1.1 A plan of the proof of Theorem 0.1

In order to prove Theorem 0.1, one has to give an explicit definition of the open set ${\cal F}\subset {\cal P}$ . This definition consists of two groups of conditions, which should be satisfied by the polynomials $f_1,\dots , f_k$ at every point $o\in {\mathbb P}^{M+k}$ at which they all vanish. The first group of conditions is about the singularities of the complete intersection $F(\underline {f})$ : they can be quadratic or multi-quadratic of a rank bounded from below. The corresponding definitions and facts are given in Subsection 1.2. Assuming that the conditions of the first group are satisfied, we get that the scheme of common zeros of the polynomials $f_1,\dots , f_k$ is an irreducible reduced factorial variety $F=F(\underline {f})\subset {\mathbb P}^{M+k}$ with terminal singularities, and so $\mathop {\mathrm {Pic}} F={\mathbb Z} H_F$ and $K_F=-H_F$ , so that the question, Is it divisorially canonical?, makes sense.

Assuming that F is not divisorially canonical, let us fix an effective divisor $D_F\sim n(D_F)H_F$ , where $n(D_F)\geqslant 1$ , such that the pair

$$ \begin{align*}\left(F,\frac{1}{n(D_F)}D_F\right) \end{align*} $$

is not canonical; that is, there is an exceptional divisor E over F, satisfying the Noether-Fano inequality:

$$ \begin{align*}\mathop{\mathrm{ord}}\nolimits_E D_F>n(D_F)\, a(E). \end{align*} $$

We have to show that the existence of such a divisor leads to a contradiction. Let $B\subset F$ be the centre of the exceptional divisor E on F. The information about the singularities of the varieties F makes it possible to easily exclude the option when $\mathop {\mathrm {codim}}(B\subset F)=2$ . This is done in Subsection 1.3.

After that, in Subsection 1.4, we produce the second group of local conditions for the tuple of polynomials $\underline {f}\in {\cal F}$ : now they are the regularity conditions. Assuming that they are satisfied at every point $o\in F$ , we exclude the option $B\not \subset \mathop {\mathrm {Sing}}F$ in Subsection 1.5, and in Subsection 1.6, the option that the point $o\in B$ of general position is a quadratic singularity of F. In Subsection 1.7 we describe the procedure of excluding the multi-quadratic case, when the point $o\in B$ of general position is a multi-quadratic singularity of the type $2^l$ , $l\in \{2,\dots ,k\}$ . This is the hardest part of the work, which is completed in the subsequent sections.

1.2 Multi-quadratic singularities

Let $o\in {\mathbb P}^{M+k}$ be a point at which $f_1,\dots ,f_k$ all vanish. Let us consider a system of affine coordinates $z_*=(z_1,\dots ,z_{M+k})$ with the origin at the point o on an affine chart ${\mathbb A}^{M+k}\subset {\mathbb P}^{M+k}$ , containing that point. Write down

$$ \begin{align*}\begin{array}{ccccc} f_1=f_{1,1}+f_{1,2}+ & \dots & + f_{1,d_1}, & &\\ f_2=f_{2,1}+f_{2,2}+ & \dots & & + f_{2,d_2}, & \\ & \dots & & &\\ f_k=f_{k,1}+f_{k,2}+ & \dots & & & +f_{k,d_k}, \end{array} \end{align*} $$

where we use the same symbols $f_i$ for the non-homogeneous polynomials in $z_*$ , corresponding to the original polynomials $f_i$ , and $f_{i,a}$ is a homogeneous polynomial of degree a in $z_*$ . Obviously, if the linear forms $f_{1,1},\dots ,f_{k,1}$ are linearly independent, then in a neighborhood of the point o, the scheme of common zeros of the polynomials $f_1,\dots ,f_k$ is a non-singular complete intersection of codimension k. In order to give the definition of a multi-quadratic singularity, we will need the concept of the rank of a tuple of quadratic forms.

Definition 1.1 [Reference Pukhlikov2].

The rank of the tuple of quadratic forms $q_1,\dots ,q_l$ in N variables is the number

$$ \begin{align*}\mathop{\mathrm{rk}}(q_1,\dots,q_l)=\mathop{\mathrm{min}}\{\mathop{\mathrm{rk}}(\lambda_1 q_1+\dots+\lambda_l q_l)\,|\,(\lambda_1,\dots,\lambda_l)\neq (0,\dots,0)\}. \end{align*} $$

Obviously, $\mathop {\mathrm {rk}}(q_1,\dots ,q_l)\leqslant N$ . For that reason, in the sequel, the inequality $\mathop {\mathrm {rk}}(q_*)\geqslant a$ means implicitly that the forms $q_i$ depend on a sufficient $(\geqslant a)$ number of variables.

Take $l\in \{1,2,\dots ,k\}$ .

Now the first condition, defining the subset ${\cal F}\subset {\cal P}$ , is stated in the following way.

(MQ1) For every point $o\in {\mathbb P}^{M+k}$ , such that

$$ \begin{align*}f_1(o)=\dots=f_k(o)=0, \end{align*} $$

either the linear forms $f_{1,1},\dots ,f_{k,1}$ are linearly independent or $\underline {f}$ has at the point o a multi-quadratic singularity of type $2^l$ , where $l\in \{1,2,\dots ,k\}$ , of rank

$$ \begin{align*}\geqslant 2l+4k+2\varepsilon(k)-1. \end{align*} $$

Theorem 1.1. Assume that $\underline {f}$ satisfies the condition (MQ1). Then the scheme of common zeros of the polynomials $f_1,\dots ,f_k$ is an irreducible reduced factorial variety F=F(f) – a complete intersection of codimension k with terminal singularities, and, moreover,

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}F\subset F)\geqslant 4k+2\varepsilon(k). \end{align*} $$

Proof is given in §4 (Subsections 4.1-4.3).

Assume that $\underline {f}$ satisfies the condition (MQ1). For a point $o\in F=F(\underline {f})$ , the symbol $T_oF$ stands for the subspace $\{f_{1,1}=\dots =f_{k,1}=0\}\subset {\mathbb C}^{M+k}$ . For the proof of Theorem 0.1, we will need one more property of the tuple $\underline {f}$ , which we include in the definition of the subset ${\cal F}$ .

(MQ2) For any point $o\in F$ , which is a multi-quadratic of type $2^l$ , where $l\geqslant 2$ , the rank of the tuple of quadratic forms

$$ \begin{align*}f_{1,2}|_{T_oF},\dots,f_{k,2}|_{T_oF} \end{align*} $$

is at least $10k^2+8k+2\varepsilon (k)+5$ .

The condition (MQ2) for multi-quadratic points of type $2^l$ with $l\geqslant 2$ implies the condition (MQ1) because the rank of a quadratic form, restricted to a hyperplane, drops at most by 2; however, for the convenience of references, we state the conditions (MQ1) and (MQ2) independently of each other. These conditions are used in the proof of Theorem 0.1 in different ways.

So every tuple $\underline {f}\in {\cal F}$ satisfies (MQ1) and (MQ2).

1.3 Subvarieties of codimension 2

Following the plan, given in Subsection 1.1, let us fix an effective divisor $D_F\sim n(D_F)H_F$ , $n(D_F)\geqslant 1$ , such that the pair $(F,\frac {1}{n(D_F)}D_F)$ is not canonical. By the symbol

$$ \begin{align*} \mathop{\mathrm{CS}}\left(F,\frac{1}{n(D_F)}D_F\right), \end{align*} $$

we denote the union of the centres on F of all exceptional divisors over F, satisfying the Noether-Fano inequality (that is to say, of all non-canonical singularities of that pair). This is a closed subset of F. Let B be an irreducible component of maximal dimension of that set.

Proof. Assume the converse: $\mathop {\mathrm {codim}}(B\subset F)=2$ . Then $B\not \subset \mathop {\mathrm {Sing}}F$ . Moreover, let $P\subset {\mathbb P}^{M+k}$ be a general linear subspace of dimension $2k+2$ . Theorem 1.1 implies that $P\cap \mathop {\mathrm {Sing}}F=\emptyset $ , so that $F\cap P$ is a non-singular complete intersection of type $\underline {d}$ in $P\cong {\mathbb P}^{2k+2}$ . Furthermore, the pair

$$ \begin{align*}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) \end{align*} $$

is not canonical, and the irreducible subvariety $B\cap P$ is an irreducible component of maximal dimension of the set

$$ \begin{align*}\mathop{\mathrm{CS}}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right), \end{align*} $$

so that (as $F\cap P$ is non-singular)

$$ \begin{align*} \mathop{\mathrm{mult}}\nolimits_{B\cap P}D_F|_{F\cap P}>n(D_F). \end{align*} $$

However, $D_F|_{F\cap P}\sim n(D_F)H_{F\cap P}$ (where $H_{F\cap P}$ is the class of a hyperplane section of $F\cap P$ ), so that by [Reference Pukhlikov17, Proposition 3.6] or [Reference Suzuki18], we get a contradiction, proving the proposition. Q.E.D.

1.4 Regularity conditions

In order to continue the proof of Theorem 0.1, we need a second group of conditions defining the set ${\cal F}$ . Let $o\in F$ be a point. We use the notations of Subsection 1.2. By the symbol $T_oF$ we denote the linear tangent space

$$ \begin{align*}\{f_{1,1}=\dots=f_{k,1}=0\}\subset{\mathbb C}^{M+k}, \end{align*} $$

and by the symbol ${\mathbb P}(T_oF)$ its projectivization. Let ${\cal S}=(h_1,\dots ,h_M)$ be the sequence of homogeneous polynomials

$$ \begin{align*}f_{i,j}|_{{\mathbb P}(T_oF)}, \end{align*} $$

where $j\geqslant 2$ , placed in the lexicographic order: $(i_1,j_1)$ precedes $(i_2,j_2)$ , if $j_1<j_2$ or $j_1=j_2$ , but $i_1<i_2$ . By the symbol ${\cal S}[-m]$ denote the sequence ${\cal S}$ with the last m members removed. Finally, the symbol ${\cal S}[-m]|_{\Pi }$ stands for the restriction of that sequence (that is, the restriction of each its member) onto a linear subspace $\Pi \subset {\mathbb P}(T_oF)$ . The regularity conditions depend on the type of the singularity $o\in F$ .

First, let the point $o\in F$ be non-singular, so that ${\mathbb P}(T_oF)\cong {\mathbb P}^{M-1}$ . In that case, the regularity condition is stated in the following way.

(R1) The sequence

$$ \begin{align*}{\cal S}[-(k+\varepsilon(k)+3)]|_{\Pi} \end{align*} $$

is regular for every subspace $\Pi \subset {\mathbb P}(T_oF)$ of codimension $k+\varepsilon (k)-1$ .

The condition (R1) is assumed for every non-singular point $o\in F$ . It implies the following key fact.

Theorem 1.2. Let $P\subset {\mathbb P}^{M+k}$ be an arbitrary linear subspace of codimension $\leqslant k+ \varepsilon (k)-1$ . Then for every non-singular point $o\in F\cap P$ and every prime divisor $Y\sim n(Y)H_{F\cap P}$ on $F\cap P$ , the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oY\leqslant2n(Y) \end{align*} $$

holds.

Proof is given in §7 (Subsections 7.1, 7.2).

Now let $o\in F$ be a quadratic singularity (this case corresponds to the value $l=1$ in Definition 1.2). Here ${\mathbb P}(T_oF)\cong {\mathbb P}^M$ . In this case, the regularity condition is stated as follows.

