Birationally rigid varieties with a pencil of Fano double covers. I

Aleksandr V. Pukhlikov

Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn GERMANY

e-mail: pukh@mpim-bonn.mpg.de

Steklov Institute of Mathematics

Gubkina 8 117966 Moscow RUSSIA e-mail: pukh@mi.ras.ru

Division of Pure Mathematics Department of Mathematical Sciences

M&O Building, Peach Street The University of Liverpool Liverpool L69 7ZL ENGLAND e-mail: pukh@liv.ac.uk

October 3, 2003

Abstract

We prove that a general Fano fibration π: V → ℙ^{1}, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on V there are no other structures of a rationally connected fibration. The proof is based on the method of maximal singularities.

CONTENTS

Introduction 0.1. Birationally rigid varieties 0.2. Varieties with a pencil of double covers 0.3. The main result 0.4. Historical remarks 0.5. Acknowledgements

1. The method of maximal singularities and the regularity conditions 1.1. A criterion of birational rigidity 1.2. An explicit construction of the fibration V/ℙ^{1} 1.3. The regularity conditions outside the branch divisor 1.4. The regularity conditions on the branch divisor 1.5. Start of the proof of Theorem 1

2. Singularity of a fiber outside the branch divisor 2.1. Hypertangent divisors and linear systems 2.2. Scheme of the proof of the condition (vs) 2.3. Movable families of curves

3. Singularity of a fiber on the branch divisor 3.1. Notations and discussion of the regularity condition 3.2. Start of the proof of the condition (vs) 3.3. Hypertangent divisors and tangent cones 3.4. Constructing new cycles 3.5. Degrees and multiplicities

References

Introduction

In this paper we study birational geometry of higher-dimensional algebraic varieties with a pencil of Fano double covers. The main result of the paper, that is, the theorem on birational superrigidity of these varieties provided they are sufficiently twisted over the base, is formulated below in Sec. 0.3. The paper presents the outcome of the first stage of the research; the second part of this work, dealing with a relaxation of the twistedness condition, will be published in the subsequent paper.

0.1 Birationally rigid varieties

A rationally connected projective variety V with at most ℚ-factorial terminal singularities is said to be birationally rigid, if for any birational map

χ: V − − → V′,

where V′ belongs to the same class of varieties, and any moving linear system Σ′ on V′ there exists a birational self-map χ^{∗} ∈ Bir V, providing the following inequality

c(Σ) ≤ c(Σ′),  (1)

where Σ = (χ ∘ χ^{∗})_{∗}Σ′ is the strict transform of the linear system Σ′ on V with respect to the birational map

χ ∘ χ^{∗}: V − − →^{χ^{∗}} V − − →^{χ} V′,

and the symbol c(·) stands for the threshold of the canonical adjunction of the linear system | · |,

c(Λ) = sup{ε ∈ ℚ_{+} | D + εK ∈ A_{+}^{1}(·)},

D ∈ Λ is an arbitrary divisor of the linear system Λ, K stands for the canonical class of the variety, A_{+}^{1}(·) ⊂ A^{1}(·) ⊗ ℝ means the closed cone of effective cycles on the variety under consideration. The variety V is said to be birationally superrigid, if the inequality (1) is always true for χ^{∗} = id_{V}.

Let π: V → ℙ^{1} be a fibration into rationally connected varieties, where V has ℚ-factorial terminal singularities. (By the theorem of Graber-Harris-Starr [5], in this case the variety V is itself automatically rationally connected.) The next question is of crucial importance for understanding birational geometry of V:

are there other (that is, different from the original map π: V → ℙ^{1}) structures of a rationally connected fibration on V?

Assume that V is non-singular, Pic V = ℤK_{V} ⊕ π^{∗} Pic ℙ^{1} and the following condition holds:

K_{V} ∉ Int A_{+}^{1}V.  (2)

Let F_{t} = π^{−1}(t) be the fiber over a point t ∈ ℙ^{1}, F ∈ Pic V the class of a fiber. The condition (2) means that if D ∼ −nK_{V} + lF is an effective divisor on V, then l ∈ ℤ_{+}. Conditions of this type for three-dimensional Mori fiber spaces are discussed in [2]. The following fact is well-known (see [17,20-22]).

Proposition 0.1. In the assumptions above let V be a birationally superrigid variety. Then:

(i) there is only one non-trivial structure of a fibration into rationally connected varieties on V, that is, the morphism π; in other words, if τ: W → T is a fibration into rationally connected (or just uniruled) varieties and χ: V − − → W is a birational map, then χ transform fibers into fibers, that is, the following diagram

V − − →^{χ} W
π ↓  ↓ τ
ℙ^{1} − − →^{α} T

commutes for a certain map α: ℙ^{1} → T.

(i) If τ: W → T is another fibration of the same type, that is, Pic W = ℤK_{W} ⊕ τ^{∗} Pic ℙ^{1}, and χ: V − − → W is a (fiber-wise) birational map, then χ is an isomorphism of fibers of general position.

Thus the property of being birationally rigid reduces birational geometry of the variety V to biregular geometry of the fibration V/ℙ^{1}. That is why the word “rigidity” has been chosen: the variety V does not admit any birational modifications inside the natural class of Fano fibrations with the relative Picard number one.

Nowadays quite a few classes of birationally (super)rigid Fano varieties are known (see, for instance, [3,14,18,19,23-25]). The known examples make it possible to conjecture that birational (super)rigidity is a typical property in dimension 3 and higher. Much less is known about Fano fibrations, their birational geometry is harder to investigate. A brief history of the theory of birational rigidity for Fano fibrations see below in Sec. 0.4. The aim of the present paper is to prove birational superrigidity of fibrations V/ℙ^{1}, the fibers of which are Fano double hypersurfaces of index 1 [19].

0.2 Varieties with a pencil of double covers

The symbol ℙ stands for the projective space ℙ^{M+1} over the field of complex numbers ℂ. Let

G = ℙ(H^{0}(ℙ, O_{ℙ}(m)))

be the space of all Fano hypersurfaces of degree m, 3 ≤ m ≤ M − 1, W the space of all hypersurfaces of degree 2l in ℙ, where m + l = M + 1. Let

F = {F | σ: F →^{2:1} G}

be the class of algebraic varieties realized as double covers of hypersurfaces G ∈ G branched over W ∩ G, W ∈ W. Set F_{sm} ⊂ F to be the set of smooth double hypersurfaces corresponding to pairs (G, W ∩ G) of smooth varieties. Obviously, F ∈ F_{sm} is a smooth Fano variety of index 1 with the Picard group Pic F = ℤK_{F}. Let F_{sm}^{reg} ⊂ F_{sm} be the smooth subset consisting of varieties F ∈ F_{sm} satisfying the regularity condition of Sec. 1.3,1.4 below (which is identical to the regularity condition of Sec. 1.3 in [19]). Recall that in [19] the following fact was proved.

Theorem A. (i) Any variety F ∈ F_{sm}^{reg} is birationally superrigid.

(ii) The set F_{sm}^{reg} is non-empty. Moreover, the following estimate holds:

codim_{F_{sm}}(F_{sm} \ F_{sm}^{reg}) ≥ 2.

Set F_{sing} = F \ F_{sm}, codim_{F} F_{sing} = 1. Let F_{sing}^{reg} be the open subset in F_{sing}, consisting of all singular double hypersurfaces satisfying the regularity condition of Sec. 1.3,1.4 below. We note in Sec. 1.3,1.4 that the following inequality holds:

codim_{F}(F_{sing} \ F_{sing}^{reg}) ≥ 2.

In the present paper we study Fano fibrations V/ℙ^{1}, each fiber F_{t} = π^{−1}(t), t ∈ ℙ^{1} of which is a variety from the family F. Set

F^{reg} = F_{sm}^{reg} ∪ F_{sing}^{reg}.

By what was said above, codim_{F}(F \ F^{reg}) ≥ 2. Since the fibration V/ℙ^{1} can be looked at as a morphism ℙ^{1} → F, that is, a curve in F, for a general variety V/ℙ^{1} we get:

F_{t} ∈ F^{reg}

for all points t ∈ ℙ^{1}. If this is the case, we say that the fibration V/ℙ^{1} is regular. A general construction of regular Fano fibrations V/ℙ^{1} is described below in Sec. 1.2.

0.3 The main result

Theorem 1. Assume that a regular fibration V/ℙ^{1} satisfies the K^{2}-condition:

K_{V}^{2} ∉ Int A_{+}^{2}V,

where A_{+}^{2}V ⊂ A^{2}V ⊗ ℝ is the closed cone of effective cycles of codimension two. The the fibration V/ℙ^{1} is birationally superrigid.

The symbol A^{i}♯ stands, as usual, for the group of classes of codimension i cycles on the variety ♯ modulo numerical equivalence.

Corollary 0.1. (i) For a fibration V/ℙ^{1} of general position the following equality holds:

Bir V = Aut V = ℤ/2ℤ = {id, τ},

where τ ∈ Aut V is the Galois involution of the double cover V/Q.

(ii) The variety V is non-rational.

Part (i) follows from Proposition 0.1 and Theorem A. Part (ii) is obvious.

We prove Theorem 1 in a few steps. Below in Sec. 1.1 we formulate a sufficient condition of birational superrigidity for an arbitrary Fano fibration V/ℙ^{1} in terms of numerical geometry of fibers (Theorem 2). Essentially this fact was proved in [17,20], although the papers that we have just mentioned discussed Fano fibrations of a certain particular type. We will not repeat these arguments here, just making reference to [17,20].

Now to prove Theorem 1 we need to check that the fibers of the fibration V/ℙ^{1}, that is, the regular Fano hypersurfaces of index 1 (in the sense of the regularity conditions formulated below in Sec. 1.3,1.4) satisfy the conditions of Theorem 2. This verification makes our proof. It is carried out in Sections 2 and 3. Birational geometry of varieties with a pencil of Fano double covers that do not satisfy the K^{2}- condition will be studied in the next paper, the second part of the present research.

