 1.4, see Subsections 4.2, 4.3] almost word for word. We will go through the main steps of these arguments, dwelling on the necessary modifications. Whenever possible, we use the same notations as in [9, Subsections 4.2, 4.3].

### 4.2 Preliminary constructions

*P*of the hypersurface

*X*by a general 5-dimensional linear subspace, containing the point

*o*. Obviously, \(P\subset {{\mathbb {P}}}^5\) is a factorial hypersurface, \(o\in P\) is an isolated quadratic singularity of the maximal rank. Let Open image in new window be the strict transform of the hypersurface

*P*, so that \(E_P=P^+\cap E\) is a non-singular 3-dimensional quadric. Set Open image in new window. Obviously, by the inversion of adjunction the pair \((P,\frac{1}{n}D_P)\) has the point

*o*as an isolated centre of a non-log canonical singularity. Since \(a(E_P)=2\) and \(D^+_P\sim nH_P-\alpha nE_P\) (where \(H_P\) is the class of a hyperplane section of the hypersurface \(P\subset {{\mathbb {P}}^5}\)), and moreover \(\alpha <2\), we conclude that the pair Open image in new window is not log canonical and the unionof 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 Open image in new window, the inequality

### Proposition 4.3

The case Open image in new window is impossible.

### Proof

*E*by a hypersurface of degree \(d_S\geqslant 1\), that is, \(S\sim d_SH_E\), where \(H_E\) is the class of a hyperplane section of the quadric

*E*. We have Open image in new window, so that

*E*. Let \(\Delta \in |H|\) be the uniquely determined hyperplane section of the hypersurface

*X*, such that \(\Delta \ni o\) and Open image in new window. For the effective divisor \(D_{\Delta }\) the inequalityholds. Taking into account that Open image in new window, we get a contradiction with condition (S3), which by assumption is satisfied for the hypersurface

*X*. \(\blacksquare \)

### 4.3 The case of codimension 2

*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 4.4

- (a)
Open image in new window is a hyperplane section of the quadric

*E*, on which*S*is cut out by a hypersurface of degree \(d_S\geqslant 2\), - (b)
Open image in new window is the section of the quadric

*E*by a linear subspace of codimension 2.

### Proof

The proof repeats the proof of [9, Lemma 4.1], and we do not give it here. (The key point in the arguments is that due to the inequality \(\alpha <2\) every line \(L=[p,q]\subset E\), joining some point \(p,q\in S\) and lying on *E*, is contained in \(D^+_X\), because \(\mathrm{mult}_SD^+_X>n\).) \(\blacksquare \)

### Proposition 4.5

Option (b) does not take place.

### Proof

Assume the converse: case (b) takes place. Let \(P\subset X\) be the section of the hypersurface *X* by the linear subspace of codimension 2 in \({{\mathbb {P}}}^N\), that is uniquely determined by the conditions \(P\ni o\) and \(P^+\cap E=S\).

*X*, containing

*P*. For a general divisor \(\Delta \in |H-P|\) we have the equality

*G*is an effective divisor on \(\Delta \), not containing

*P*as a component. Obviously, \(G\in |mH_{\Delta }|\), where \(m=n-a\) and \(H_{\Delta }\) is the class of a hyperplane section of \(\Delta \subset {{\mathbb {P}}}^{N-1}\). The symbols \(G^+\) and \(\Delta ^+\) stand for the strict transforms of

*G*and \(\Delta \) on \(X^+\), respectively. Now,

*E*and, besides,

*X*. \(\blacksquare \)

### 4.4 The hyperplane section \(\Delta \)

*E*, where \(\Lambda \subset D^+_X\). Set

### Lemma 4.6

### 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. \(\blacksquare \)

*S*is cut out on the quadric \(\Lambda \) by a hypersurface of degree \(d_S\geqslant 2\), we obtain the inequalitySince Open image in new window, we getConsider the blow-up \(\sigma _S:{\widetilde{\Delta }}\rightarrow \Delta ^+\) of the subvariety \(S\subset \Delta ^+\) of codimension 2 and denote its exceptional divisor \(\sigma _S^{-1}(S)\) by the symbol \(E_S\).

### Proposition 4.7

### Proof

This is a well-known fact, see [6, Proposition 9]. (Note that the subvariety *S* is, generally speaking, singular, however \(\Delta ^+\) is non-singular at the general point of *S* and \({\widetilde{\Delta }}\) is non-singular at the general point of \(S_1\).) \(\blacksquare \)

### 4.5 End of the proof

the case of general position \(S_1\ne E_S\cap {\widetilde{\Lambda }}\), so that \(S_1\not \subset {\widetilde{\Lambda }}\),

the special case \(S_1=E_S\cap {\widetilde{\Lambda }}\).

The proof of Proposition 4.2 is now complete. \(\square \)

### Proof of Proposition 2.2

Let us check that the operation of reduction, described in Sect. 4.1, can be \(\rho \) times applied to the hypersurface \(\Gamma \subset {{\mathbb {P}}}^M\). Consider the hypersurface \(\Gamma _i\subset {{\mathbb {P}}}^{M-i}\), where \(i\in \{0,\dots ,\rho -1\}\). Let us show, in the first place, that \(\Gamma _i\) satisfies condition (S1). Let \(p\in \Gamma _i\) be an arbitrary singularity. If \(i=0\), then by condition (R2.1), the point *p* is a quadratic singularity of rank \(\geqslant 8\). If \(i\geqslant 1\), then there are two options: either \(p\in \Gamma \) is a non-singular point, or \(p\in \Gamma \) is a singularity (recall that \(\Gamma _i\) is a section of the hypersurface \(\Gamma \) by a linear subspace of codimension *i* in \({{\mathbb {P}}}^M\)). In the second case by condition (R2.1) the point *p* is a quadratic singularity of \(\Gamma \) of rank \(\geqslant 2\rho +6\geqslant 2i+8\), since \(\rho \geqslant i+1\). Since a hyperplane section of a quadric of rank \(r\geqslant 3\) is a quadric of rank \(\geqslant r-2\), we conclude that \(p\in \Gamma _i\) is a quadratic singularity of rank \(\geqslant 8\), so that condition (S1) is satisfied at that point (for the hypersurface \(\Gamma _i\)).

*p*is non-singular on \(\Gamma \), so that \(\Gamma _i\) is a section of \(\Gamma \) by a linear subspace of codimension

*i*, which is contained in the tangent hyperplane Open image in new window. By condition (R1.4) the point \(p\in \Gamma _i\) is a quadratic singularity of rank(one should take into account that the cutting subspace is of codimension \(i-1\) in Open image in new window). Therefore, condition (S1) is satisfied in any case.

*Y*on the section of the hypersurface \(\Gamma _i\) by a linear subspace \(P^*\) of codimension 2 in \({{\mathbb {P}}}^{M-i}\). Assume the converse:In some affine coordinates with the origin at the point

*o*on the subspace Open image in new window the equation of the hypersurface \(P^*\cap \Gamma _i\) has the form

*d*,

*l*and \(\rho \) under consideration is higher than 1, which gives a contradiction with assumption (9) and proves that the hypersurface \(\Gamma _i\) satisfies condition (S3).

Note that all singular points of \(\Gamma _i\) are quadratic singularities of rank \(\geqslant 8\), 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.\(\square \)

## Notes

### Acknowledgements

The author is 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.

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