(R2) The sequence

$$ \begin{align*}{\cal S}[-4]|_{\Pi} \end{align*} $$

is regular for every hyperplane $\Pi \subset {\mathbb P}(T_oF)$ .

The condition (R2) is assumed for every quadratic singular point $o\in F$ and implies the following key fact.

Theorem 1.3. Let $o\in F$ be a quadratic singularity and $W\ni o$ the section of F by a hyperplane that is not tangent to F at the point o, and $Y\sim n(Y)H_W$ a prime divisor on W. Then the following inequality holds:

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oY\leqslant 4n(Y). \end{align*} $$

Proof is given in §7 (Subsection 7.3).

(The symbol $H_W$ stands for the class of a hyperplane section of the variety W; the linear form, defining the hyperplane that cuts out W, is not a linear combination of the forms $f_{1,1},\dots ,f_{k,1}$ .)

Now let $o\in F$ be a multi-quadratic point of type $2^l$ , where $l\in \{2,\dots ,k\}$ . Here we will need two regularity conditions. In the first of them, the symbol $T_oF$ means the projective closure of the linear subspace

$$ \begin{align*}\{f_{1,1}=\dots=f_{k,1}=0\}\subset{\mathbb C}^{M+k} \end{align*} $$

in ${\mathbb P}^{M+k}$ .

(R3.1) For every subspace $P\subset T_oF$ of codimension $\varepsilon (k)$ , containing the point o, the scheme of common zeros of the polynomials

$$ \begin{align*}f_1|_P,\dots,f_k|_P,\quad f_{i,2}|_P\quad\mbox{for all}\quad i: d_i\geqslant 3 \end{align*} $$

is an irreducible reduced subvariety of codimension $k+k_{\geqslant 3}$ in P, where

$$ \begin{align*}k_{\geqslant 3}=\sharp\{i=1,\dots,k\,|\, d_i\geqslant 3\}. \end{align*} $$

Note that in the condition (R3.1), the homogeneous polynomials $f_{i,2}$ in the affine coordinates $z_*$ are considered as quadratic forms in homogeneous coordinates on ${\mathbb P}^{M+k}$ .

In the second regularity condition for multi-quadratic points, the symbol $T_oF$ means a linear subspace in ${\mathbb C}^{M+k}$ .

(R3.2) For every linear subspace $\Pi \subset {\mathbb P}(T_oF)$ of codimension $\varepsilon (k)$ , the sequence

$$ \begin{align*}{\cal S}[-m^*]|_{\Pi} \end{align*} $$

is regular, where $m^*=\mathop {\mathrm {max}}\{\varepsilon (k)+4-l,0\}$ .

The conditions (R3.1) and (R3.2) are assumed for every multi-quadratic singular point $o\in F$ . They imply the following key inequality. In Theorem 1.4, stated below, the symbol $T_oF$ stands for the projective closure of the embedded tangent space – that is, a linear subspace in ${\mathbb P}^{M+k}$ , containing the point o.

Theorem 1.4. Let $P\subset T_oF$ be an arbitrary linear subspace of codimension $\leqslant \varepsilon (k)$ and $Y\ni o$ a prime divisor on $F\cap P$ , $Y\sim n(Y)H_{F\cap P}$ . Then the following inequality holds:

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_o Y\leqslant\frac32\cdot 2^kn(Y). \end{align*} $$

Proof is given in §7 (Subsections 7.3).

(The symbol $H_{F\cap P}$ stands for the class of a hyperplane section of the variety $F\cap P$ ; we will show below – see §4 – that $F\cap P$ is an irreducible factorial complete intersection.)

Summing up, let us give a complete definition of the subset ${\cal F}\subset {\cal P}$ : it consists of the tuples $\underline {f}$ , satisfying the conditions (MQ1,2), the condition (R1) at every non-singular point $o\in F(\underline {f})$ , the condition (R2) at every quadratic point $o\in F(\underline {f})$ and the conditions (R3.1,2) at every multi-quadratic point $o\in F(\underline {f})$ .

The inequality for the codimension of the complement ${\cal P}\setminus {\cal F}$ , given in Theorem 0.1, is shown in §8.

1.5 Exclusion of the non-singular case

We carry on with the proof of divisorial canonicity of the variety $F\in {\cal F}$ . In the notations of Subsection 1.3, assume that the point of general position $o\in B$ is a non-singular point of F. We know (Proposition 1.1) that $\mathop {\mathrm {codim}}(B\subset F)\geqslant 3$ . Consider a general subspace $P\ni o$ of dimension $k+3$ . Then $F\cap P$ is a non-singular three-dimensional variety, and the point o is a connected component of the set

$$ \begin{align*}\mathop{\mathrm{CS}}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) \end{align*} $$

(if $\mathop {\mathrm {codim}}(B\subset F)\geqslant 4$ , then $\mathop {\mathrm {CS}}$ can be replaced, by inversion of adjunction, by $\mathop {\mathrm {LCS}}$ ); that is, outside the point o in a neighborhood of that point, the pair

(6) $$ \begin{align} \left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) \end{align} $$

is canonical. It is well known (see [Reference Pukhlikov19, Proposition 3] or [Reference Pukhlikov20, Chapter 7, Proposition 2.3]), and it follows from here that either the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oD_F>2n(D_F) \end{align*} $$

holds or on the exceptional divisor $E\cong {\mathbb P}^{M-1}$ of the blow-up $F^+\to F$ of the point o, there is a hyperplane $\Theta \subset E$ (uniquely determined by the pair (6)), such that the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oD_F+\mathop{\mathrm{mult}}\nolimits_{\Theta}D^+_F> 2n(D_F) \end{align*} $$

holds, where $D^+_F$ is the strict transform of $D_F$ on $F^+$ .

The first option is impossible as it contradicts Theorem 1.2. In the second case, denote by the symbol $|H-\Theta |$ the projectively k-dimensional linear system of hyperplane sections of F, a general element of which $W\ni o$ is non-singular at the point o and satisfies the equality

$$ \begin{align*}W^+\cap E=\Theta. \end{align*} $$

The restriction $D_W=(D_F\circ W)$ is an effective divisor on W, and $n(D_W)=n(D_F)$ and the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oD_W\geqslant\mathop{\mathrm{mult}}\nolimits_oD_F+\mathop{\mathrm{mult}}\nolimits_{\Theta}D^+_F>2n(D_W) \end{align*} $$

holds, which again contradicts Theorem 1.2. We have shown the following fact.

1.6 Exclusion of the quadratic case

Again, let $o\in B$ be a point of general position.

Proof. Assume the converse: the point o is a quadratic singularity of F. Let $P\ni o$ be a general $(k+3)$ -dimensional linear subspace in ${\mathbb P}^{M+k}$ . By the condition (MQ1) and Theorem 1.1, the intersection $F\cap P$ is a three-dimensional variety with the unique singular point o, which is a non-degenerate quadratic singularity. This intersection can be constructed in two steps: first, we consider the intersection $F\cap P'$ with a general linear subspace $P'\subset {\mathbb P}^{M+k}$ , $P'\ni o$ of dimension

$$ \begin{align*}k+\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}F\subset F), \end{align*} $$

and after that the intersection with a general subspace $P\subset P'$ , $P\ni o$ of dimension $(k+3)$ . Now we get the following: the pair

$$ \begin{align*}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right) \end{align*} $$

is not log-canonical, but canonical outside the point o. Let us consider the blow-up

$$ \begin{align*}\varphi_P\colon P^+\to P \end{align*} $$

of the point o with the exceptional divisor ${\mathbb E}_P\cong {\mathbb P}^{k+2}$ and let $(F\cap P)^+\subset P^+$ be the strict transform of $F\cap P$ on $P^+$ , so that $(F\cap P)^+\to F\cap P$ is the blow-up of the quadratic singularity o with the exceptional divisor $E_P=(F\cap P)^+\cap {\mathbb E}_P$ , which is a non-singular two-dimensional quadric in the three-dimensional subspace $\langle E_P\rangle \subset {\mathbb E}_P$ . Obviously, $a(E_P, F\cap P)=1$ , so that, writing down

$$ \begin{align*}D_P=D_F|_{F\cap P}\sim n(D_F)H_{F\cap P} \end{align*} $$

and $D_P^+\sim n(D_F)H_{F\cap P}-\nu E_P$ (the strict transform of $D_P$ on $(F\cap P)^+$ ), we obtain two options:

• • either $\nu>2n(D_F)$ , so that $E_P$ is a non-log-canonical singularity of the pair $(F\cap P,\frac {1}{n(D_F)} D_P)$ ,

• • or $n(D_F)<\nu \leqslant 2n(D_F)$ , and then the closed set

  $$ \begin{align*}\mathop{\mathrm{LCS}}\left(\left(F\cap P,\frac{1}{n(D_F)} D_P\right), (F\cap P)^+\right) \end{align*} $$
  – the union of the centres of all non-log-canonical singularities of the original pair $(F\cap P,\frac {1}{n(D_F)} D_P)$ on $(F\cap P)^+$ – is a connected closed subset of the non-singular quadric $E_P$ , which can be either a (possibly reducible) connected curve $C_P\subset E_P$ or a point $x_P\in E_P$ .

(It is well known – see, for instance, [Reference Pukhlikov20, Chapter 2, Proposition 3.7] – that the inequality $\nu \leqslant n(D_F$ ) is impossible.) In the case $\nu>2n(D_F)$ , we get

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_o D_P=\mathop{\mathrm{mult}}\nolimits_o D_F> 4n(D_F), \end{align*} $$

which contradicts Theorem 1.3, so that this case is impossible. Coming back to the original variety F, let us consider the blow-ups $\varphi _{\mathbb P}\colon ({\mathbb P}^{M+k})^+\to {\mathbb P}^{M+k}$ and $\varphi \colon F^+\to F$ of the point o, where $F^+$ is identified with the strict transform of F on $({\mathbb P}^{M+k})^+$ , with the exceptional divisors ${\mathbb E}$ and E, respectively, so that $E=F^+\cap {\mathbb E}$ is a quadratic hypersurface E in the subspace $\langle E\rangle \subset {\mathbb E}$ of codimension $(k+1)$ . By the condition for the rank (MQ1), the case of a point $x_P\in E_P$ is impossible: in that case, the quadric E would contain a linear subspace of codimension 2 (with respect to E), which cannot happen. Now, arguing word for word as in [Reference Pukhlikov2, Subsection 3.2] and using [Reference Pukhlikov2, Theorem 3.1], we get that on the quadric E, there is a hyperplane section $\Lambda \subset E$ , such that

$$ \begin{align*}\nu+\mathop{\mathrm{mult}}\nolimits_{\Lambda}D^+_F>2n(D_F). \end{align*} $$

Taking the linear system $|H_F-\Lambda |$ (of the projective dimension $(k-1)$ ) of hyperplane sections of the variety F, a general divisor $W\in |H_F-\Lambda |$ in which contains the point o and its strict transform $W^+$ cuts out $\Lambda $ on E (that is, $W^+\cap E=\Lambda $ ), we set $D_W=(D_F\circ W)$ and obtain the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_oD_W=2(\nu+\mathop{\mathrm{mult}}\nolimits_{\Lambda}D^+_F)>4n(D_F)=4n(D_W), \end{align*} $$

which contradicts Theorem 1.3. This completes the proof of Proposition 1.3.