Remark. Superrigidity of Fano fibrations V/ℙ^{1}, the fibers of which are double spaces (m = 1) and double quadrics (m = 2) of index 1, is proved in [17] and [21,22], respectively, and for this reason these varieties are not considered in this paper.

0.4 Historical remarks

Investigating structures of a fibration into rationally connected (uniruled) varieties is a very old subject. The classical proof of the Noether theorem on the Cremona group of the plane, presented by Yu.I.Manin in [1], can be looked at in this way: step by step a certain pencil of rational curves on ℙ^{2} is modified, thus one goes over from one structure of a ℙ^{1}-fibration on the plane to another. Birational geometry of varieties of dimension higher than two, on which there are a lot of various structures of a rationally connected fibration, is very hard to study. Apart from a few exceptional types, these varieties are still out of reach for the modern technique. However, on rationally connected varieties which are in a certain sense general there is only one structure of a rationally connected fibration with the minimality condition that it is equivalent to a Fano fibration with the relative Picard number one. This is the very phenomenon of birational rigidity.

In the modern birational geometry birationally rigid varieties first come to the light in the papers of Yu.I.Manin in the form of del Pezzo surfaces over non-closed fields [15,16]. The first theorems on birational rigidity of non-trivial rationally connected fibrations were proved by V.A.Iskovskikh as theorems on uniqueness of a pencil of rational curves for some surfaces over non-closed fields. These theorems continue the above-mentioned work of Yu.I.Manin. The study of the absolute and relative cases in dimension two over a non-closed field prepared the basis for working in higher dimensions.

In the classical paper of V.A.Iskovskikh and Yu.I.Manin [14] the test class technique was developed that made it possible to prove (in the modern terminology) birational superrigidity of the smooth three-dimensional quartic V_{4} ⊂ ℙ^{4} (actually, in [14] birational superrigidity of the double space branched over a sextic was proved as well and also the crucial step was made in the proof of birational rigidity of the double quadric of index one [11]). After that attempts were made to use this technique to obtain similar results in the relative case for fibrations over a non-trivial base. For one class of varieties the study was successful: V.G.Sarkisov’s theorem proves that the given structure of a conic bundle is unique provided the discriminant divisor is sufficiently big [26,27]. The proof of Sarkisov’s theorem is based on the following two technical principles:

(1) the test class technique of V.A.Iskovskikh and Yu.I.Manin,

(2) the fiber-wise modifications.

The possibility of making fiber-wise modifications of conic bundles remaining at the same time in the class of smooth varieties is an exclusive property of these varieties. In a sense this special feature comes from the fact that the group of automorphisms of the fiber Aut ℙ^{1} is very big: say, for a typical Fano variety of dimension higher than two this group is finite. Thus there is no hope to use similar arguments for higher dimensional Fano fibrations.

After Sarkisov’s papers [26,27] had been published, there remained only one class of rationally connected three-folds, birational geometry of which was a terra incognita, that is, the class of fibrations into del Pezzo surfaces over ℙ^{1}. The attempts to use fiber-wise modifications similar to Sarkisov’s theorem proved unsuccessful, since immediately converted the variety under consideration into a singular one and, moreover, the acquired singularities were out of control. However, the abovementioned test class technique also refused to work. Since mid-80s and up to mid-90s attempts were made to construct at least some examples of three-dimensional del Pezzo fibrations similar to Sarkisov’s rigid conic bundles, but without any success. This activity was summed up in [12]: the only outcome of the almost decade-long work and an immense amount of completed computations were some conjectures — and no essential progress in their proof. One can see from [12] that there was no understanding why the test class technique that works so impeccably in the absolute case (the three-dimensional quartic [14]) does not allow a single step forward in the case of del Pezzo fibrations: the test class simply refused to be constructed.

The situation changed radically when the paper [17] appeared. It became immediately clear why the test class technique refused to generalize to the relative case: as it turned out, the desired class just did not exist. For the three-dimensional quartic the test class technique is equivalent to the technique of counting multiplicities introduced in [17,18,23]. However, the technique of counting multiplicities is much more flexible, since it describes properties of a certain effective cycle of codimension two (the self-intersection of the moving linear system defining the birational map under consideration), whereas the test class gives just a number, the intersection number of this cycle with the test class. In the relative case, when the base of the fibration is non-trivial, any effective cycle can be decomposed into the vertical and horizontal components. Informally speaking, each of them requires its own test class.

The methods developed in the paper [17] were later used for proving birational rigidity of big classes of higher-dimensional Fano fibrations [20-22]. In these papers (and in the present paper as well) birational rigidity is derived from the K^{2}- condition. However when the K^{2}-condition is somewhat weakened the methods of these papers still work well and make it possible to give a complete description of birational geometry of the variety under consideration. See [29,30] and the series of papers [6-8], where birational rigidity is proved for a few classes of del Pezzo fibrations over ℙ^{1}. These classes were not considered in [17] because they do not satisfy the K^{2}-condition. However, when the deviation from the K^{2}-condition grows too strong, the methods fail to work.

Note also that in spite of the progress in the general theory of factorization of birational maps between three-fold Mori fiber spaces (the Sarkisov program [4,28]), all attempts either to improve Sarkisov’s results or to prove the rationality criterion for conic bundles have been unsuccessful up to this day [13]. See the recent paper [2] on this point. We will discuss it in the next papers.

0.5 Acknowledgements

One part of the present research (corresponding to the second section of this paper) was carried out by the author during his stay at the University of Bayreuth in 2001 as a Humboldt Research Fellow. The crucial step (investigation of singularities coming from a double point on the branch divisor of a fiber, making the contents of the third section of the paper) was made during my work at Max-Planck-Institut für Mathematik in Bonn in 2003. The author is very grateful to Alexander von Humboldt Stiftung, Mathematisches Institut der Universität Bayreuth (in the first place, to Prof. Th.Peternell) and Max-Planck-Institut für Mathematik in Bonn for hospitality, the excellent conditions of work and general support.

1 The method of maximal singularities and the regularity conditions

1.1 A criterion of birational rigidity

Let π: V → ℙ^{1} be a smooth standard Fano fibration, that is, V be a smooth variety with

Pic V = ℤK_{V} ⊕ ℤF,

where F is the class of a fiber. Define the degree of a horizontal subvariety Y ⊂ V, π(Y) = ℙ^{1}, by the formula

deg Y = (Y · F · (−K_{V})^{dim Y −1}),

and the degree of a vertical subvariety Y ⊂ π^{−1}(t) by the formula

deg Y = (Y · (−K_{V})^{dim Y}).

By this definition the degree of the variety V itself coincides with the degree of a fiber, deg V = deg F.

Smooth Fano fibrations, the fibers of which are complete intersections in weighted projective spaces, satisfy also the following property: their fibers have at most isolated singularities.

We say that the Fano fibration V/ℙ^{1} satisfies

condition (v), if for any irreducible vertical subvariety Y of codimension 2, Y ⊂ π^{−1}(t) = F_{t}, and any smooth point o ∈ F_{t} the following estimate holds:

\frac{mult_{o}}{deg} Y ≤ \frac{2}{deg V} ;

condition (vs), if for any vertical subvariety Y ⊂ F_{t} of codimension 2 (with respect to V, that is, a prime divisor on F_{t}), any singular point o ∈ F_{t} and any infinitely near point x ∈ F̃_{t}, where φ: F̃_{t} → F_{t} is the blow up of the point o, φ(x) = o, Ỹ ⊂ F̃_{t} the strict transform of the subvariety Y on F̃_{t}, the following estimates hold:

\frac{mult_{o}}{deg} Y ≤ \frac{4}{deg V} , \frac{mult_{x} Ỹ}{deg Y} ≤ \frac{2}{deg V} ;

condition (h), if for any horizontal subvariety Y of codimension 2 and any point o ∈ Y the following estimate holds

\frac{mult_{o}}{deg} Y ≤ \frac{4}{deg V} .

Assume that dim V ≥ 4 and the variety V satisfies the condition

A^{2}V = ℤK_{V}^{2} ⊕ ℤH_{F} ,

where H_{F} = (−K_{V} · F) and a fiber F = F_{t} ⊂ V of general position satisfies the condition A^{2}F = ℤ(H_{F} · H_{F})_{F}. Set A_{ℝ}^{2}V = A^{2}V ⊗ ℝ ≅ ℝ^{2} and define the cone of effective cycles A_{+}^{2}V ⊂ A_{ℝ}^{2}V as the closure (in the real topology) of the set

{λΔ | λ ∈ ℝ_{+}, Δ is the class of an effective cycle}.

Definition 1.1 We say that the Fano fibration V/ℙ^{1} satisfies the K^{2}-condition, if

K_{V}^{2} ∉ Int A_{+}^{2}V.

Remark. It is easy to see that the K^{2}-condition is equivalent to the following claim: for any a ≥ 1 and b ≥ 1 the class

Δ(a, b) = aK_{V}^{2} − bH_{F}

is not effective. Indeed, H_{F} ∈ A_{+}^{2}V, so that K_{V}^{2} ∈ Int A_{+}^{2}V if and only if Δ(N, 1) ∈ A_{+}^{2}V for some N ≥ 1. This implies immediately that both conditions are equivalent.

Theorem 2. Assume that the smooth standard Fano fibration V/ℙ^{1} satisfies the K^{2}-condition and the conditions (v), (vs) and (h). Then V/ℙ^{1} is birationally superrigid.

For the proof see [17,20].

1.2 An explicit construction of the fibration V/ℙ^{1}

Let us describe an explicit construction of regular fibrations V/ℙ^{1}. For each fiber F ∈ F^{reg} (singular or smooth) the anticanonical linear system | − K_{F} | determines precisely the double cover σ_{F}: F → G ⊂ ℙ. For this reason, π_{∗}O(−K_{V}) is a locally free sheaf of rank M + 2 on ℙ^{1}. It gives a locally trivial ℙ-fibration over ℙ^{1}. The variety V is realized as a double cover of a smooth divisor Q on ℙ(π_{∗}O(−K_{V})). Namely, let

E = ⨁_{i=0}^{M+1} O_{ℙ^{1}}(a_{i})

be a locally free sheaf, normalized by the condition that

a_{0} = 0 ≤ a_{1} ≤ … ≤ a_{i} ≤ a_{i+1} ≤ … ≤ a_{M+1}.