1.7 Exclusion of the multi-quadratic case

This is the hardest and the longest part of our work. Fix a point $o\in B$ of general position, which by what was proven is a multi-quadratic singularity of type $2^l$ , satisfying the conditions (MQ1,2). The pair $(F,\frac {1}{n(D_F)}D_F)$ has a non-canonical singularity, the centre B of which is a component of the maximal dimension of the set $\mathop {\mathrm {CS}}(F,\frac {1}{n(D_F)}D_F)$ , so that in a neighborhood of the point o, this pair is canonical outside B. We will show that this is impossible. This will be done in a few steps, and now we describe the scheme of the proof and state the key intermediate claims.

Definition 1.3. A pair $[X,o]$ , where

$$ \begin{align*}X\subset{\mathbb P}(X)={\mathbb P}^{N(X)} \end{align*} $$

is an irreducible reduced factorial complete intersection of type $\underline {d}$ in the projective space ${\mathbb P}(X)$ , $\mathop {\mathrm {dim}}X=N(X)-k\geqslant 3$ , and $o\in X$ is a point, is called a complete intersection with a marked point or, for brevity, a marked complete intersection of level $(l_X,c_X)$ ), where $l_X$ , $c_X$ are positive integers, satisfying the inequalities

$$ \begin{align*}2\leqslant l_X\leqslant k\quad\mbox{and} \quad c_X\geqslant l_X+4, \end{align*} $$

if the following conditions are satisfied:

(MC1) the inequality

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}X\subset X)\geqslant c_X \end{align*} $$

holds,

(MC2) the point $o\in X$ is a multi-quadratic singularity of type $2^{l_X}$ , the rank of which satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in X)\geqslant 2l_X+c_X-1, \end{align*} $$

(MC3) the non-singular part $X\setminus \mathop {\mathrm {Sing}}X$ of the variety X satisfies the condition of divisorial canonicity,

$$ \begin{align*}\mathop{\mathrm{ct}}(X\setminus\mathop{\mathrm{Sing}}X)\geqslant 1; \end{align*} $$

that is, for every effective divisor $A\sim aH_X$ , we have $\mathop {\mathrm {CS}}(X,\frac {1}{a}A)\subset \mathop {\mathrm {Sing}}X$ .

The non-singular set of integers

$$ \begin{align*}I_X=[k+l_X+3,k+c_X-1]\cap{\mathbb Z} \end{align*} $$

is called the admissible set of the marked complete intersection $[X,o]$ .

Proposition 1.4. The following inequality holds:

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}E_X\subset E_X)\geqslant c_X. \end{align*} $$

Remark 1.2. Proposition 1.4 implies the estimate

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}E_X\subset{\mathbb E}_X)\geqslant k+c_X. \end{align*} $$

Therefore, for every $m\leqslant k+c_X$ and a general subspace $P\ni o$ of dimension m in ${\mathbb P}(X)$ , the strict transform $P^+\subset {\mathbb P}(X)^+$ does not meet the set $\mathop {\mathrm {Sing}}E_X$ , since $P^+\cap {\mathbb E}_X$ is a general linear subspace of dimension $m-1\leqslant k+c_X-1$ in ${\mathbb E}_X$ . Therefore, for $m=k+c_X$ an isolated, and for $m\leqslant k+c_X-1$ the unique singularity o of the variety $X\cap P$ is resolved by the blow-up of that point, and moreover, the exceptional divisor

$$ \begin{align*}E_{X\cap P}=P^+\cap E_X \end{align*} $$

of that blow-up is a non-singular complete intersection of $l_X$ quadrics in the linear subspace of codimension $(k-l_X)$ in ${\mathbb E}_{X\cap P}=P^+\cap {\mathbb E}_X$ . The discrepancy of that exceptional divisor is

$$ \begin{align*}a(E_{X\cap P})=a(E_{X\cap P}, X\cap P)=m-1-k-l_X, \end{align*} $$

so that for $m=k+l_X+3$ , we have $a(E_{X\cap P})=2$ . The meaning of the lower end of the admissible set is in that equality.

In the following definition, we use the notations of Remarks 1.1 and 1.2. We continue to consider a marked complete intersection $[X,o]$ of level $(l_X,c_X)$ .

Definition 1.4. A triple $(X,D,o)$ , where $D\sim n(D)H_X$ is an effective divisor on X, $n(D)\geqslant 1$ , is called a working triple, if for a general subspace $P\ni o$ of dimension $k+c_X-1$ in ${\mathbb P}(X)$ , the pair

(7) $$ \begin{align} \left(X\cap P,\frac{1}{n(D)}D|_{X\cap P}\right) \end{align} $$

is not log-canonical at the point o.

Remark 1.3. Since the point o is the unique singularity of the variety $X\cap P$ , and by (MC3) the pair (7) is canonical outside the point o, there is a non-log-canonical singularity of that pair, the centre of which on $X\cap P$ is precisely the point o. By inversion of adjunction, the same is true for a general subspace $P\ni o$ of dimension $m\leqslant k+c_X-2$ .

Let us introduce one more notation. For the strict transform $D^+$ of the divisor D on $X^+$ , write

$$ \begin{align*}D^+\sim n(D)H_X-\nu(D)E_X \end{align*} $$

(in order to simplify the notations, the pullback of the divisorial class $H_X$ on $X^+$ is denoted by the same symbol $H_X$ ). Respectively, for a general subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $m\leqslant k+c_X-1$ , we have

$$ \begin{align*}D_P=D|_{X\cap P}\sim n(D)H_{X\cap P} \end{align*} $$

and

$$ \begin{align*}D^+_P\sim n(D)H_{X\cap P}-\nu(D)E_{X\cap P}, \end{align*} $$

where $H_{X\cap P}=H_X|_{X\cap P}$ is the class of a hyperplane section of the variety $X\cap P\subset P\cong {\mathbb P}^m$ .

Proof is given in §3 (Subsection 3.2).

Let us come back to the task of excluding the multi-quadratic case. Recall that $F\in {\cal F}$ , so that we can use the conditions (MQ1,2) and the statement of Theorem 1.4. We fix a point of general position $o\in B$ , where B is an irreducible component of the maximal dimension of the closed set $\mathop {\mathrm {CS}}(F,\frac {1}{n(D_F)}D_F)$ .

Proof is given in §3 (Subsection 3.1).

Assume now that $l\leqslant k-1$ . The symbol $T_oF$ stands again for a subspace of codimension $(k-l)$ of the projective space ${\mathbb P}^{M+k}$ . Set

$$ \begin{align*}T=F\cap T_oF. \end{align*} $$

This is subvariety of codimension $(k-l)$ in F and a complete intersection of type $\underline {d}$ in ${\mathbb P}(T)=T_oF$ .

If $l=k$ , then for uniformity of notations, we set $T=F$ .

Proof is given in §3 (Subsection 3.4).

Proposition 1.5 (taking into account Remark 1.4) implies that $\nu (D_T)>n(D_T)$ . Now the main stage in the exclusion of the multi-quadratic case (and thus in the proof of Theorem 0.1) is given by the following claim.

Proposition 1.8. There is a sequence of marked complete intersections

$$ \begin{align*}[R_0=T,o],\quad [R_1,o],\quad\dots,\quad [R_a,o], \end{align*} $$

where $a\leqslant \varepsilon (k)$ and ${\mathbb P}(R_{i+1})$ is a hyperplane in ${\mathbb P}(R_{i})$ , containing the point o, and of effective divisors $D_i\sim n(D_i)H_{R_i}$ on $R_i$ , $n(D_i)\geqslant 1$ , such that $D_0=D_T$ and

$$ \begin{align*}(R_0,D_0,o),\quad (R_1,D_1,o),\quad\dots,\quad (R_a,D_a,o) \end{align*} $$

are working triples, and moreover, for every $i=0,\dots ,a-1$ , the inequality

$$ \begin{align*}2-\frac{\nu(D_{i+1})}{n(D_{i+1})}<\frac{1}{1+\frac{1}{k}}\left( 2-\frac{\nu(D_i)}{n(D_i)}\right) \end{align*} $$

holds and $\nu (D_a)>\frac 32 n(D_a)$ .

Proof is given in §3 (Subsection 3.5) and §5.

Now let us complete the exclusion of the multi-quadratic case. The variety $R_a$ is a section of $T=F\cap T_oF$ by a subspace of codimension $\leqslant \varepsilon (k)$ , containing the point o, and $D_a$ is an effective divisor on $R_a$ , satisfying the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_o D_a=2^k\nu(D_a)>\frac32\cdot 2^kn(D_a). \end{align*} $$

This contradicts Theorem 1.4.

The contradiction completes the proof of divisorial canonicity of the variety $F\in {\cal F}$ .

2.1 The mobile linear system $\Sigma $

Assume that the Fano-Mori fibre space $\pi \colon V\to S$ satisfies all conditions of Theorem 0.2. Fix a fibre space $\pi '\colon V'\to S'$ that belongs to one of the two classes: either the class of rationally connected fibre spaces (and then we say that the rationally connected case is being considered) or the class of Mori fibre spaces in the sense of Subsection 0.2 (and then we say that the case of a Mori fibre space is being considered). We will study both cases simultaneously.

In the rationally connected case, let $Y'\in \mathop {\mathrm {Pic}}S'$ be a very ample class. Set

$$ \begin{align*}\Sigma'=|(\pi')^*Y'|=|-mK_V'+(\pi')^*Y'|, \end{align*} $$

where $m=0$ . This is a mobile complete linear system on $V'$ (it defines the morphism $\pi '$ ).

In the case of a Mori fibre space, let

$$ \begin{align*}\Sigma'=|-m K_V'+(\pi')^*Y'| \end{align*} $$

be a complete linear system on $V'$ , where $m\geqslant 0$ and $Y'$ is a very ample divisorial class on $S'$ , and moreover, for $m\geqslant 1$ , the system $\Sigma '$ is very ample.

In both cases, set

$$ \begin{align*}\Sigma=(\chi^{-1})_*\Sigma'\subset|-n K_V+{\pi}^*Y| \end{align*} $$

to be the strict transform of $\Sigma '$ on V with respect to the birational map $\chi \colon V\dashrightarrow V'$ . Note that if $m=0$ and $n=0$ , then by construction of these linear systems, the map $\chi $ is fibre-wise.

2.2 A fibre-wise birational modification of the fibre space $V/S$

Let $\sigma _S\colon S^+\to S$ be a composition of blow-ups with non-singular centres,

$$ \begin{align*}S^+=S_N\stackrel{\sigma_{S,N}}{\to}S_{N-1} \to\dots\stackrel{\sigma_{S,1}}{\to}S_0=S, \end{align*} $$

where $\sigma _{S,i+1}\colon S_{i+1}\to S_i$ blows up a non-singular subvariety $Z_{S,i}\subset S_i$ . Set $V_i=V\times _SS_i$ and $\pi _i\colon V_i\to S_i$ ; by the assumption on the stability with respect to birational modifications of the base, $V_i/S_i$ is a Fano-Mori fibre space. Obviously,

$$ \begin{align*}V_{i+1}=V_i\times_{S_i}S_{i+1} \end{align*} $$

is the result of the blow-up $\sigma _{i+1}\colon V_{i+1}\to V_i$ of the subvariety $Z_i=\pi ^{-1}_i(Z_{S,i})\subset V_i$ . Therefore, we get the commutative diagram

$$ \begin{align*}\begin{array}{ccccccccccccccc} V^+ & = & V_N & \stackrel{\sigma_N}{\to} & \dots & \to & V_{i+1} & \stackrel{\sigma_{i+1}}{\to} & V_i & \to & \dots & \stackrel{\sigma_1}{\to} & V_0 & = & V \\ & & \downarrow & & \dots & & \downarrow & & \downarrow & & \dots & & \downarrow & & \\ S^+ & = & S_N & \stackrel{\sigma_{S,N}}{\to} & \dots & \to & S_{i+1} & \stackrel{\sigma_{S,i+1}}{\to} & S_i & \to & \dots & \stackrel{\sigma_{S,1}}{\to} & S_0 & = & S, \end{array} \end{align*} $$

where the vertical arrows $\pi \colon V_i\to S_i$ are Fano-Mori fibre spaces. The symbol $\Sigma ^i$ stands for the strict transform of the system $\Sigma $ on $V_i$ , $\Sigma ^+=\Sigma ^N$ . In these notations, let us consider a sequence of blow-ups $\sigma _{S,*}$ such that for every $i=0,1,\dots ,N-1$ ,

$$ \begin{align*}Z_i\subset\mathop{\mathrm{Bs}}\Sigma^i, \end{align*} $$

and the base set of the system $\Sigma ^+$ contains entirely no fibre $\pi ^{-1}_+(s_+)$ , where $s_+\in S^+$ and $\pi _+=\pi _N$ . (If this is true already for the original system $\Sigma $ , then we set $\sigma _S=\mathop {\mathrm {id}}_S$ , $S^+=S$ and $V^+=V$ , and there is no need to make any blow-ups; but we will soon see that this case is impossible.)