In particular, E is generated by global sections. Set X = ℙ(E) to be its Proj in the sense of Grothendieck, π_{X}: X → ℙ^{1} the natural projection, L_{X} the tautological sheaf, Q ⊂ X a smooth divisor on X, corresponding to a section

s_{Q} ∈ H^{0} (X, L_{X}^{⊗m} ⊗ π_{X}^{∗}O_{ℙ^{1}}(a_{Q})) ,

a_{Q} ∈ ℤ_{+}. The symbol π_{Q}: Q → ℙ^{1} stands for the projection π_{X}|_{Q}. Obviously, Q/ℙ^{1} is a smooth fibration into Fano hypersurfaces of degree m in ℙ. Let W ⊂ X be an irreducible hypersurface, corresponding to a section

s_{W} ∈ H^{0} (X, L_{X}^{⊗2l} ⊗ π_{X}^{∗}O_{ℙ^{1}}(2a_{W})) ,

a_{W} ∈ ℤ_{+}, whereas W_{Q} = W ∩ Q is a smooth divisor on Q. We denote the fiber π_{Q}^{−1}(t) over a point t ∈ ℙ^{1} by the symbol G_{t} (or just G, when it is clear which point is meant or when it is inessential). Finally, set

σ: V → Q

to be the double cover, branched over W_{Q}. The natural projection onto ℙ^{1} will be denoted by π, the fiber π^{−1}(t) by the symbol F_{t} (or just F). It is easy to see that

Pic V = ℤK_{V} ⊕ ℤF

and up to twisting by an invertible sheaf O_{ℙ^{1}}(k), k ∈ ℤ, the sheaves E and π_{∗}O(−K_{V}) on ℙ^{1} coincide.

More precisely, let L_{X} ∈ Pic X be the class of the tautological sheaf L_{X}, L_{Q} = L_{X}|_{Q} its restriction to Q, so that

Pic Q = ℤL_{Q} ⊕ ℤG.

Set L_{V} = σ^{∗}L_{Q}. It is easy to see that

K_{V} = −L_{V} + (a_{1} + … + a_{M} − 2 + a_{Q} + a_{M})F.

By the Lefschetz theorem

A^{2}V = ℤK_{V}^{2} ⊕ ℤH_{F} ,

where H_{F} = (−K_{V} · F) is the class of a hyperplane section. The symbol H_{F} is used in the present paper in two different meanings: as a class of codimension two on V and as the hyperplane section of the fiber, that is, an element of A^{1}F. Every time it is clear which of the two concepts is meant.

It is easy to compute that

(K_{V}^{2} · L^{M−1}) = 2m(4 − a_{1} − … − a_{M+1} − a_{Q} − a_{W}) + 2a_{Q}.

Since (H_{F} · L^{M−1}) = 2m and the linear system |L_{V} | is free, the inequality (K_{V}^{2} · L^{M−1}) ≤ 0 implies, that K_{V}^{2} ∉ Int A_{+}^{2}V, where A_{+}^{2}V ⊂ A^{2}V ⊗ ℝ is the closed cone of effective cycles of codimension two.

1.3 The regularity conditions outside the branch divisor

Let σ: F → G ⊂ ℙ be a Fano double hypersurface of index 1, F ∈ F. The variety F is realized as a complete intersection of codimension two in the weighted projective space

ℙ(\underbrace{1, 1, …, 1,}_{M+2} l),

see [19]: F is of type m · 2l and given by the pair of equations

f̃(x_{0}, …, x_{M+1}) = 0, u^{2} = g̃(x_{0}, …, x_{M+1}),

where x_{∗} are the coordinates of weight 1, u is the coordinate of weight l, f̃ is the equation of the hypersurface G ⊂ ℙ = ℙ(1, …, 1), g̃ is the equation of the hypersurface W ∩ ℙ.

Let o ∈ F be an arbitrary point. First of all, we draw the reader’s attention to the following obvious fact:

o ≠ (\underbrace{0, 0, …, 0,}_{M+2} 1).

Thus we may assume that the point o lies in one of the standard affine charts 𝔸^{M+2} with the coordinates

z_{i} = x_{i}/x_{0}, i = 1, …, M + 1, y = u/x_{0}^{l}

and its z_{∗}-coordinates are (0, …, 0). With respect to the coordinate system (z_{∗}, y) the affine part of the variety F is given by the pair of equations

f = q_{1} + … + q_{M} = 0, y^{2} = g = w_{0} + … + w_{2l},

where q_{i} and w_{j} are homogeneous polynomials in z_{∗} of degrees i and j, respectively. Set p = σ(o) ∈ G. The point p lies on the branch divisor W if and only if w_{0} = 0. If p ∉ W, then we normalize the second equation and assume that w_{0} = 1.

Let us formulate first the regularity conditions outside the branch divisor. In this case the fiber F is given with respect to the affine coordinate system (z_{∗}, y) with the origin of the z_{∗}-system at p = σ(o) by the equations

\{ \begin{matrix}f = q_{a} + … + q_{m} = 0, \\ y^{2} = g = 1 + w_{1} + … + w_{2l}, \end{matrix}

where a ≥ 1.

Set

\sqrt{g} = (1 + w_{1} + … + w_{2l})^{1/2} = 1 + ∑_{i=1}^{∞} γ_{i}(w_{1} + … + w_{2l})^{i} =

= 1 + ∑_{i=1}^{∞} Φ_{i}(w_{1}, …, w_{2l}),

where Φ_{i}(w_{1}(z_{∗}), …, w_{2l}(z_{∗})) are homogeneous in z_{∗} of degree i ≥ 1,

γ_{i} = (−1)^{i−1} \frac{(2i − 3)!!}{2^{i}i!} = (−1)^{i−1} \frac{(2i − 3)!}{2^{2i−2}i!(i − 2)!}

is the standard i-th coefficient of the Taylor expansion of the function (1 + s)^{1/2} at the point s = 0. Obviously,

Φ_{i}(w_{∗}) = w_{i} + A_{i}(w_{1}, …, w_{i−1})

for i ≤ 2l. For i ≥ 1 set

[\sqrt{g}]_{i} = 1 + ∑_{j=1}^{i} Φ_{i}(w_{∗}), g^{(i)} = g − [\sqrt{g}]_{i}^{2}.

It is easy to see that the first non-zero component of the polynomial g^{(i)} is of degree i + 1. More precisely, this component is equal to

g_{i+1} = 2Φ_{i+1}(w_{1}(z_{∗}), …, w_{i+1}(z_{∗})).

The regularity condition at a smooth point p ∈ G (R1.1):

The sequence

q_{1}, …, q_{m}, g_{l+1}, …, g_{2l−1}

is regular in O_{p,ℙ}. Here a = 1.

The regularity condition at a double point p ∈ G (R1.2):

If 2l ≥ m + 1, then the system of M − 1 homogeneous polynomials

q_{2}, …, q_{m}, g_{l+1}, …, g_{2l−1},

whereas if 2l ≤ m, then the system of homogeneous polynomials

q_{2}, …, q_{m−1}, g_{l+1}, …, g_{2l}

defines a curve in ℙ^{M} = ℙ(T_{p}ℙ), neither component of which is contained in a hyperplane.

Furthermore, the system of M homogeneous equations

q_{2} = … = q_{m−1} = g_{l+1} = … = g_{2l} = 0  (3)

defines a non-zero subscheme Z_{∗} in ℙ^{M}, such that for any hyperplane P ⊂ ℙ^{M}

deg(P ∩ Z_{∗}) < λ_{m,l} = \frac{m!(2l − 1)!}{6(l − 1)!}

for m ≥ 4 and

deg(P ∩ Z_{∗}) < λ_{3,l} = 12\frac{(2l − 1)!}{(l + 1)!}(l − 2)

for m = 3. If the scheme Z_{∗} is reduced, then this condition means simply that any set of λ_{m,l} points is not contained in a hyperplane.

Remark. Since w_{0} = y(0) = 1, in a neighborhood of the singular point o ∈ F the equations

y − [\sqrt{g}]_{i} = 0 and σ^{∗}g^{(i)} = 0

define the same divisor. Consider the system of equations (3) on the fiber F (and not on the projectivized tangent space ℙ(T_{p}ℙ)). The system defines an effective 1-cycle C_{∗} on F. By construction, its degree is equal to

deg C_{∗} = 2m!\frac{(2l − 1)!}{(l − 1)!},

whereas its multiplicity at the point o ∈ F satisfies the estimate

mult_{o} C_{∗} ≥ m!\frac{(2l)!}{l!} = deg C_{∗},  (4)

so that what we actually have in (4) is an equality and C_{∗} is an algebraic sum of lines on F, that is, curves of the form L ∋ o, the image σ(L) ⊂ ℙ of which is a line, and moreover the morphism σ: L → σ(L) is an isomorphism. Considering the zero-dimensional scheme Z_{∗} as an effective zero-dimensional cycle, we get by construction:

Z_{∗} = ℙ(T_{o}C_{∗}).

In particular, for any hyperplane P ⊂ ℙ the one-dimensional part of the scheme

{q_{2} = … = q_{m} = g_{l+1} = … = g_{2l} = 0} ∩ σ^{−1}(P)

is of degree not higher than λ_{m,l} − 1. In other words, if all components of the cycle C_{∗} are of multiplicity 1, then no more than λ_{m,l} − 1 of these lines are contained in σ^{−1}(P).

1.4 The regularity conditions on the branch divisor

In this case the variety F is given with respect to the affine coordinate system (z_{∗}, y) by the system of equations

\{ \begin{matrix}f = q_{1} + … + q_{m} = 0, \\ y^{2} = g = w_{1} + … + w_{2l}. \end{matrix}

The regularity condition at a smooth point o ∈ F (R2.1):

the sequence of homogeneous polynomials

q_{1}, …, q_{m}

is regular in O_{p,ℙ} and the quadratic form q_{2} does not vanish identically on the plane {q_{1} = w_{1} = 0}.

Note that since the point o ∈ F is smooth, this plane is of codimension exactly two, that is, the linear forms q_{1} and w_{1} are linearly independent: the plane {q_{1} = w_{1} = 0} is the tangent plane to the branch divisor W ∩ G of the morphism σ_{F}.

The regularity condition at a double point o ∈ F (R2.2):

In this case we have the double cover σ_{F}: F → G, branched over the divisor W_{G} = W ∩ G. The first regularity condition is smoothness of the hypersurface G at the point p = σ(o), that is, q_{1} ≠ 0. Furthermore, the divisor W_{G} should have at the point p a non-degenerate quadratic singularity:

w_{1} = λq_{1},

λ ∈ ℂ. For convenience of notations assume that q_{1} = z_{M+1}. The quadratic polynomial

w̄_{2} = w_{2}|_{{z_{M+1}=0}}

is of the maximal rank. Let E_{G} ≅ ℙ^{M−1} be the exceptional divisor of the blow up φ_{G}: G̃ → G of the point p. Take z_{1}, …, z_{M} for homogeneous coordinates on E_{G} and set

W_{E} = {w̄_{2} = 0}.

It is a non-singular quadratic hypersurface in E_{G}. Denote by the symbol q̄_{i} the restriction of the homogeneous polynomial q_{i} onto the hyperplane q_{M+1} = 0. Now the remaining part of the condition (R2.2) looks as follows:

the system of homogeneous equations

q̄_{2} = … = q̄_{m} = 0

defines in E_{G} ≅ ℙ_{(z_{1}:...:z_{M})}^{M−1} an irreducible subvariety Z_{2·...·m}, which is an irreducible reduced complete intersection of codimension (m − 1). The quadric

q̄_{2} = 0

is smooth and distinct from W_{E}.

Definition 1.2. A Fano double hypersurface F ∈ F is regular, if each smooth point on it is regular in the sense of the corresponding condition (R1.1) or (R2.1) and each of its singular points is regular in the sense of the corresponding condition (R1.2) or (R2.2). Notation: F ∈ F^{reg}.

The conditions (R1.1) and (R2.1) coincide with the regularity conditions of the paper [19] (Definitions 1 and 2 in Sec. 1.3). In [19, Sec. 4.3] it was shown that non-regular smooth double spaces form a closed subset of codimension at least two in the set of all smooth double hypersurfaces F_{sm}. Moreover, it follows from the computations of Sec. 4.3 in [19] that the set of Fano double hypersurfaces F with at least one smooth non-regular point o ∈ F is of codimension at least two in F. Thus a general singular double hypersurface F ∈ F_{sing} has exactly one singular point whereas all its smooth points are regular. The singular point o ∈ F is a nondegenerate double point. If p = σ(o) ∉ W_{G}, then the fact that the condition (R1.2) is open implies that in a neighborhood of F ∈ F the following estimate holds

codim_{F_{sing}}(F_{sing} \ F_{sing}^{reg}) ≥ 1  (5)

and thus

codim_{F}(F_{sing} \ F_{sing}^{reg}) ≥ 2.  (6)

If p = σ(o) ∈ W_{G}, then in a similar way the fact that the condition (R2.2) is open implies the estimate (5) in a neighborhood of F ∈ F. Thus the estimates (5) and (6) are global.

1.5 Start of the proof of Theorem 1

Let us check that the regular fibration V/ℙ^{1} satisfies the conditions (v) and (h). Assume that the opposite inequality holds:

\frac{mult_{o}}{deg} Y > \frac{2}{deg V} ,

where o ∈ F = F_{t} is a smooth point, Y ⊂ F is a prime divisor. Let

T = σ^{−1}(T_{p}G ∩ G)

be the tangent divisor, p = σ(o). By the regularity conditions, mult_{o} T = 2. Since T ⊂ F is a hyperplane section, we get deg T = deg V, so that

\frac{mult_{o}}{deg} T = \frac{2}{deg V}

and thus Y ≠ T. Both subvarieties Y, T are irreducible, so that the intersection Y ∩ T is of codimension two with respect to F and the effective cycle Z = (Y ∘ T) of the scheme-theoretic intersection of Y and T is well defined. Obviously, the cycle Z satisfies the inequality

\frac{mult_{o}}{deg} Z > \frac{4}{deg V} .  (7)

However it was proved in [19], Sec. 3, that for a regular point o ∈ F it is impossible. This proves the condition (v).

Let us prove that the condition (h) holds. To begin with, let us consider first the smooth case, where o ∈ F is a smooth point. Assume that an irreducible horizontal subvariety Y ⊂ V of codimension two satisfies the inequality

\frac{mult_{o}}{deg} Y > \frac{4}{deg V} .

Since π(Y) = ℙ^{1}, we get Y ≠ F, so that Z = (Y ∘ F) is an effective cycle of codimension two on the fiber F, satisfying the inequality (7). As it was pointed out above, this is impossible. The condition (h) is proved in the smooth case.

Now let o ∈ F be a double point. Arguing in the same way as in the smooth case, let us construct the effective cycle Z = (Y ∘F) of codimension two on the fiber

F. Since \frac{mult_{o}}{deg} F = 2, the cycle Z satisfies the inequality

\frac{mult_{o}}{deg} Z > \frac{8}{deg V} .

Let us show that this is impossible. Without loss of generality assume that Z ⊂ F is an irreducible subvariety of codimension two. Its image on G satisfies the estimate

\frac{mult_{p}}{deg} σ(Z) > \frac{4}{deg G}.

Now if p ∈ G is a smooth point, then the arguments of the paper [20] (they work without any modifications for an arbitrary degree deg G ≤ dim G + 1) show that this is impossible. If p ∈ G is a double point, then by the condition (R1.2) the homogeneous polynomials q_{2}, …, q_{m} make a regular sequence, so that the standard arguments of [20] give a contradiction once again (see Sec. 3.1 in [20]).

This completes the proof of the condition (h).

2 Singularity of a fiber outside the branch divisor

2.1 Hypertangent divisors and linear systems

Let φ = φ_{F,o}: F̃ → F be the blow up of the fiber at an arbitrary point o, φ_{G} = φ_{G,p}: G̃ → G the blow up of the fiber G at the point p = σ(o), E = E_{F} ⊂ F̃ and E_{G} ⊂ G̃ the exceptional divisors.

Definition 2.1. The linear system

φ_{∗}(|kH_{F} − (k + 1)E|)

of divisors on F (respectively, the linear system

(φ_{G})_{∗}(|kH_{G} − (k + 1)E_{G}|)

of divisors on G) is called the k-th hypertangent linear system and denoted by the symbol Λ_{k} = Λ_{k}^{F} (respectively, Λ_{k}^{G}).

One can say that Λ_{k} is the largest linear subsystem of the system |kH_{F} |, the strict transform of which satisfies the property

Λ̃_{k} ⊂ |kH_{F} − (k + 1)E|,

and similarly for G. In the general case one cannot assert that

σ^{∗}Λ_{k}^{G} ⊂ Λ_{k},  (8)

since if p ∈ W_{G} is a smooth point of the branch divisor, then the double cover σ: F → G does not extend to a double cover F̃ → G̃ (there is a rational map of degree two between these varieties; this rational map has a fairly simple structure, however it is not a finite morphism). But if p ∉ W_{G} or p ∈ W_{G} is a double point of the branch divisor, then the inclusion (8) holds.

The symbol

Λ_{k}^{E}

stands for the corresponding linear system on the exceptional divisor:

Λ_{k}^{E} = Λ̃_{k}|_{E} or Λ_{k}^{E} = Λ̃_{k}^{G}|_{E_{G}},

depending on the context. It is easy to see that

({ \atop Bs Λ_{k}} ∘ E) = Bs Λ_{k}^{E}

in the scheme-theoretic sense, in particular, the corresponding effective algebraic cycles are equal, that is, the equality respects multiplicities.

Abusing out notations, we sometimes use the notion of a hypertangent system for a certain special subsystem of the hypertangent system, which permits an explicit description. In practice it is these special subsystems that we use. Let p ∈ G be a point, z_{1}, …, z_{M+1} a system of linear coordinates with the origin at p, and assume that the hypersurface G is given by the equation

f = q_{a} + q_{a+1} + … + q_{m} = 0,

a = 1 or 2. Then

Λ_{k}^{G} ⊃ |∑_{i=a}^{k} s_{k−i}f_{i}| ,  (9)

where

f_{i} = q_{a} + … + q_{i},

k ≥ a and s_{j} means an arbitrary homogeneous polynomial of degree j in the variables z_{∗}. The inclusion (9) is obvious, since

f_{i}|_{G} = (−q_{i+1} − … − q_{m})|_{G}.

Now assume that p ∉ W_{G}. Let us construct the hypertangent system Λ_{k}. Obviously, Λ_{k} ⊃ σ^{∗}Λ_{k}^{G}, but in fact the system Λ_{k} is much larger. Following [19,22,25], let us describe the construction of hypertangent divisors, associated with the double cover σ. Since p ∉ W_{G}, we may assume that the hypersurface

W_{t} = W ∩ ℙ_{t} ⊂ ℙ

is given by the equation

g(z_{∗}) = 1 + w_{1} + … + w_{2l} = 0,

w_{i}(z_{∗}) are homogeneous of degree i. Setting formally

\sqrt{g} = 1 + ∑_{i=1}^{∞} Φ_{i}(w_{1}, …, w_{2l}),  (10)

where Φ_{i}(w_{1}(z_{∗}), …, w_{2l}(z_{∗})) are homogeneous polynomials of degree i in z_{∗}, write for j ≥ 1

[\sqrt{g}]_{j} = 1 + ∑_{i=1}^{j} Φ_{i}(w_{∗}(z_{∗})).