By the assumptions on the fibre space $V/S$ , the fibre $\pi ^{-1}_+(s_+)$ is isomorphic to the fibre $F_s=\pi ^{-1}(s)$ of the original fibre space, where $s=\sigma _S(s_+)$ . Let ${\cal T}$ be the set of all prime $\sigma _S$ -exceptional divisors on $S^+$ . We get

$$ \begin{align*}\Sigma^+\subset\left|-n\sigma^*K_V+\pi^*_+\left(\sigma^*_SY-\sum_{T\in{\cal T}}b_TT\right)\right|= \end{align*} $$

$$ \begin{align*}=\left|-nK^++\pi^*_+\left(\sigma^*_SY+\sum_{T\in{\cal T}} (na_T-b_T)T\right)\right|, \end{align*} $$

where $\sigma \colon V^+\to V$ is the composition of the morphisms $\sigma _i$ , $K^+=K_{V^+}$ , $b_T\geqslant 1$ and $a_T\geqslant 1$ for all $T\in {\cal T}$ , $a_T=a(T,S)$ is the discrepancy of T with respect to S.

Let $\varphi \colon \widetilde {V}\to V^+$ be the resolution of singularities of the composite map $\chi _+=\chi \circ \sigma \colon V^+\dashrightarrow V'$ , ${\cal E}$ the set of prime $\varphi $ -exceptional divisors on $\widetilde {V}$ and $\psi =\chi \circ \sigma \circ \varphi \colon \widetilde {V}\to V'$ is a birational morphism.

Proof. Assume that this is not the case. Then there is an exceptional divisor $E\in {\cal E}$ , satisfying the Noether-Fano inequality

$$ \begin{align*}\mathop{\mathrm{ord}}\nolimits_ED^+=\mathop{\mathrm{ord}}\nolimits_E\Sigma^+>na(E,V^+) \end{align*} $$

(we write $D^+,\Sigma ^+$ instead of $\varphi ^*D^+,\varphi ^*\Sigma ^+$ for simplicity). Set $B=\varphi (E)\subset V^+$ .

There are two options:

(1) $\pi _+(B)=S^+$ ,

(2) $\pi _+(B)$ is a proper irreducible closed subset $S^+$ .

If (1) is the case, then the fibre $F=F_s$ of general position intersects B. The restriction

$$ \begin{align*}\Sigma^+_F=\Sigma^+|_F\subset|-nK_F| \end{align*} $$

is a mobile linear system, and moreover, the pair $(F,\frac {1}{n}D^+_F)$ is not canonical for $D^+_F=D^+|_F$ . This contradicts the condition $\mathop {\mathrm {mct}}(F)\geqslant 1$ .

Therefore, (2) is the case. Let $p\in B$ be a point of general position and $F=\pi ^{-1}_+(\pi _+(p))$ , so that $p\in F$ . Since $F\not \subset \mathop {\mathrm {Bs}}\Sigma ^+$ , the restriction $D^+_F=D^+|_F$ is well defined (although the linear system $\Sigma ^+_F$ may have fixed components). By inversion of adjunction, the pair $(F,\frac {1}{n}D^+_F)$ is not log-canonical. This contradicts the condition $\mathop {\mathrm {lct}}(F)\geqslant 1$ . Q.E.D. for the proposition.

Denote by the symbol $\widetilde {\Sigma }$ the strict transform of the system $\Sigma ^+$ on $\widetilde {V}$ . Obviously,

(8) $$ \begin{align} \widetilde{\Sigma}=\psi^*\Sigma'=|-m\psi^*K'+\psi^*(\pi')^*Y'|, \end{align} $$

where $K'=K_{V'}$ ; that is, $\widetilde {\Sigma }$ is a complete linear system. We have another presentation for this linear system:

$$ \begin{align*}\widetilde{\Sigma}=\left|\varphi^*D^+-\sum_{E\in{\cal E}}b_EE\right|= \end{align*} $$

(9) $$ \begin{align} =\left|-n\widetilde{K}+\varphi^*\pi^*_+\left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)+\sum_{E\in{\cal E}}(na_E-b_E)E\right|, \end{align} $$

where $\widetilde {K}=K_{\widetilde {V}}$ , $D^+\in \Sigma ^+$ is a general divisor and $a_E=a(E,V^+)$ is the discrepancy.

2.3 The mobile system of curves

Take a family of irreducible curves ${\cal C'}$ on $V'$ , contracted by the projection $\pi '$ , sweeping out a Zariski dense subset of the variety $V'$ and not meeting the set where the birational map $\psi ^{-1}$ is not determined. Assume that for a general pair of points $p,q$ in a fibre of general position of the projection $\pi '$ , there is a curve $C'\in {\cal C'}$ containing the both points. In the rationally connected case, the curves of the family ${\cal C'}$ are rational (the existence of such family is shown in [Reference Kollár24, Chapter II]); in the case of a Mori fibre space, we do not require this. For a curve $C'\in {\cal C'}$ , set $\widetilde {C}=\psi ^{-1}(C')$ (at every point of the curve $C'$ , the map $\psi ^{-1}$ is an isomorphism); thus, we get a family $\widetilde {{\cal C}}$ of irreducible curves on $\widetilde {V}$ . Both in the rationally connected case and the case of a Mori fibre space, the inequality

$$ \begin{align*}(C'\cdot K')<0 \end{align*} $$

holds, so that $(\widetilde {C}\cdot \widetilde {K})=(C'\cdot K')<0$ . Furthermore,.

$$ \begin{align*}(\widetilde{C}\cdot\widetilde{D})=(C'\cdot D')=-m(C'\cdot K')\geqslant 0, \end{align*} $$

and $(\widetilde {C}\cdot \widetilde {D})=0$ if and only if $m=0$ (since obviously $(C'\cdot (\pi ')^*Y')=0$ ).

Let ${\cal C}^+=\varphi _*\widetilde {{\cal C}}$ be the image of the family $\widetilde {{\cal C}}$ on $V^+$ and ${\cal C} =\sigma _*{\cal C}^+$ its image on V.

Proof. Assume the converse: $\pi (C)$ is a point on S. By the construction of the family ${\cal C}'$ this means that the map $\chi ^{-1}$ is fibre-wise: there is a rational dominant map $\beta '\colon S'\dashrightarrow S$ , such that the diagram

$$ \begin{align*}\begin{array}{ccc} V & \stackrel{\phantom{xxx}\chi^{-1}}{\dashleftarrow} & V'\\ \downarrow & & \downarrow \\ S & \stackrel{\phantom{xx}\beta'}{\dashleftarrow} & S' \end{array} \end{align*} $$

is commutative, and moreover, $\dim S'> \dim S$ (otherwise, $\beta '$ is birational and then $\chi $ is fibre-wise, contrary to our assumption). In that case, for a point $s\in S$ of general position, the fibre $F_s=\pi ^{-1}(s)$ is birational to $(\pi ')^{-1}(\beta ')^{-1}(s)$ . Here, $\dim (\beta ')^{-1}(s)\geqslant 1$ and either the fibre $(\pi ')^{-1}(s')$ for a point $s'\in (\beta ')^{-1}(s)$ of general position is rationally connected or the anti-canonical class of the variety $(\pi ')^{-1}(\beta ')^{-1}(s)$ is $\pi '$ -ample, and we get a contradiction with the condition $\mathop {\mathrm {mct}} (F_s)\geqslant 1$ (the fibre $F_s$ is a birationally superrigid Fano variety). Q.E.D. for the proposition.

For a general curve $C\in {\cal C}$ , set

$$ \begin{align*}\pi_*C=d_C\overline{C}, \end{align*} $$

where $d_C\geqslant 1$ . Replacing, if necessary, the family $\cal {C}'$ by some open subfamily, we may assume that the integer $d_C$ does not depend on C. For the corresponding curve $C^+\in \cal {C}^+$ , we have $(\pi _+)_*C^+=d_C\overline {C}^+$ , where $\overline {C}^+$ is the strict transform of the curve $\overline {C}$ on $S^+$ .

2.4 Proof of Theorem 2.1

Recall that we assume that $n>m$ . Using the two presentations (8) and (9) for the class of a divisor $\widetilde {D}\in \widetilde {\Sigma }$ , we get

$$ \begin{align*}d_C\left(\overline{C}^+\cdot\left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)\right)+\sum_{E\in{\cal E}}(na_E-b_E)(\widetilde{C}\cdot E)= (n-m)(\widetilde{C}\cdot\widetilde{K})<0, \end{align*} $$

whence, taking into account the inequalities $b_E\leqslant na_E$ for all $E\in {\cal E}$ (Proposition 2.1), it follows that

$$ \begin{align*}\left(\overline{C}^+\cdot \left(\sigma^*_SY+\sum_{T\in{\cal T}}(na_T-b_T)T\right)\right)<0. \end{align*} $$

However, by the K-condition (the assumption (ii) in Theorem 0.2), the class Y is pseudo-effective, so that

$$ \begin{align*}(\overline{C}^+\cdot\sigma^*_S Y)=(\overline{C}\cdot Y)\geqslant 0, \end{align*} $$

and $(\overline {C}^+\cdot T)\geqslant 0$ for all $T\in {\cal T}$ , so that ${\cal T}\neq \emptyset $ , and for some $T\in {\cal T}$ , such that $(\overline {C}^+\cdot T)>0$ , the inequality $b_T>na_T$ holds. Since $a_T\geqslant 1$ for all $T\in {\cal T}$ , we conclude that

$$ \begin{align*}\left(\overline{C}^+\cdot\left(\sigma^*_SY-\sum_{T\in{\cal T}} b_T T \right)\right)< -n\left(\overline{C}^+\cdot \sum_{T\in{\cal T}} a_T T \right)\leqslant -n. \end{align*} $$

For a general curve $\overline {C}^+$ , consider the algebraic cycle of the scheme-theoretic intersection

$$ \begin{align*}(D^+\circ \pi_+^{-1}(\overline{C}^+))=\left(\left(\sigma^* D-\pi_+^*\left(\sum_{T\in{\cal T}}b_T T\right)\right)\circ\pi_+^{-1}(\overline{C}^+)\right). \end{align*} $$

The numerical class of that effective cycle is

$$ \begin{align*}n(\sigma^*(-K_V)\cdot\pi^{-1}_+(\overline{C}^+))+\left(\overline{C}^+ \cdot\left(\sigma^*_S Y-\sum_{T\in{\cal T}}b_TT\right)\right)F \end{align*} $$

(where F is the class of a fibre of the projection $\pi _+$ ) and the class of the effective cycle $\sigma _*(D^+\circ \pi ^{-1}_+(\overline {C}^+ ))$ in the numerical Chow group is

$$ \begin{align*}-n(K_V\cdot\pi^{-1}(\overline{C}))+bF, \end{align*} $$

where $b<-n$ . This contradicts the condition (iii) of Theorem 0.2. The proof of Theorem 2.1 is complete. Therefore, in both cases (that of a rationally connected fibre space and of a Mori fibre space), the map $\chi $ is fibre-wise. The first claim of Theorem 0.2 (in the rationally connected case) is shown. It remains to prove the birational rigidity.

2.5 Proof of birational rigidity

Starting from this moment, we assume that $V'/S'$ is a Mori fibre space and the birational map $\chi \colon V\dashrightarrow V'$ is fibre-wise; however, the corresponding map of the bases $\beta \colon S\dashrightarrow S'$ is not birational: $\mathop {\mathrm {dim}}S>\mathop {\mathrm {dim}}S'$ and the fibres $\beta ^{-1}(s')$ for $s'\in S'$ are of positive dimension. We have to obtain a contradiction, showing that this case is impossible.

First of all, let us consider the fibre-wise modification of the fibre space $V/S$ (Subsection 2.2). Now we will need a composition of blow-ups $\sigma _S\colon S^+\to S$ with non-singular centres such that as in Subsection 2.2, none of the fibres of the Fano-Mori fibre space $V^+/S^+$ are contained in the base set $\mathop {\mathrm {Bs}}\Sigma ^+$ , and, in addition, $\sigma _S$ resolves the singularities of the rational dominant map $\beta \colon S\dashrightarrow S'$ ; that is,

$$ \begin{align*}\beta_+=\beta\circ\sigma_S\colon S^+\to S' \end{align*} $$

is a morphism. (So that the inclusion $Z_i=\pi ^{-1}_i(Z_{S,i})\subset \mathop {\mathrm {Bs}}\Sigma ^i$ – see Subsection 2.2 – no longer takes place for all $i=0,\dots ,N-1$ .)

The fibre $\beta ^{-1}_+(s')$ over a point $s'\in S'$ of general position is an irreducible non-singular subvariety of positive dimension. Set $G(s')=(\pi ')^{-1}(s')$ and let $G^+(s')$ be the strict transform of $G(s')$ on $V^+$ . Obviously,

$$ \begin{align*}G^+(s')=\pi^{-1}_+(\beta^{-1}_+(s')) \end{align*} $$

is a union of fibres of the projection $\pi _+$ over the points of the variety $\beta ^{-1}_+(s')$ .

Since $\pi '\colon V'\to S'$ is a Mori fibre space, we have the equality $\rho (V')=\rho (S')+1$ . Let ${\cal E}'$ be the set of all $\psi $ -exceptional divisors $E'\in {\cal E}'$ , satisfying the equality $\pi '(\psi (E'))=S'$ . Furthermore, let ${\cal Z}\subset \mathop {\mathrm {Pic}}\widetilde {V}\otimes {\mathbb Q}$ be the subspace, generated by the subspace $\psi ^*(\pi ')^*\mathop {\mathrm {Pic}}S'\otimes {\mathbb Q}$ and the classes of all $\psi $ -exceptional divisors on $\widetilde {V}$ , the images of which on $V'$ do not cover $S'$ . Then the equality

$$ \begin{align*}\mathop{\mathrm{Pic}}\widetilde{V}\otimes{\mathbb Q}={}\mathbb Q\widetilde{K}\oplus\left(\bigoplus_{E'\in{\cal E}'}{\mathbb Q}E'\oplus {\cal Z}\right)\end{align*} $$

holds; in particular, the subspace in brackets is a hyperplane in $\mathop {\mathrm {Pic}}\widetilde {V}\otimes {\mathbb Q}$ . Writing down the class $\widetilde {K}$ with respect to the morphisms $\varphi $ and $\psi $ , we get the equality

(10) $$ \begin{align} \varphi^*K^++\sum_{E\in{\cal E}}a^+_EE=\psi^* K'+\sum_{E'\in{\cal E}'}a'(E')E'+Z_1, \end{align} $$

where $Z_1\in {\cal Z}$ is some effective class, $a^+_E=a(E,V^+)$ for $\varphi $ -exceptional divisors $E\in {\cal E}$ and $a'(E')=a(E',V')$ for $\psi $ -exceptional divisors $E'\in {\cal E}'$ , covering $S'$ . Here, all $a^+_E\geqslant 1$ and $a'(E')>0$ . Using the $\psi $ -presentation (8) and the $\varphi $ -presentation (9) of the divisorial class $\widetilde {D}$ and expressing $K'$ from the formula (10), we get the following equality in $\mathbb {\mathrm { Pic}}\widetilde {V}\otimes {\mathbb Q}$ :

(11) $$ \begin{align} (m-n)\varphi^*K^++\varphi^*\pi^*_+Y_++\sum_{E\in{\cal E}}(ma^+_E-b_E)E=m\sum_{E'\in{{\cal E}}'}a'(E')E'+Z_2, \end{align} $$

where $Y_+=\sigma ^*_SY+\sum _{T\in {\cal T}}(na_T-b_T)T$ and $Z_2=mZ_1+\psi ^*(\pi ')^*Y'\in {\cal Z}$ is an effective class. Applying to both sides of (11) $\varphi _*$ and restricting onto a fibre of general position of the projection $\pi _+$ , we get that

$$ \begin{align*}(m-n)K^+|_{\pi_+^{-1}(s_+)} \end{align*} $$

is an effective class. Since $m\geqslant n$ and the fibre $\pi _+^{-1}(s_+)$ is a Fano variety, we conclude that $m=n$ and (11) turns into

(12) $$ \begin{align} \varphi^*\pi^*_+Y_++\sum_{E\in{\cal E}}(na^+_E-b_E)E=n\sum_{E'\in{{\cal E}}'}a'(E')E'+Z_2. \end{align} $$

By Proposition 2.1, we have $b_E\leqslant na^+_E$ for all $E\in {\cal E}$ . Again, we apply $\varphi _*$ and get that the class $Y_+$ is effective on $S^+$ .

Now let us consider the defined above fibre $G=G(s')$ of general position of the morphism $\pi '$ and its strict transforms $\widetilde {G}$ on $\widetilde {V}$ and $G^+$ on $V^+$ (the symbol $s'$ for simplicity of notations is omitted). Obviously, for every $Z\in {\cal Z}$ , we have $Z|_{\widetilde {G}}=0$ . Furthermore, for any linear combination with non-negative coefficients,

$$ \begin{align*}\left.\left(\sum_{E'\in{\cal E'}}b'_{E'}E'\right)\right|_{\widetilde{G}} \end{align*} $$

is a fixed divisor on $\widetilde {G}$ . Now let $\Delta $ be a very ample divisor on $S^+$ . Then the restriction $\varphi ^*\pi ^*_+\Delta |_{\widetilde {G}}$ is mobile (recall that $\beta ^{-1}_+(s')$ is a variety of positive dimension, so that $\Delta |_{\beta ^{-1}_+(s')}$ is a mobile class). Therefore,

$$ \begin{align*}\varphi^*\pi^*_+\Delta\not\in\bigoplus_{E'\in{\cal E'}}{\mathbb Q}E'\oplus {\cal Z}, \end{align*} $$

whence we conclude that

$$ \begin{align*}\mathop{\mathrm{Pic}}\widetilde{V}\otimes{\mathbb Q}={\mathbb Q}[\varphi^*\pi^*_+\Delta]\oplus\left(\bigoplus_{E'\in{\cal E'}}{\mathbb Q}E'\oplus{\cal Z}\right). \end{align*} $$

However, this cannot be the case. Let $F^+\subset G^+$ be a fibre of general position of the morphism $\pi _+$ and $\widetilde {F}\subset \widetilde {G}$ its strict transform on $\widetilde {V}$ . Restricting (12) onto $\widetilde {F}$ , we obtain the equality

$$ \begin{align*}\sum_{E\in{\cal E}}(na_E^+-b_E)E|_{\widetilde{F}}= n\sum_{E'\in{\cal E'}}a'(E')E'|_{\widetilde{F}}, \end{align*} $$

where on the right-hand side, it is a linear combination of all divisors $E'|_{\widetilde {F}}$ , $E'\in {\cal E'}$ , with positive coefficients (it is here that we use the assumption that the singularities of the variety $V'$ are terminal; see Subsection 0.2), and on the left-hand side, it is a linear combination of $\varphi $ -exceptional divisors $E|_{\widetilde {F}}$ , $E\in {\cal E}$ , with non-negative coefficients. Since by construction $\pi ^*_+\Delta |_{F^+}=0$ , we have $\varphi ^*\pi ^*_+\Delta |_{\widetilde {F}}=0$ , whence it follows that the restriction of every divisorial class in $\mathop {\mathrm {Pic}}\widetilde {V}\otimes {\mathbb Q}$ onto $\widetilde {F}$ is fixed (is a linear combination of $\varphi $ -exceptional divisors $E|_F,E\in {\cal E}$ ), which is impossible. This contradiction completes the proof of Theorem 0.2.

3.1 The working triple $(F,D_F,o)$

Let us prove Proposition 1.6. Proposition 1.2, shown in Subsection 1.5, implies the condition (MC3). Theorem 1.1 gives the condition (MC1) for $c_F=4k+2\varepsilon (k)$ (the inequality $c_F\geqslant l+4$ is satisfied in the obvious way, since $l\leqslant k$ ). Finally, the condition (MQ1) gives precisely (MC2). Therefore, $[F,o]$ is indeed a marked complete intersection of level $(l,c_F)$ .

Consider a general subspace $P^{\sharp }\ni o$ of dimension $k+c_F$ in ${\mathbb P}^{M+k}$ . The pair

$$ \begin{align*}\left(F\cap P^{\sharp},\frac{1}{n(D_F)}D_F|_{F\cap P^{\sharp}}\right) \end{align*} $$

is not canonical. By (MC1), the singularities of the variety ${F\cap P^{\sharp }}$ are zero-dimensional, and moreover, $o\in \mathop {\mathrm {Sing}}{F\cap P^{\sharp }}$ and

$$ \begin{align*}\mathop{\mathrm{CS}}\left(F\cap P^{\sharp},\frac{1}{n(D_F)}D_F|_{F\cap P^{\sharp}}\right)\subset\mathop{\mathrm{Sing}} (F\cap P^{\sharp}), \end{align*} $$

and the point o is an (isolated) centre of some non-canonical singularity of that pair. For a general subspace $P\ni o$ of dimension $k+c_F-1$ , take a general hyperplane in $P^{\sharp }$ , containing the point o. By inversion of adjunction, we have the equalities

$$ \begin{align*}\{o\}=\mathop{\mathrm{LCS}}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right)= \mathop{\mathrm{CS}}\left(F\cap P,\frac{1}{n(D_F)}D_F|_{F\cap P}\right), \end{align*} $$

and this is precisely (7). Q.E.D. for Proposition 1.6.

As we explained in Subsection 1.7, from now, our work is constructing a certain special sequence of working triples. This sequence starts with the working triple $(F,D_F,o)$ . In order to construct the sequence, we will need certain facts about working triples.

3.2 Multiplicity at the marked point

Let us prove Proposition 1.5. We use the notations of Subsection 1.7 and work with a working triple $(X,D,o)$ , where $[X,o]$ is a marked complete intersection. Assume that $\nu (D)\leqslant 2n(D)$ (otherwise, there is nothing to prove).