Now we get

Λ_{k} ⊃ |∑_{i=a}^{k} s_{k−i}f_{i} + ∑_{i=l}^{min\{k,2l−1\}} s_{k−i}^{∗}(y − [\sqrt{g}]_{i})| ,  (11)

where s_{k−i}^{∗} are homogeneous polynomials in z_{∗} of degree k − i; if k ≤ l − 1, then the right-hand side is assumed to be equal to zero. The inclusion (11) follows from (9) and the following fact.

Lemma 2.1. In the local coordinates z_{∗} we get

(y − [\sqrt{g}]_{i})|_{F} = 2Φ_{i+1}(w_{∗}(z_{∗}))|_{F} + …,

where the dots stand for a formal series, the components of which are homogeneous polynomials of degree i + 2 and higher in the variables z_{∗}.

Proof: it is obvious, since (y^{2} −g)|_{F} ≡ 0, g(p) = 1 and the formal decomposition (10) holds.

Note that

Φ_{i}(w_{∗}) = \frac{1}{2}w_{i} + A_{i}(w_{1}, …, w_{i−1}).

Now let us consider the case when p = σ(o) ∈ W_{G}. If the branch divisor is nonsingular at the point p, then the local equation of the hypersurface W_{t} is of the form

g(z_{∗}) = w_{1} + … + w_{2l} = 0,

where the linear forms q_{1}, w_{1} are linearly independent. Since the inverse image of the divisor

{w_{1}|_{G} = 0}

on F is obviously singular, we obtain:

Λ_{k} ⊃ |∑_{i=1}^{k} s_{k−i}f_{i} + s_{k−1}w_{1}| .  (12)

However, if p = σ(o) is a singularity of the divisor W_{G}, then our methods of constructing hypertangent linear systems give at most the inclusion Λ_{k} ⊃ σ^{∗}Λ_{k}^{G}, where Λ_{k}^{G} is given by the formula (9).

The regularity conditions make it possible to get a lower bound for the codimension of the base set of hypertangent systems. In the formulae below it is assumed that the segment [a, b] ⊂ ℝ is an empty set when b < a. For an arbitrary point o ∈ F set

M = [a, m − 1] ∩ ℤ_{+} = {a, …, m − 1},

where a = mult_{p} G ∈ {1, 2}, p = σ(o), and

L = [l, 2l + a − 3] ∩ ℤ_{+} = {l, …, 2l + a − 3}.

Thus the sets M, L depend on the type of the point o. At each stage of the proof the point o is assumed to be fixed and the symbols M, L mean the sets corresponding to this point.

For e = max{m − 1, 2l − 1} we denote the hypertangent linear system Λ_{e} by the symbol Λ_{∞}.

Proposition 2.1. The following estimates hold:

(i) if p = σ(o) ∉ W_{G} is a smooth point of the hypersurface G, then

codim_{o} Bs Λ_{k} ≥ codim_{E} Bs Λ_{k}^{E} ≥ ♯[1, k] ∩ M + ♯[1, k] ∩ L,

in particular,

dim_{o} Bs Λ_{∞} ≤ 1,

(ii) if p = σ(o) ∉ W_{G} is a double point of the hypersurface G, then

codim Bs Λ_{k} ≥ codim_{E} Bs Λ_{k}^{E} ≥ ♯[2, k] ∩ M + ♯[2, k] ∩ L,

and moreover

Bs Λ_{∞} ⊂ C_{∗}

(see Sec. 1.3). More to that, let P ⊂ ℙ, P ∋ p, be an arbitrary hyperplane, P_{F} = σ^{−1}(P ∩ G) the corresponding section of the fiber F, Λ_{k}^{P} = Λ_{k}|_{P_{F}} the restriction of the linear system Λ_{k} onto P_{F}. Then for k ≤ max{m, 2l} − 2

codim_{P_{F}} Bs Λ_{k}^{P} ≥ ♯[2, k] ∩ M + ♯[2, k] ∩ L,

and

dim Bs Λ_{∞}^{P} ≤ 1,  (13)

whereas if in (13) the equality holds then the degree of the one-dimensional part of the basic subscheme Bs Λ_{∞}^{P} does not exceed λ_{m,l}.

(iii) If p = σ(o) ∈ W_{G} is a smooth point on the branch divisor W_{G}, then the following inequality holds:

codim_{o} Bs Λ_{k} ≥ codim_{E} Bs Λ_{k}^{E} ≥ ♯[1, k] ∩ M + 1,

(iv) if p = σ(o) ∈ W_{G} is a double point on the branch divisor W_{G}, then the following inequality holds:

codim_{o} Bs Λ_{k} ≥ codim_{E} Bs Λ_{k}^{E} ≥ ♯[1, k] ∩ M.

Proof. To obtain out claims, we replace the hypertangent linear systems Λ_{k} by their subsystems (9), (11) and (12), constructed above, and use the regularity conditions (Sec. 1.3,1.4). Q.E.D.

2.2 Scheme of the proof of the condition (vs)

Assume that there exists a prime divisor Y ⊂ F = F_{t}, satisfying the estimate

\frac{mult_{x} Ỹ}{deg Y} > \frac{1}{m},  (14)

where x ∈ E is an infinitely near point of the first order, that is, E ⊂ F̃ is the exceptional divisor of the blow up of the point o ∈ F, φ: F̃ → F. Here the singular point o ∈ F is generated by a singularity of the hypersurface G = G_{t}, that is, p = σ(o) ∈ G is a non-degenerate double point, p ∉ W. Set σ^{−1}(p) = {o, o^{+}} and let φ_{G}: G̃ → G be the blow up of the point p. The map σ extends in an obvious way to a morphism

σ̃: F̃ \ {o^{+}} → G̃,

whereas on the exceptional divisor E ⊂ F̃ the morphism σ̃ is an isomorphism, which makes it possible to identify E with the exceptional divisor of the blow up φ_{G} and thus consider E as embedded in 𝕋 = ℙ(T_{p}ℙ) ≅ ℙ^{M}, that is, in the exceptional divisor of the blow up

φ_{ℙ}: ℙ̃ → ℙ

of the point p ∈ ℙ. Depending on the context one of the inclusions E ⊂ F̃ or E ⊂ G̃ will be meant.

Let us show that the assumption (14) leads to a contradictions. In order to do that, we will use the method developed in [20]. The arguments break into a few steps. The first step is given by

Proposition 2.2. There exists a hyperplane P ⊂ ℙ, P ∋ p, such that σ(Y) ⊄ P and the effective algebraic cycle Y_{P} = (Y ∘_{F} P_{F}), where P_{F} = σ^{−1}(P_{G}), P_{G} = P ∩ G is a hyperplane section, satisfies the estimate

\frac{mult_{o}}{deg} Y_{P} > \frac{3}{2m}.

The symbol ∘_{F} is used to emphasize that the cycle Y_{P} is constructed in the sense of the intersection theory on F, and not on V. For a proof of the proposition see [20].

Step two. Consider the variety P_{F} ⊂ F. It is an irreducible variety of dimension M − 1 with the double point o ∈ P_{F}. Let φ_{P}: P̃ → P_{F} be the blow up of the point o, E_{P} ⊂ P̃ the exceptional divisor. Obviously, P̃ embeds into F̃, and E_{P} into E as a hyperplane section of the quadric E with respect to the embedding E ↪ 𝕋. Since the variety F̃ is factorial, the strict transform Ỹ is a Cartier divisor. Therefore, the effective cycle Ỹ_{P} = (Ỹ ∘ P̃), that is, the strict transform of the cycle Y_{P} on P̃, is a Cartier divisor,

Ỹ_{P} ∼ aH_{P} − bE_{P} ,

where H_{P} is the class of a hyperplane section. By Proposition 2.2

b > \frac{3}{2}a.

By the regularity condition we get for the tangent divisor

T = σ^{−1}(T_{p}G ∩ G)

that mult_{o} T = 6, deg T = 4m, so that for the class of its strict transform T̃ ⊂ F̃ we get T̃ ∼ 2H − 3E and thus for its restriction T̃_{P} = T̃ ∩ P̃ on P̃ we get

T̃_{P} ∼ 2H_{P} − 3E_{P} .

Proposition 2.3. Let Z ∼ αH_{P} − βE_{P} be an effective Cartier divisor on P̃. Assume that β > \frac{3}{2}α. Then Z contains T̃_{P} as a component of positive multiplicity.

Proof is given below.

Step three. Write down the effective divisor Ỹ_{P} in the following form:

Ỹ_{P} = cT̃_{P} + Z,

where c ∈ ℤ_{+} and the effective divisor Z does not contain T̃_{P} as a component. Setting Z ∼ αH_{P} − βE_{P}, we obtain from the system of equations

a = 2c + α, b = 3c + β

and the condition 2b > 3a, that

β > \frac{3}{2}α.

By Proposition 2.3 this implies that T̃_{P} is a component of positive multiplicity of the divisor Z. A contradiction.

Thus we have proved that the estimate (14) is impossible which implies that the condition (vs) holds for the case of a singular point o ∈ F outside the branch divisor. Q.E.D.

2.3 Movable families of curves

Let us prove Proposition 2.3. We use the method of the paper [20].

Lemma 2.2. The divisor T_{P} = T ∩ P_{F} is swept out by a family of curves {C_{δ}, δ ∈ Δ}, the general member of which is irreducible and satisfies the inequality

\frac{mult_{o}}{deg} C_{δ} > \frac{2}{3}.  (15)

First of all, let us obtain Proposition 2.3 from this fact. Let {C̃_{δ}, δ ∈ Δ} be the strict transform of this family of curves on P̃, T̃_{P} ⊂ P̃ the strict transform of the divisor T_{P}. Obviously,

(Z · C̃_{δ}) = α deg C_{δ} − β mult_{o} C_{δ} < 0,

since β > \frac{3}{2}α. Therefore C̃_{δ} ⊂ Z. However the curves C̃_{δ} sweep out T̃_{P}, thus Z ⊃ T̃_{P}. Q.E.D. for Proposition 2.3.