Since for a general subspace $P\ni o$ of dimension $m\in I_X$ , the inequality $a(E_{X\cap P})\geqslant 2$ holds (see Remark 1.2), the pair

$$ \begin{align*}\left((X\cap P)^+,\frac{1}{n(D)}D^+_P\right) \end{align*} $$

is not log-canonical, and moreover,

$$ \begin{align*}\mathop{\mathrm{LCS}}\left((X\cap P)^+,\frac{1}{n(D)}D^+_P\right)\subset E_{X\cap P}. \end{align*} $$

Let $B(P)\subset E_{X\cap P}$ be the centre of some non-log-canonical singularity of that pair. Then the inequality

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_{B(P)}D^+_P>n(D) \end{align*} $$

holds, and the more so,

$$ \begin{align*}\mathop{\mathrm{mult}}\nolimits_{B(P)}D^+_P|_{E_{X\cap P}}>n(D). \end{align*} $$

Considering a general subspace $P^*\ni o$ of the minimal admissible dimension $k+l_X+3$ in $I_X$ as a general subspace of codimension $\geqslant l_X$ in a general subspace $P\ni o$ of the maximal admissible dimension $k+c_X-1$ in $I_X$ (recall that by assumption $c_X\geqslant 2l_X+4$ ), we see that the centre $B(P)$ of some non-log-canonical singularity is of dimension $\geqslant l_X$ . However, $E_{X\cap P}$ is a non-singular complete intersection of $l_X$ quadrics in the projective space of dimension $l_X+c_X-2$ , and the divisor $D^+_P|_{E_{X\cap P}}$ is cut out on $E_{X\cap P}$ by a hypersurface of degree $\nu (D)$ in that projective space. Therefore, (for example, by [Reference Pukhlikov17, Proposition 3.6]), the inequality

$$ \begin{align*}\nu(D)\geqslant\mathop{\mathrm{mult}}\nolimits_{B(P)} D^+_P|_{E_{X\cap P}} \end{align*} $$

holds. Therefore, $\nu (D)>n(D)$ . Q.E.D. for Proposition 1.5.

3.3 Transversal hyperplane sections

We still work with an arbitrary working triple $(X,D,o)$ , where $[X,o]$ is a marked complete intersection of level $(l_X, c_X)$ .

Proposition 3.1. Let $R\ni o$ be the section of the variety X by a hyperplane ${\mathbb P}(R)\subset {\mathbb P}(X)$ , which is not tangent to X at the point o. Then $D\neq bR$ for $b\geqslant 1$ . Moreover, if D contains R as a component – that is,

$$ \begin{align*}D=D^*+bR, \end{align*} $$

where $b\geqslant 1$ – then $(X,D^*,o)$ is a working triple.

Proof. If $c_X\geqslant 2l_X+4$ , then the first claim (that D is not a multiple of R) follows immediately from Proposition 1.5: indeed, the hyperplane ${\mathbb P}(R)$ is not tangent to X at the point o; that is, for the strict transform $R^+$ on the blow-up of that point, we have

$$ \begin{align*}R^+\sim H_X-E_X, \end{align*} $$

so that the equality $D=bR$ implies that $n(D)=b=\nu (D)$ , which contradicts Proposition 1.5. However, we will show now that the additional assumptions for the parameters $l_X$ and $c_X$ are not needed.

By Remark 1.4, the condition (MC1) for X implies the inequality

(13) $$ \begin{align} \mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}R\subset R)\geqslant c_X-2. \end{align} $$

Since the hyperplane ${\mathbb P}(R)$ is not tangent to X at the point o, this point is a multi-quadratic singularity of the variety R of type $2^{l_X}$ , the rank of which (by Remark 1.4 and the condition (MC2)) satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in R)\geqslant 2l_X+c_X-3. \end{align*} $$

Consider a general linear subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $k+c_X-2$ . That dimension, generally speaking, does not belong to $I_X$ , and only the inequality

$$ \begin{align*}a(E_{X\cap P})\geqslant 1 \end{align*} $$

holds. The variety $X\cap P$ has a unique singularity, the point o, and its strict transform $(X\cap P)^+$ and the exceptional divisor $E_{X\cap P}$ are non-singular.

The intersection $P\cap {\mathbb P}(R)$ is a general linear subspace of dimension $k+c_X-3$ in ${\mathbb P}(R)$ , containing the point o. For that reason, $R\cap P$ has a unique singularity, the point o, and moreover, the exceptional divisor

$$ \begin{align*}E_{R\cap P}=(R\cap P)^+\cap{\mathbb E}_X=R^+\cap E_{X\cap P} \end{align*} $$

is non-singular, and the map $(R\cap P)^+\to R\cap P$ is the blow-up of the point o on $R\cap P$ , which resolves the singularities of that variety. From here, taking into account that

$$ \begin{align*}\nu(R)=1\leqslant a(E_{X\cap P}), \end{align*} $$

it follows that the pair $(X\cap P,R\cap P)$ is canonical. By inversion of adjunction, we get that for every $m\in I_X$ and a general subspace $P^{\sharp }\ni o$ of dimension m, the pair $(X\cap P^{\sharp }, R\cap P^{\sharp })$ is canonical. Therefore, $D\neq bR$ , $b\geqslant 1$ , and the first claim of the proposition is shown.

Assume now that $D=D^*+bR$ , where $b\geqslant 1$ . Then for a general subspace $P\ni o$ of dimension $k+c_X-1$ in ${\mathbb P}(X)$ , the pair $(X\cap P, \frac {1}{n(D)}D_P)$ is not log canonical at the point o. As we saw above, the pair $(X\cap P, R_P)$ is log-canonical (and even canonical). The condition of being log-canonical is linear, so we conclude that the pair

$$ \begin{align*}\left(X\cap P, \frac{1}{n(D^*)} D^*|_{X\cap P}\right) \end{align*} $$

is not log-canonical at the point o. Therefore, $(X,D^*,o)$ is a working triple. Q.E.D. for the proposition.

Proof. First of all, let us check that $[R,o]$ is a marked complete intersection of level $(k,c_R)$ . The inequality $c_R\geqslant k+4$ holds by assumption. Furthermore,

$$ \begin{align*}\mathop{\mathrm{codim}} (\mathop{\mathrm{Sing}}R\subset R) \geqslant c_X-2=c_R, \end{align*} $$

so that the condition (MC1) is satisfied. Furthermore, the point $o\in R$ is a multi-quadratic singularity, the rank of which satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in R)\geqslant\mathop{\mathrm{rk}}(o\in X)-2\geqslant 2k+c_R-1, \end{align*} $$

so that the condition (MC2) holds. The condition (MC3) holds by assumption. The bound for the codimension of the singular set $\mathop {\mathrm {Sing}}R$ guarantees that the complete intersection $R\subset {\mathbb P}(R)$ is irreducible, reduced and factorial. Therefore, $[R,o]$ is a marked complete intersection of level $(k,c_R)$ . Set

$$ \begin{align*}I_R=[2k+3,k+c_R-1]. \end{align*} $$

Obviously, $(D\circ R)\sim n(D\circ R)H_R=n(D)H_R$ , where $H_R$ is the class of a hyperplane section of R. It remains to check that for a general subspace $P\ni o$ of dimension $k+c_R-1$ in ${\mathbb P}(R)$ , the pair

(14) $$ \begin{align} \left(R\cap P, \frac{1}{n(D\circ R)}(D\circ R)|_{R\cap P}\right) \end{align} $$

is not log-canonical at the point o. In order to do this, we present P as the intersection

$$ \begin{align*}P=P^{\sharp}\cap{\mathbb P}(R), \end{align*} $$

where $P^{\sharp }\ni o$ is a general subspace of dimension

$$ \begin{align*}k+c_R=k+c_X-2 \end{align*} $$

in ${\mathbb P}(X)$ . As $k+c_R\in I_X$ , the point o is the only singularity of the variety $X\cap P^{\sharp }$ (and the only singularity of the variety $R\cap P$ ), and

$$ \begin{align*}\{o\}=\mathop{\mathrm{LCS}}\left(X\cap P^{\sharp},\frac{1}{n(D)}D|_{X\cap P^{\sharp}}\right). \end{align*} $$

The variety $R\cap P$ is the section of the variety $X\cap P^{\sharp }$ by the hyperplane $P= P^{\sharp }\cap {\mathbb P}(R)$ , containing the point o, so that by inversion of adjunction the pair (14) is not log-canonical. At the same time, it is canonical outside the point o since the subspace $P\subset {\mathbb P}(R)$ is generic, the non-singular part $R\backslash \mathop {\mathrm {Sing}}R$ is divisorially canonical and the equality $\{o\}=\mathop {\mathrm {Sing}}(R\cap P)$ holds. Therefore, the pair (14) is not log-canonical precisely at the point o, which completes the proof of Theorem 3.1.

3.4 Tangent hyperplane sections

Now let us consider a marked complete intersection $[X,o]$ of level $(l_X,c_X)$ , where $l_X\leqslant k-1$ . Let R be the section of the variety X by a hyperplane ${\mathbb P}(R)\subset {\mathbb P}(X)$ , which is tangent to X at the point o. By the symbol ${\mathbb P}(R)^+$ , denote the strict transform of the hyperplane ${\mathbb P} (R)$ on ${\mathbb P}(X)^+$ and set

$$ \begin{align*}{\mathbb E}_R={\mathbb P}(R)^+\cap {\mathbb E}_X. \end{align*} $$

Obviously, ${\mathbb E}_R\cong {\mathbb P}^{N(X)-2}$ is the exceptional divisor of the blow-up of the point o on the hyperplane ${\mathbb P}(R)$ . Set also $E_R=R^+\cap {\mathbb E}_X$ . Obviously, $E_R\subset {\mathbb E}_R$ , and

$$ \begin{align*}\mathop{\mathrm{codim}}(E_R\subset{\mathbb E}_R)=k. \end{align*} $$

Proposition 3.2. Assume that $c_X\geqslant l_X+5$ and the point $o\in R$ is a multi-quadratic singularity of type $2^{l_X+1}$ , and moreover, the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} (o\in R)\geqslant 2l_X+c_X-2 \end{align*} $$

holds. Then $D\neq bR$ for $b\geqslant 1$ . Moreover, if the divisor D contains R as a component – that is,

$$ \begin{align*}D=D^*+bR, \end{align*} $$

where $b\geqslant 1$ – then $(X,D^*,o)$ is a working triple.

Proof is completely similar to the proof of Proposition 3.1, but we give it in full details because there are some small points where the two arguments are different. The inequality (13) holds in this case again. Let us use the additional assumption about the singularity $o\in R$ . Consider a general linear subspace $P\ni o$ in ${\mathbb P}(X)$ of dimension $k+l_X+3$ (it is the minimal admissible dimension). We get the equality $a(E_{X\cap P})=2$ . Obviously,

$$ \begin{align*}R^+\sim H_X-2E_X \end{align*} $$

and, respectively, on $(X\cap P)^+$ , we have

$$ \begin{align*}(R\cap P)^+\sim H_{X\cap P}-2E_{X\cap P}. \end{align*} $$

Arguing as in the transversal case, we note that the intersection $P\cap {\mathbb P}(R)$ is a general subspace of dimension $k+l_X+2$ in ${\mathbb P}(R)$ . Taking into account that by the inequality (13), the inequality

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}R\subset{\mathbb P}(R))\geqslant k+c_X-2 \end{align*} $$

holds, and that by assumption $c_X\geqslant l_X+5$ , we see that $R\cap P$ has a unique singularity, the point o. Furthermore, by the assumption about the rank of the singular point $o\in R$ , we get the inequality

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}E_R\subset E_R)\geqslant c_X-3\geqslant l_X+2, \end{align*} $$

so that

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}E_R\subset{\mathbb E}_R)\geqslant k+l_X+2. \end{align*} $$

The exceptional divisor $E_{R\cap P}$ is the section of the subvariety $E_R\subset {\mathbb E}_R$ by a general linear subspace of dimension $k+l_X+1$ , whence we conclude that the variety $E_{R\cap P}$ is non-singular. Thus, we have shown that the singularity $o\in R\cap P$ is resolved by one blow-up. Therefore, the pair

$$ \begin{align*}((X\cap P)^+,(R\cap P)^+) \end{align*} $$

is canonical, so that the pair

$$ \begin{align*}(X\cap P, R\cap P) \end{align*} $$

is canonical, too. We have shown that $D\neq bR$ for $b\geqslant 1$ .