Proof of Lemma 2.2. The variety P_{F} is of dimension m + l − 2, the divisor T_{P} ⊂ P_{F} is of dimension m + l − 3. We construct the required family of curves (C_{δ}, δ ∈ Δ), intersecting T_{P} with m + l − 4 hypertangent divisors. To order the construction procedure, let us introduce some new notations:

M = {2, …, m − 1}, L = {l, …, 2l − 1},
c_{e} = ♯[4, e] ∩ M + ♯[3, e] ∩ L, e ∈ ℤ_{+}.  (16)

Here and below we assume silently that the segment [a, b] ⊂ ℝ is the empty set when b < a. For e ≤ 2 we get c_{e} = 0, for e ≥ max{m, 2l} − 1 we get that c_{e} = m + l − 4, provided that l ≥ 3. Let us assume that this is the case and that m ≥ 4. Note that for m = 4

♯[4, e] ∩ M = 0,

since this set is empty. The cases l = 2 and m = 3 we will treat separately. Obviously,

c_{e+1} ≥ c_{e}.

Define the ordering function

χ: {1, …, m + l − 4} → ℤ_{+}

by the formula

χ([c_{e−1} + 1, c_{e}] ∩ ℤ_{+}) = e.  (17)

In accordance with our remark above, if c_{e−1} = c_{e}, then the formula (17) is meaningless, since the set [c_{e−1} + 1, c_{e}] is empty. Note that

c_{e+1} − c_{e} ∈ {0, 1, 2}

by the definition (16). It is easy to check that (17) gives a correct definition of an integer-valued function χ.

Denote by the symbol Λ_{i}^{P} the restriction of the hypertangent system Λ_{i} onto P_{F}. Set

Λ^{P} = ∏_{i=1}^{m+l−4} Λ_{χ(i)}^{P}.

Note that in this product the hypertangent system Λ_{e} can appear at most twice, see (16). Let

𝔻 = {D_{i} ∈ Λ_{χ(i)}^{P}, i = 1, …, m + l − 4} ∈ Λ^{P}

be a general set of hypertangent divisors.

Definition 2.2. We say that a family of closed algebraic sets (Γ_{u}, u ∈ U) of (co)dimension i on an algebraic variety Z is a dense movable family if for a general u ∈ U all irreducible components of the set Γ_{u} are of (co)dimension i and these components form a family of irreducible algebraic varieties sweeping out Z.

Lemma 2.3. For i = 1, …, m + l − 4 the closed algebraic set

R_{i}(𝔻) = ⋂_{j=1}^{i} D_{i} ∩ T_{P}

is for a general 𝔻 ∈ Λ^{P} of codimension i in T_{P}. For i = 1, …, m + l − 5 the family of cycles

(R_{i}(𝔻), 𝔻 ∈ Λ^{P})

is a dense movable family of cycles of codimension i on T.

Proof. Set R_{0}(𝔻) = T and argue by induction on i = 1, …, m + l − 4. Assume that the claim of the lemma is proved for i ≤ j ≤ m + l − 5 (if j = 0, then there is nothing to prove). Set χ(j + 1) = e. By definition,

R_{j+1}(𝔻) = R_{j}(𝔻) ∩ D_{j+1},

where D_{j+1} ∈ Λ_{e}^{P} is a general divisor. By definition of the function χ we get

j + 1 ∈ [c_{e−1} + 1, c_{e}].

By Proposition 2.1, the following inequality holds:

codim_{P_{F}} Bs Λ_{e}^{P} ≥ c_{e} + 1,  (18)

so that

codim_{T_{P}} Bs Λ_{e}^{P} |_{T_{P}} ≥ c_{e},

whereas

codim_{T_{P}} R_{j}(𝔻) = j ≤ c_{e} − 1

by (16), (17). Therefore, neither of the irreducible components of the closed subset R_{j}(𝔻) is contained in the base set of the hypertangent system Λ_{e}^{P}. In particular,

R_{j}(𝔻) ⊄ D_{j+1}

and therefore R_{j+1}(𝔻) is a closed subset of pure codimension j + 1 in T_{P}, which proves the first claim of the lemma. Now assume that j ≤ m + l − 6. Then either

e ≤ max{m, 2l} − 2,

so that by Proposition 2.1 we get the estimate

codim_{P_{F}} Bs Λ_{e}^{P} ≥ c_{e} + 2,

which is stronger than the inequality (18), or e = max{m, 2l} − 1, but in this case c_{e} = c_{e−1} + 2, since

j + 2 ∈ [c_{e−1} + 1, c_{e}],

so that

codim_{T_{P}} R_{j}(𝔻) = j = c_{e} − 2.

In any case for each irreducible component Z of the set R_{j}(𝔻) for j ≤ m + l − 6 we get

codim_{Z} Bs(Λ_{e}^{P} |_{Z}) ≥ 2,

so that the linear system Λ_{e}^{P} |_{Z} is movable. This proves the second claim of Lemma 2.3.

Consider the family of closed one-dimensional sets

(R(𝔻) = R_{m+l−4}(𝔻), 𝔻 ∈ Λ^{P}).

We can no longer assert that irreducible components of the set R(𝔻) form a movable family of curves: at the last step, that is, when we make curves from surfaces, some fixed components can appear. However, in any case the following decomposition holds:

R(𝔻) = (T_{P} ∘ D_{1} ∘ … ∘ D_{m+l−4}) = ∑_{δ_{i}∈Δ} C_{δ_{i}} + Φ,  (19)

where (C_{δ}, δ ∈ Δ) is a movable family of curves, Φ an effective 1-cycle, that is, the fixed part of the family of curves R(𝔻), 𝔻 ∈ Λ^{P}. We get the equality of 1-cycles

Φ = Bs Λ_{∞}^{P}.

The family (C_{δ}, δ ∈ Δ) sweeps out T_{F}, if it is non-empty. However, by construction

deg R(𝔻) = 4m ∏_{j=1}^{m+l−4} χ(j) = 4m (∏_{j=4}^{m−1} j) (∏_{j=l}^{2l−1} j) =

= \frac{2m!(2l − 1)!}{3(l − 1)!},

whereas by the regularity condition

deg Φ < λ_{m,l} = \frac{m!(2l − 1)!}{6(l − 1)!} < deg R(𝔻).

Therefore, the family of irreducible curves (C_{δ}, δ ∈ Δ) is non-empty and sweeps out the divisor T.

Let us, finally, estimate the ratio mult_{o} / deg for a general curve C_{δ}. As we mentioned above, mult_{o} Φ = deg Φ (see Sec. 1.3). Besides, for a general set 𝔻 ∈ Λ^{P} the ratio

\frac{mult_{o}}{deg} C_{δ_{i}}

(in the sense of the formula (19)) does not depend on i. Consequently,

\frac{mult_{o}}{deg} C_{δ} = \frac{mult_{o} R(𝔻) − deg Φ}{deg R(𝔻) − deg Φ}.

However, by construction

mult_{o} R(𝔻) ≥ 6 ∏_{j=1}^{m+l−4}(χ(j) + 1) = 6 (∏_{j=5}^{m} j) ( ∏_{j=l+1}^{2l} j) =

= \frac{m!}{4} · \frac{(2l)!}{l!},

whence

\frac{mult_{o}}{deg} C_{δ} ≥ \frac{\frac{m!}{4} · \frac{(2l)!}{l!} − λ_{m,l}}{\frac{2m!}{3} · \frac{(2l − 1)!}{(l − 1)!} − λ_{m,l}} > \frac{2}{3}

in accordance with the choice of the number λ_{m,l}. This proves the lemma for m ≥ 5, l ≥ 3.

If l = 2, then the arguments presented above work with the only modification: instead of (16) one should use the formula

c_{e} = ♯[3, e] ∩ M + ♯[3, e] ∩ L,

e ∈ ℤ_{+}. In this case an independent hypertangent divisor adds into the linear system Λ_{2}^{P} and thus the codimension of its base set (and the codimension of the base set of all the subsequent hypertangent systems Λ_{j}^{P}, j ≥ 3) exceeds by one the corresponding codimension in the just considered case l ≥ 3. It is this fact that makes it possible to change the definition of the number c_{e} and accordingly shift by one the function χ. The rest of the arguments are completely similar to the case l ≥ 3 discussed above.

The case m ≥ 3 is slightly harder. In order to obtain the needed codimension of the base set of a hypertangent system, one should use the following set of hypertangent divisors:

𝔻 = {D_{i} ∈ Λ_{l+i}^{P} | i = 1, …, l − 1}.

We draw the reader’s attention to the fact that the first divisor in this set is taken from the linear system Λ_{l+1}^{P}, that is, in contrast to the case m ≥ 4, which we considered above, we skip the system Λ_{l}^{P}. As a result, we obtain once again a movable family of closed algebraic sets

R_{k}(𝔻) = (⋂_{j=1}^{k} D_{j}) ∩ T

for k ≤ 2l − 2, whereas irreducible components of the sets R_{k}(𝔻) form a family and sweep out T. Again we modify the family of curves R_{2l−1}(𝔻), deleting the fixed

part Φ of degree deg Φ < λ_{3,l} = 12(l − 2)\frac{(2l − 1)!}{(l + 1)!} and obtain a family of irreducible

curves (C_{δ}, δ ∈ Δ), sweeping out T and satisfying the estimate (15). Proof of Lemma 2.2 is now complete.

3 Singularity of a fiber on the branch divisor

3.1 Notations and discussion of the regularity condition

We have the double cover

F = F_{t} →^{σ} G = G_{t} ⊂ ℙ = ℙ^{M+1},

G ⊂ ℙ is a smooth hypersurface of degree m ≤ M − 1. At the point p ∈ G the branch hypersurface W_{G} = W ∩ G has an isolated quadratic singularity, so that o = σ^{−1}(p) ∈ F is an (isolated) non-degenerate double point of the fiber F. We get the following commutative diagram of maps

E = E_{F} ⊂ F̃ →^{σ̃} G̃ ⊃ E_{G}
φ_{F} ↓  ↓ φ_{G}
F →^{σ} G,

where φ_{F} and φ_{G} are the blow ups of the points o ∈ F and p ∈ G, respectively, E_{F} and E_{G} are the exceptional divisors, σ̃ the double cover, branched over W̃_{G} ⊂ G̃, that is, over the strict transform of the divisor W_{G}. Besides,

σ̃_{E} = σ̃|_{E}: E → E_{G} ≅ ℙ^{M−1}

is the double cover, branched over the quadric

W_{E} = W̃_{G} ∩ E_{G}.