By inversion of adjunction for every $m\in I_X$ and a general subspace $P^{\sharp }\ni o$ of dimension m, the pair $(X\cap P^{\sharp }, R\cap P^{\sharp })$ is canonical (recall that $n(R)=1$ ). Repeating the arguments given in the transversal case (the proof of Proposition 3.1) word for word, we complete the proof of Proposition 3.2.

Remark 3.1. If for $l_X\leqslant k-1$ the intersection $X\cap T_oX$ has the point o as a multi-quadratic singularity of type $2^k$ , the rank of which satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in X\cap T_oX)\geqslant 2l_X+c_X-2, \end{align*} $$

the assumption about the rank $\mathop {\mathrm {rk}}(o\in R)$ in the statement of Proposition 3.2 holds automatically for every tangent hyperplane at the point o.

Theorem 3.2 (on the tangent hyperplane section).

Let $[X,o]$ be a marked complete intersection of level $(l_X,c_X)$ , where $2\leqslant l_X\leqslant k-1$ and $c_X\geqslant l_X+7$ , and $(X,D,o)$ a working triple. Let R be the section of X by a hyperplane which is tangent to X at the point o, and assume that R is not a component of the divisor D. Assume that the point $o\in R$ is a multi-quadratic singularity of type $2^{l_R}$ , where $l_R=l_X+1$ , the rank of which satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in R)\geqslant 2l_R+c_R-1=2l_X+c_X-1, \end{align*} $$

where $c_R=c_X-2$ , and also that the inequality $\mathop {\mathrm {ct}}(R\backslash \mathop {\mathrm {Sing}}R)\geqslant 1$ holds. Then $(R,(D\circ R),o)$ is a working triple on the marked complete intersection $[R,o]$ of level $(l_R,c_R)$ .

Proof is similar to the transversal case (Theorem 3.1), and we just emphasize the necessary modifications. The fact that $[R,o]$ is a marked complete intersection of level $(l_R,c_R)$ is checked in the tangent case even easier than in the transversal one, because the assumption about the singularity $o\in R$ is among the assumptions of the theorem.

A general subspace $P\ni o$ of dimension $k+c_R-1=k+c_X-3$ in ${\mathbb P}(R)$ is again presented as the intersection $P=P^{\sharp }\cap {\mathbb P}(R)$ , where $P^{\sharp }\ni o$ is a general subspace of dimension $k+c_R\in I_X$ in ${\mathbb P}(X)$ , and now, repeating the arguments in the transversal case and using inversion of adjunction, we get that $(R,(D\circ R),o)$ is a working triple. Q.E.D. for the theorem.

Proof of Proposition 1.7.

We assume that $l\leqslant k-1$ . Recall that the symbol T stands for the intersection $F\cap T_oF$ ; this is a subvariety of codimension $(k-l)$ in F. Let us construct a sequence of subvarieties

$$ \begin{align*}T_0=F\supset T_1\supset\dots\supset T_{k-l}=T, \end{align*} $$

where $T_{i+1}$ is the section of $T_i\ni o$ by some hyperplane ${\mathbb P}(T_{i+1})=\langle T_{i+1}\rangle \ni o$ , which is tangent to $T_i$ at the point o. Theorem 1.1 implies that the inequality

$$ \begin{align*}c_F\geqslant l+3(k-l)+4 \end{align*} $$

holds (the inequality of Theorem 1.1 for the codimension $c_F$ is much stronger, but for the clarity of exposition, we give the weakest estimate that is sufficient for the proof of Proposition 1.7; this remark also applies to the estimate of the rank of the multi-quadratic singularity $o\in T$ below). Furthermore, the condition (MQ2) implies that $o\in T$ is a multi-quadratic singularity of type $2^k$ , and moreover, the inequality

(15) $$ \begin{align} \mathop{\mathrm{rk}}(o\in T)\geqslant 2k+c_F-1 \end{align} $$

holds. Finally, by Theorem 1.2 for every hyperplane section W of every subvariety $T_i$ , $i=0,1,\dots ,k-l$ , every non-singular point $p\in W$ and every prime divisor Y on W the inequality

(16) $$ \begin{align} \frac{\mathop{\mathrm{mult}}_p}{\mathop{\mathrm{deg}}}Y\leqslant\frac{2}{\mathop{\mathrm{deg}} F} \end{align} $$

holds. Then for all $i=0,1,\dots ,k-l$ , the pair $[T_i,o]$ is a marked complete intersection of level

$$ \begin{align*}(l_i=l+i,c_i=c_F-2i). \end{align*} $$

Indeed, the inequality $c_i\geqslant l_i+4$ is true by the definition of the numbers $l_i$ , $c_i$ , the condition (MC1) follows from Remark 1.4, the point $o\in T_i$ by construction is a multi-quadratic singularity of type $2^{l+1}$ , and moreover, by (15), we have

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in T_i)\geqslant 2l+c_F-1=2l_i+c_i-1, \end{align*} $$

and, finally, repeating the proof of Proposition 1.1 and the arguments of Subsection 1.5 word for word, we get that by (16), the condition (MC3) holds. Therefore, $[T,o]$ is a marked complete intersection of level $(k,c_{k-l})$ , where $c_{k-l}=c_F-2(k-l)\geqslant k+4$ . Recall (Proposition 1.6) that $c_F=4k+2\varepsilon (k)$ . Since $l\geqslant 2$ , the inequality

$$ \begin{align*}c_{k-l}\geqslant c_T=2k+2\varepsilon(k)+4 \end{align*} $$

holds, so that $[T,o]$ is a marked complete intersection of level $(k,c_T)$ , as we claimed.

It remains to construct the working triple $(T,D_T,o)$ . We will construct a sequence of working triples $(T_i,D_i,o)$ , where $i=0,1,\dots ,k-l$ and $D_0=D_F$ . Assume that $(T_i,D_i,o)$ is already constructed and $i\leqslant k-l-1$ . Let us check that all assumptions that allow us to apply Proposition 3.2 are satisfied.

Indeed, the fact that $i\leqslant k-l-1$ implies the inequality $c_i\geqslant l_i+7$ . The point $o\in T_{i+1}$ is a multi-quadratic singularity of type $2^{l_i+1}$ , the rank of which satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in T_{i+1})\geqslant 2l_{i+1}+c_{i+1}-1=2l_i+c_i-1 \end{align*} $$

(see above). Applying Proposition 3.2, we remove $T_{i+1}$ from the effective divisor $D_i$ (if it is necessary) and obtain the working triple $(T_i,D^*_i,o)$ , where the effective divisor $D^*_i$ does not contain $T_{i+1}$ as a component.

It is easy to see that we have all assumptions of Theorem 3.2. Set

$$ \begin{align*}D_{i+1}=(D^*_i\circ T_{i+1}). \end{align*} $$

Now $(T_{i+1}, D_{i+1}, o)$ is a working triple. Proof of Proposition 1.7 is complete.

3.5 Plan of the proof of Proposition 1.8

Recall that by the condition (MQ2), the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} (o\in T)\geqslant 10k^2+8k+2\varepsilon(k)+5 \end{align*} $$

holds. The pair $[T,o]$ is a marked complete intersection of level $(k,c_T)$ , where $c_T=2k+2\varepsilon (k)+4$ . Let

$$ \begin{align*}R_0=T,R_1,\dots,R_a, \end{align*} $$

where $a\leqslant \varepsilon (k)$ , be an arbitrary sequence of subvarieties in T, where $R_{i+1}$ is the section of $R_i$ by the hyperplane ${\mathbb P}(R_{i+1})$ in ${\mathbb P}(R_i)$ , containing the point o. Set $c_i=c_T-2i$ , where $i=0,1,\dots ,a$ .

Proof. Since $a\leqslant \varepsilon (k)$ , the inequality $c_i\geqslant k+4$ holds in an obvious way (in fact, $c_i\geqslant 2k+4)$ . The condition (MC1) holds by Remark 1.4. The condition (MC3) is obtained by repeating the proof of Proposition 1.1 and the arguments of Subsection 1.5 word for word, taking into account Theorem 1.2. Finally, again by Remark 1.4, the inequality

$$ \begin{align*}10k^2+8k+2\varepsilon(k)+4\geqslant 2k+c_i+2i-1 \end{align*} $$

implies the condition (MC2). Q.E.D. for the proposition.

Now let us construct for every $i=0,1,\dots ,a$ an effective divisor $D_i$ on $R_i$ in the same way as we did it in Subsection 3.4 in the proof of Proposition 1.7, applying instead of Proposition 3.2, its ‘transversal’ analog, Proposition 3.1, and Theorem 3.1 instead of Theorem 3.2. More precisely, if the effective divisor $D_i$ , where $i\leqslant a-1$ , is already constructed, we remove from this divisor all components that are hyperplane sections (if there are such components), and obtain an effective divisor $D^*_i$ that does not contain hyperplane sections as components, and such that $(R_i,D^*_i,o)$ is a working triple (Proposition 3.1).

Proposition 3.4. The following inequality holds:

$$ \begin{align*}\frac{\nu(D^*_i)}{n(D^*_i)}\geqslant\frac{\nu(D_i)}{n(D_i)}. \end{align*} $$

Proof. It is sufficient to consider the case when $D^*_i$ is obtained from $D_i$ by removing one hyperplane section $Z\ni o$ . Write down

$$ \begin{align*}D_i=D^*_i+bZ, \end{align*} $$

where $b\geqslant 1$ . Since $c_i\geqslant 2k+4$ , we can apply Proposition 1.5: $\nu (D_i)>n(D_i)$ . However, $\nu (Z)=n(Z)=1$ . Set $\nu (D_i)=\alpha n(D_i)$ , where $\alpha>1$ . We get

$$ \begin{align*}\frac{\nu(D^*_i)}{n(D^*_i)}= \frac{\alpha n(D^*_i)+(\alpha-1)b}{n(D^*_i)}>\alpha, \end{align*} $$

which proves the proposition. Q.E.D.

(If we remove from $D_i$ , a hyperplane section that does not contain the point o, the claim of Proposition 3.4 is trivial.)