The symbol H_{E} stands for the hyperplane section of the quadric E with respect to the standard embedding E ↪ ℙ^{M}, Pic E = ℤH_{E}.

Let W_{t} = W ∩ ℙ_{t} be given by the equation

h = w_{1} + w_{2} + … + w_{2l} = 0,

and G by the equation

f = q_{1} + q_{2} + … + q_{m} = 0

with respect to the affine coordinates z_{∗} = (z_{1}, …, z_{M+1}) with the origin at the point p. The divisor W_{G} has at p a non-degenerate quadratic singularity, so that w_{1} = λq_{1}, for simplicity of notations assume that q_{1} = z_{M+1}. The quadratic polynomial w̄_{2} = w_{2}|_{{z_{M+1}=0}} is of the maximal rank. Take z_{1}, …, z_{M} for homogeneous coordinates on E_{G}, then

σ̃_{E}: E → E_{G} ≅ ℙ^{M−1}

is branched over the non-singular quadric W_{E} = {w̄_{2} = 0}. For an arbitrary point y ∈ E_{G} \ W_{E} let C(y) ⊂ E_{G} be the cone consisting of all lines L ⊂ E_{G} that contain y and touch W_{E}. More formally, let

π_{y}: E_{G} \ {y} → ℙ^{M−2}

be the projection from the point y. Its restriction onto the quadric W_{E},

π_{y}|_{W_{E}}: W_{E} → ℙ^{M−2}

is a double cover, branched over a quadric Q(y) ⊂ ℙ^{M−2}. Now

C(y) = \overline{π_{y}^{−1}(Q(y))}.

Obviously, C(y) is a quadric cone with the vertex at the point y. Since the quadric W_{E} is non-singular, the cone C(y) has only one singularity, that is, the point y.

Denote the restriction of the polynomial q_{i} onto the hyperplane q_{M+1} = 0 by the symbol q̄_{i}. By the regularity condition, the system of homogeneous equations

q̄_{2} = … = q̄_{m} = 0

defines in E_{G} an irreducible reduced complete intersection of codimension (m−1), an irreducible subvariety Z_{2·...·m}. Moreover, the quadric q̄_{2} = 0 is smooth and distinct from W_{E}.

Lemma 3.1. Assume that the condition (R2.2) holds. Then the subvariety Z_{2·...·m} is not contained in a quadric cone C(y), y ∈ E_{G}\W_{E}, and in a tangent plane T_{y}W_{E}, y ∈ W_{E}.

Proof. Set

Z_{2·...·j} = {z ∈ ℙ^{M−1} | q̄_{2} = … = q̄_{j} = 0}.

It is easy to see that Z_{2·...·j} is an irreducible reduced complete intersection of codimension j. From the long exact cohomology sequence we obtain that

h^{0}(O_{Z_{2}}(2)) = … = h^{0}(O_{Z_{2·...·j}}(2)) = … = h^{0}(O_{Z_{2·...·m}}(2)),

and moreover, the restriction map

H^{0}(O_{ℙ^{M−1}}(2)) → H^{0}(O_{Z_{2·...·m}}(2))

is surjective. This implies that Z_{2·...·m} is contained in one and only one quadric Z_{2} and thus is not contained in any quadric cone C(y), y ∈ E_{G} \ W_{E}. In a similar way, the restriction map

H^{0}(O_{ℙ^{M−1}}(1)) → H^{0}(O_{Z_{2·...·m}}(1))

is an isomorphism, so that Z_{2·...·m} is not contained in a hyperplane, in particular, in a hyperplane of the form T_{y}W_{E}, y ∈ W_{E}.

Now fix a prime divisor R ⊂ F, and let R̃ ⊂ F̃ be its strict transform. Fix also an arbitrary point x ∈ E, lying outside the branch divisor of the cover σ̃_{E}, that is, σ̃(x) ∉ W_{E}. (For a point x ∈ E on the branch divisor the arguments given below work automatically with simplifications. The arguments of Sec. 2 can be also used in this case, in contrast to the situation outside the branch divisor W_{E}.)

Proposition 3.1. The following estimate holds:

μ = mult_{x} R̃ ≤ \frac{1}{m} deg R.  (20)

Remark. For some k ≥ 1 we have R ∼ H_{F}, where H_{F} = σ^{∗}H_{G} is a hyperplane section. Since obviously deg R = 2mk, the estimate (20) takes the form of the following inequality:

μ ≤ 2k.

3.2 Start of the proof of the condition (vs)

Assume the converse: μ > 2k. We have the presentation

R̃ ∼ kφ_{F}^{∗} H_{F} − νE,

whereas mult_{o} R = 2ν.

Lemma 3.2. The following inequality holds: ν ≤ 2k.

Proof. Assume the converse: ν > 2k. Then

\frac{mult_{o}}{deg} R > \frac{2}{m}.

Set R̄ = σ(R) ⊂ G. It is a prime divisor on the smooth hypersurface G ⊂ ℙ. Since σ: R → R̄ is a finite morphism, we get the inequality

\frac{mult_{o}}{deg} R̄ > \frac{2}{m}.

However, this is impossible, since p ∈ G is a regular point. Indeed, the tangent divisor T_{1}^{+} = T_{p}G ∩ G satisfies the equality

\frac{mult_{p}}{deg} T_{1}^{+} = \frac{2}{m},

so that R̄ ≠ T_{1}^{+} and (R̄∘T_{1}^{+}) is an effective cycle of codimension two (T_{1}^{+} is obviously irreducible).

Since mult_{p} T_{1}^{+} = 2, we get the inequality

mult_{p}(R̄ ∘ T_{1}^{+}) ≥ 2 mult_{p} R̄.

Taking into account that deg(R̄ ∘ T_{1}^{+}) = deg R̄, we conclude that there exists an irreducible component Y_{2} of the cycle (R̄ ∘ T_{1}^{+}), satisfying the estimate

\frac{mult_{p}}{deg} Y_{2} ≥ 2\frac{mult_{p}}{deg} R̄.

As usual, let f = q_{1} + q_{2} + … + q_{m} be the polynomial defining the hypersurface G with respect to the coordinate system z_{∗} with the origin at the point p. Setting

f = q_{1} + q_{2} + … + q_{i},

let us construct the hypertangent systems

Λ_{i}^{G} = |∑_{j=1}^{i} f_{j}s_{i−j}|_{G} = 0| ,

and consider the standard hypertangent divisors

T_{i}^{+} = {f_{i}|_{G} = 0} ∈ Λ_{i}^{G}.

Set

T_{i} = σ^{∗}T_{i}^{+}, Λ_{i} = σ^{∗}Λ_{i}^{G}.

These divisors and linear systems will be of crucial importance below. At the moment, note that by the regularity condition we get

codim_{G} Bs Λ_{i}^{G} = i

(in fact, Bs Λ_{i}^{G} = T_{1}^{+} ∩ … ∩ T_{i}^{+}). Let

𝔻 = (D_{1}, …, D_{m−1}) ∈ ∏_{j=1}^{m−1} Λ_{i}^{G}

be a general set of divisors. Let us construct by induction a sequence of irreducible subvarieties Y_{i}, i = 1, …, m − 1, satisfying the following properties:

(i) Y_{1} = R̄, Y_{2} was constructed above, codim_{G} Y_{i} = i;

(ii) Y_{i+1} ⊂ Y_{i}, Y_{i} ⊄ D_{i+1}, Y_{i+1} is an irreducible component of the closed set Y_{i} ∩ D_{i+1};

(iii) the estimate

\frac{mult_{p}}{deg} Y_{i+1} ≥ \frac{i + 2}{i + 1} · \frac{mult_{p}}{deg} Y_{i}

holds.

It is possible to construct this sequence because

codim_{G} Λ_{i+1}^{G} = i + 1 > codim_{G} Y_{i},

so that for a general divisor D_{i+1} ⊂ Λ_{i+1}^{G} we have Y_{i} ⊄ D_{i+1}. One can ensure that the property (iii) holds since Λ_{j}^{G} ⊂ |jH_{G}| and mult_{p} Λ_{j}^{G} = j + 1.

Now for an irreducible subvariety Y = Y_{m−1} we get the estimate

1 ≥ \frac{mult_{p}}{deg} Y ≥ \underbrace{\frac{m}{m − 1} · \frac{m − 1}{m − 2} · · · · · \frac{4}{3} · \frac{2}{1}}_{∥ ; \frac{2m}{3}} ·\frac{mult_{p}}{deg} R̄,

whence we get

\frac{mult_{p}}{deg} R̄ ≤ \frac{3}{2m}.

Therefore the ratio mult_{p} / deg attains its maximum at the tangent divisor T_{p}G ∩ G and this maximum is equal to 2. A contradiction. Q.E.D. for the lemma.

3.3 Hypertangent divisors and tangent cones

Thus ν ≤ 2k < μ. On the other side,

μ = mult_{x} R̃ ≤ mult_{x}(R̃ ∘ E) ≤ deg(R̃ ∘ E) = 2ν.

Set B = T_{x}E ∩ E, where the quadric E is considered as embedded in ℙ^{M} in the standard way. By Lemma 5 from Sec. 3.5 in [20],

mult_{B} R̃ ≥ \frac{1}{2}(μ − ν),

whereas for the effective cycle R_{E} = (R̃ ∘ E) we get

mult_{B} R_{E} ≥ μ − ν.  (21)

Set T̃_{i}, Λ̃_{i} to be the strict transforms of the divisors T_{i} and linear systems Λ_{i} on the blow up F̃ of the fiber F. It is easy to see that T̃_{i} ⊂ Λ̃_{i}, since by construction

mult_{o} Λ_{i} = mult_{o} T_{i}.