Now we apply Theorem 3.1, setting $D_{i+1}=(D^*_i\circ R_{i+1})$ . This cycle of the scheme-theoretic intersection is well defined as an effective divisor on $R_{i+1}$ , and moreover, $(R_{i+1}, D_{i+1}, o)$ is a working triple and

$$ \begin{align*}\frac{\nu(D_{i+1})}{n(D_{i+1})}\geqslant\frac{\nu(D^*_i)}{n(D^*_i)} \geqslant\frac{\nu(D_i)}{n(D_i)}. \end{align*} $$

We emphasize that $R_1,\dots , R_a$ is an arbitrary sequence of consecutive hyperplane sections. By Remark 1.4, for all $i=0,1,\dots ,a$ , the inequality $c_i\geqslant 2k+4$ holds, and the rank of the multi-quadratic singularity $o\in R_i$ of type $2^k$ is at least $10k^2+8k+5$ . By Theorem 1.4 and Proposition 1.5, we have the inequalities

$$ \begin{align*}n(D_i)<\nu(D_i)\leqslant\frac32n(D_i). \end{align*} $$

Therefore, at every step of our construction, the assumptions of the following claim are satisfied.

Theorem 3.3 (on the special hyperplane section).

Let $[X,o]$ be a marked complete intersection of level $(k,c_X)$ , where $c_X\geqslant 2k+4$ and the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(o\in X)\geqslant 10k^2+8k+5 \end{align*} $$

holds. Let $(X,D,o)$ be a working triple, where the effective divisor D does not contain hyperplane sections and satisfies the inequalities

$$ \begin{align*}n(D)<\nu(D)\leqslant\frac32n(D). \end{align*} $$

Then there is a section $R\ni o$ of the variety X by a hyperplane ${\mathbb P}(R)=\langle R\rangle \subset {\mathbb P}(X)= {\mathbb P}^{N(X)}$ , such that the effective divisor $D_R=(R\circ D)$ on R satisfies the inequality

$$ \begin{align*}2-\frac{\nu(D_R)}{n(D_R)}<\frac{1}{1+\frac{1}{k}}\left(2- \frac{\nu(D)}{n(D)}\right). \end{align*} $$

Now by the definition of the integer $\varepsilon (k)$ and what was said above, Theorem 3.3 immediately implies Proposition 1.8.

Proof of Theorem 3.3 is given in §5.

4.1 The definition and the first properties

Let ${\cal X}$ be an (irreducible) algebraic variety, $o\in {\cal X}$ a point.

Definition 4.1. The point o is a multi-quadratic singularity of the variety ${\cal X}$ of type $2^l$ and rank $r\geqslant 1$ , if in some neighborhood of this point, ${\cal X}$ can be realized as a subvariety of a non-singular $N=(\mathop {\mathrm {dim}} {\cal X}+ l)$ -dimensional variety ${\cal Y}\ni o$ , and for some system $(u_1, \dots , u_N)$ of local parameters on ${\cal Y}$ at the point o, the subvariety ${\cal X}$ is the scheme of common zeros of regular functions

$$ \begin{align*}\alpha_1,\dots,\alpha_l\in{\cal O}_{o,{\cal Y}}\subset{\mathbb C}[[u_1,\dots,u_N]], \end{align*} $$

which are represented by the formal power series

$$ \begin{align*}\alpha_i=\alpha_{i,2}+\alpha_{i,3}+\dots, \end{align*} $$

where $\alpha _{i,j}(u_1,\dots ,u_N)$ are homogeneous polynomials of degree j and

$$ \begin{align*}\mathop{\mathrm{rk}}(\alpha_{1,2},\dots,\alpha_{l,2})=r. \end{align*} $$

(Obviously, the order of the formal power series, representing $\alpha _i$ , and the rank of the tuple of quadratic forms $\alpha _{i,2}$ do not depend on the choice of the local parameters on ${\cal Y}$ at the point o.)

It is convenient to work in a more general context. Assume that in a neighborhood of the point o, the variety ${\cal X}$ is realized as a subvariety ${\cal X}\subset {\cal Z}$ , where $\mathop {\mathrm {dim}}{\cal Z}=\mathop {\mathrm {dim}}{\cal X}+e=N({\cal Z})$ , and for a certain system of local parameters $(v_1,\dots ,v_{N({\cal Z})})$ on ${\cal Z}$ at the point o, the subvariety ${\cal X}$ is the scheme of common zeros of regular functions

$$ \begin{align*}\beta_1,\dots,\beta_e\in{\cal O}_{o,{{\cal Z}}}\subset{\mathbb C}[[v_*]], \end{align*} $$

which are represented by the formal power series

$$ \begin{align*}\beta_i=\beta_{i,1}+\beta_{i,2}+\dots, \end{align*} $$

where $\beta _{i,j}(v_*)$ are homogeneous polynomials of degree j. Assume that for some $l\in \{0,1,\dots ,e\},$

$$ \begin{align*}\mathop{\mathrm{dim}}\langle\beta_{1,1},\dots,\beta_{e,1}\rangle=e-l, \end{align*} $$

where we assume (for the convenience of notations) that the linear forms $\beta _{j,1}$ for $l+1\leqslant j\leqslant e$ are linearly independent, so that for $1\leqslant i\leqslant l$ and $l+1\leqslant j\leqslant e$ , there are uniquely determined numbers $a_{i,j}$ , such that

$$ \begin{align*}\beta_{i,1}=\sum^e_{j=l+1}a_{i,j}\beta_{j,1}. \end{align*} $$

Set ${\cal Y}=\{\beta _j=0\,|\, l+1\leqslant j\leqslant e\}$ and

$$ \begin{align*}\beta_i^*=\beta_i-\sum_{j=l+1}^e a_{i,j}\beta_j. \end{align*} $$

Then (in a neighborhood of the point o) the variety ${\cal Y}$ is non-singular, and ${\cal X}\subset {\cal Y}$ is realized as the scheme of common zeros of the regular functions $\beta ^*_i$ , $1\leqslant i\leqslant l$ . Set

$$ \begin{align*}T_o{\cal Y}=T_o{\cal X}=\{\beta_{j,1}=0\,|\, l+1\leqslant j\leqslant e\}. \end{align*} $$

If

$$ \begin{align*}\mathop{\mathrm{rk}}\left(\beta^*_{i,2}|_{T_o{\cal X}}\, |\, 1\leqslant i\leqslant l\right)=r, \end{align*} $$

then obviously $o\in {\cal X}$ is a multi-quadratic singularity of rank r.

The rank of the multi-quadratic point $o\in {\cal X}$ is denoted by the symbol $\mathop {\mathrm {rk}} (o\in {\cal X})$ or just $\mathop {\mathrm {rk}}(o)$ , if it is clear which variety is meant. For uniformity of notations, we treat a non-singular point as a multi-quadratic one of type $2^0$ .

Proof. Using the notations for the embedding ${\cal X}\subset {\cal Z}$ introduced above, with $e=l$ (so that $\beta _{i,1}=0$ for all $i=1,\dots ,l$ ) and setting $N({\cal Z})=N$ , consider an open set $U\subset {\cal Z}$ , $U\ni o$ , such that for every point $p\in U$ , the ‘shifted’ functions

$$ \begin{align*}v^{(p)}_i=v_i-v_i(p),\quad i=1,\dots,N \end{align*} $$

form a system of local parameters at the point p, and in the formal expansion

$$ \begin{align*}\beta_i=\beta^{(p)}_{i,0}+\beta^{(p)}_{i,1}+\beta^{(p)}_{i,2}+\dots \end{align*} $$

with respect to the system of parameters $v^{(p)}_*$ , the quadratic components satisfy the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(\beta^{(p)}_{i,2}\,|\, 1\leqslant i\leqslant l)\geqslant r. \end{align*} $$

If the point p is a common zero of $\beta _1,\dots ,\beta _l$ , then $\beta ^{(p)}_{i,0}=0$ for $1\leqslant i\leqslant l$ . Set

$$ \begin{align*}T_p{\cal X}=\{\beta^{(p)}_{i,1}=0\,|\, 1\leqslant i\leqslant l\} \end{align*} $$

and assume (for the convenience of notations) that the forms $\beta ^{(p)}_{i,1}$ for $b+1\leqslant i\leqslant l$ are linearly independent, where

$$ \begin{align*}\mathop{\mathrm{dim}}\langle\beta^{(p)}_{i,1}\,|\,\,1\leqslant i\leqslant l\rangle=l-b. \end{align*} $$

Since $\mathop {\mathrm {codim}}(T_p{\cal X}\subset T_p{\cal Z})=l-b$ , by Remark 1.4, the inequality

$$ \begin{align*}\mathop{\mathrm{rk}}(\beta^{(p)}_{i,2}|_{T_p{\cal X}}\,|\, 1\leqslant i\leqslant l) \geqslant r - 2(l-b) \end{align*} $$

holds. It is easy to see from the construction of the quadratic forms $\beta ^{(p)*}_{i,2}$ , $1\leqslant i\leqslant b$ that every linear combination of these forms with coefficients $(\lambda _1,\dots ,\lambda _b)\neq (0,\dots ,0)$ is a linear combination of the original forms $\beta ^{(p)}_{i,2}$ , $1\leqslant i\leqslant l$ , not all coefficients in which are equal to zero. Therefore, the point p is a multi-quadratic singularity of rank $\geqslant r-2(l-b)$ , as we claimed. Q.E.D. for the proposition.

4.2 Complete intersections of quadrics

In the notations of Definition 4.1, let ${\cal Y}^+\to {\cal Y}$ be the blow-up of the point o with the exceptional divisor $E_{\cal Y}\cong {\mathbb P}^{N-1}$ and ${\cal X}^+\subset {\cal Y}^+$ the strict transform of ${\cal X}$ on ${\cal Y}^+$ , so that ${\cal X}^+\to {\cal X}$ is the blow-up of the point o on ${\cal X}$ with the exceptional divisor $E_{\cal Y}|_{{\cal X}^+}=E_{\cal X}$ . Therefore, $E_{\cal X}$ is the scheme of common zeros of the quadratic forms $\alpha _{i,2}$ , $i=1,\dots ,l$ , on $E_{\cal Y}\cong {\mathbb P}^{N-1}$ .

Let $q_1,\dots ,q_l$ be quadratic forms on ${\mathbb P}^{N-1}$ , where $N\geqslant l+4$ . By the symbol $q_{[1,l]}$ , we denote the tuple $(q_1,\dots ,q_l)$ .

Proposition 4.2. (i) Assume that the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} q_{[1,l]}\geqslant 2l+3 \end{align*} $$

holds. Then the scheme of common zeros of the forms $q_1,\dots ,q_l$ is an irreducible non-degenerate factorial variety $Q\subset {\mathbb P}^{N-1}$ of codimension l – that is, a complete intersection of type $2^l$ .

(ii) Assume that for some $e\geqslant 4$ , the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} q_{[1,l]}\geqslant 2l+e-1 \end{align*} $$

holds. Then the following inequality is true:

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}Q\subset Q)\geqslant e. \end{align*} $$

Proof is given below in Subsection 4.4.

Corollary 4.1. (i) Assume that the rank of the tuple $\alpha _{*,2}=(\alpha _{1,2},\dots ,\alpha _{l,2})$ of quadratic forms satisfies the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} (\alpha_{*,2})\geqslant 2l+3. \end{align*} $$

Then in a neighborhood of the point o, the scheme of common zeros of the regular functions $\alpha _1,\dots ,\alpha _l$ is an irreducible reduced factorial subvariety ${\cal X}$ of codimension l in ${\cal Y}$ .

(ii) Assume that for some $e\geqslant 4$ , the inequality

$$ \begin{align*}\mathop{\mathrm{rk}} (\alpha_{*,2})\geqslant 2l+e-1 \end{align*} $$

holds. Then in a neighborhood of the point o, the following inequality is true:

$$ \begin{align*}\mathop{\mathrm{codim}}(\mathop{\mathrm{Sing}}{\cal X}\subset{\cal X})\geqslant e. \end{align*} $$