Set also

𝕋_{i} = (T̃_{i} ∘ E) = T̃_{i} ∩ E

to be the projectivized tangent cone to the divisor T_{i} at the point o. Recall that the quadric E is realized as the double cover σ̃_{E}: E → E_{G} ≅ ℙ^{M−1}, branched over the quadric W_{E}. For a system (z_{1}, …, z_{M+1}) of affine coordinates on ℙ with the origin at the point p we may assume that q_{1} ≡ z_{M+1} and therefore (z_{1}, …, z_{M}) can be taken for homogeneous coordinates on the projective space E_{G}. In terms of these coordinates the hypersurface 𝕋_{i} ⊂ E is given by the equation

(σ̃_{E})^{∗}q_{i+1}|_{E_{G}}.

Finally, set

Λ_{i}^{E} = Λ̃_{i}|_{E}

to be the projectivized tangent system of the linear system Λ_{i} at the point o. Equations of divisors of this linear system are obtained by pulling back to E via σ̃_{E} the equations

∑_{j=1}^{i} q̄_{j+1}s̄_{i−j},  (22)

where ¯♯ means the restriction of the polynomial ♯ onto the hyperplane z_{M+1} = 0. Obviously,

𝕋_{i} ∼ (i + 1)H_{E}, Λ_{i}^{E} ⊂ |(i + 1)H_{E}|,

besides the equations (22) imply directly that

Bs Λ_{i} = T_{1} ∩ … ∩ T_{i}, Bs Λ_{i}^{E} = 𝕋_{1} ∩ … ∩ 𝕋_{i},

both equalities in the scheme-theoretic sense.

3.4 Constructing new cycles

By the regularity condition the set 𝕋_{1} ∩ … ∩ 𝕋_{i} is irreducible and not contained in the divisor B for all i = 1, …, m − 1. Let

L = (L_{2}, …, L_{m−1}) ∈ Λ_{2} × … × Λ_{m−1}

be a general set of hypertangent divisors. We denote the strict transform of the hypertangent divisor L_{j} on F̃ by the symbol L̃_{j} and its projectivized tangent cone at the point o ∈ F by the symbol

𝕃_{j} = (L̃_{j} ∘ E).

For a general divisor L_{j} ∈ Λ_{j} we get 𝕃_{j} = L̃_{j} ∩ E.

Lemma 3.3. (i) Let Y ⊂ F be a fixed irreducible subvariety of codimension l ≤ m − 2. For a general divisor L_{l+1} ∈ Λ_{l+1} we have Y ⊄ L_{l+1}.

(ii) Let Y ⊂ E be a fixed irreducible subvariety of codimension l ≤ m − 2. For a general divisor L_{l+1} ∈ Λ_{l+1} we have Y ⊄ 𝕃_{l+1}.

Proof. By the regularity condition

codim_{F} Bs Λ_{l+1} = l + 1, codim_{E} Bs Λ_{l+1}^{E} = l + 1

and for a general divisor L_{j} ∈ Λ_{j} we have 𝕃_{j} ∈ Λ_{j}^{E}. Q.E.D. for the lemma. Corollary 3.1. For a general set L we have

codim_{F} (R ∩ L_{2} ∩ … ∩ L_{m−1}) = m − 1,
codim_{E}(R_{E} ∩ 𝕃_{2} ∩ … ∩ 𝕃_{m−1}) = m − 1.

From this fact we obtain that the following effective algebraic cycles of codimension m − 1 are well defined on F and E, respectively:

R^{+} = (R ∘ L_{2} ∘ … ∘ L_{m−1})

and

R_{E}^{+} = (R_{E} ∘ 𝕃_{2} ∘ … ∘ 𝕃_{m−1}),

whereas (for a general set L)

R_{E}^{+} = (R̃^{+} ∘ E)

is the projectivized tangent cone to the cycle R^{+} at the point o. Let us describe the structure of these effective cycles. First of all we get

deg R^{+} = 2km · (m − 1)! = 2km!,
mult_{o} R^{+} = deg R_{E}^{+} = 2ν · 3 · . . . · m = νm!.

Lemma 3.4. Let Y be an irreducible component of the cycle R^{+}. If Y ⊂ T_{1}, then

Y = T_{1} ∩ T_{2} ∩ … ∩ T_{m−1}.

Proof. By construction, the equation of the divisor L_{i} is of the form

f_{1}s_{i−1} + f_{2}s_{i−2} + … + f_{i}s_{0},

where s_{j} is a homogeneous polynomial of degree j in the coordinates z_{∗}. Since the hypertangent divisors L_{i} are assumed to be general, we may assume that s_{0} ≠ 0 and thus normalize the equation by the condition that s_{0} = 1. Assume that Y ⊂ T_{1}. Then the following polynomials vanish on Y:

f_{1},
f_{1}s_{2,1}  + f_{2},
f_{1}s_{3,2}  + f_{2}s_{3,1} + f_{3},
…
f_{1}s_{m−1,m−2} +  …  + f_{m−2}s_{m−1,1} + f_{m−1},

where s_{i,j} is a homogeneous polynomial of degree j. Thus

f_{1}|_{Y} ≡ f_{2}|_{Y} ≡ … ≡ f_{m−1}|_{Y} ≡ 0,

so that Y ⊂ T_{1} ∩ T_{2} ∩ … ∩ T_{m−1}, but the latter set is irreducible and of the same dimension as Y. This proves Lemma 3.4.

3.5 Degrees and multiplicities

Set

T = T_{1} ∩ T_{2} ∩ … ∩ T_{m−1}, 𝕋 = 𝕋_{1} ∩ 𝕋_{2} ∩ … ∩ 𝕋_{m−1}.

Taking into consideration that T = (T_{1} ∘ … ∘ T_{m−1}) and 𝕋 = (𝕋_{1} ∘ … ∘ 𝕋_{m−1})_{E}, it is easy to verify that

deg T = mult_{o} T = deg 𝕋 = 2m!.

Now write down

R^{+} = aT + R^{♯}, R_{E}^{+} = a𝕋 + R_{E}^{♯},  (23)

where a ∈ ℤ_{+}, the effective cycle R^{♯} is uniquely defined by the condition that it does not contain the subvariety T as a component, and

R_{E}^{♯} = (R̃^{♯} ∘ E)

is the projectivized tangent cone to R^{♯} at the point o. Note that the irreducible subvariety 𝕋, generally speaking, can come into the effective cycle R_{E}^{♯} as a component.

Lemma 3.5. The following estimate holds:

2 mult_{o} R^{♯} ≤ deg R^{♯}.

Proof. Let Y be an irreducible component of the cycle R^{♯}. By construction, Y ≠ T; therefore by Lemma 3.4 Y ⊄ T_{1}. Thus the closed subset

T_{1} ∩ Supp R^{♯}

is of codimension m, so that the effective cycle

R^{∗} = (R^{♯} ∘ T_{1})

is well defined. Now we have a standard chain of estimates:

2 mult_{o} R^{♯} ≤ mult_{o} R^{∗} ≤ deg R^{∗} = deg R^{♯},

which is what we need.

As in Corollary 3.1, Lemma 3.5 implies that the set

B ∩ 𝕃_{2} ∩ … ∩ 𝕃_{m−1}

is of codimension m−1 in E. Denote by B^{+} the part of the effective equidimensional cycle R_{E}^{+}, the support of which is contained in B:

R_{E}^{+} = ∑_{i∈I} r_{i}Y_{i}, B^{+} = ∑_{i∈I,Y_{i}⊂B} r_{i}Y_{i}.

Lemma 3.6. The following estimate holds:

deg B^{+} ≥ (μ − ν)m!

Proof. Indeed, by (21) we get

R_{E} = (μ − ν)B + Δ,

where Δ is an effective cycle. Furthermore,

deg(B ∘ 𝕃_{2} ∘ … ∘ 𝕃_{m−1}) = 2 · 3 · . . . · m = m!,

which proves the lemma.

Lemma 3.7. Let Y ⊂ 𝕃_{2}∩…∩𝕃_{m−1} be an irreducible subvariety of codimension m − 1 in E. If Y ⊂ 𝕋_{1}, then Y = 𝕋.

Proof. The equation of the divisor 𝕃_{i} with respect to the homogeneous coordinates z_{∗} is of the form

q_{2}s_{i−1} + … + q_{i+1},

where s_{j} is a homogeneous polynomial of degree j. If Y ⊂ 𝕋_{1}, then the following polynomials vanish on Y:

q_{2},
q_{2}s_{2,1}  + q_{3},
…
q_{2}s_{m−1,m−2} +  …  + q_{m−1}s_{m−1,1} + q_{m},

where deg s_{i,j} = j. Consequently,

q_{2}|_{Y} ≡ q_{3}|_{Y} ≡ … ≡ q_{m}|_{Y} ≡ 0,

that is, Y ⊂ 𝕋, and since the dimensions coincide, Y = 𝕋. Q.E.D. for the lemma. Corollary 3.2. None of the components of the closed set

B ∩ 𝕃_{2} ∩ … ∩ 𝕃_{m−1}

is contained in 𝕋_{1}.

Proof. Let Y be such component and Y ⊂ 𝕋_{1}. By the previous lemma, Y = 𝕋. Thus 𝕋 ⊂ B: a contradiction with the regularity condition. Q.E.D. for the corollary.

Let us complete, at long last, the proof of Proposition 3.1. From the presentations (23) we get

deg R^{+} = 2km! = 2am! + deg R^{♯},
mult_{o} R^{+} = νm! = 2am! + mult_{o} R^{♯}.

By Corollary 3.2 the effective cycle B^{+} lies entirely in R_{E}^{♯}. In particular,

deg R_{E}^{♯} ≥ deg B^{+} ≥ (μ − ν)m!.  (24)

However, deg R_{E}^{♯} = mult_{o} R^{♯}. Applying Lemma 3.5, we obtain:

2(νm! − 2am!) ≤ 2km! − 2am!.

Let us rewrite the inequality (24) in the form

νm! − 2am! ≥ (μ − ν)m!.

Easy computations give us the two inequalities

k + a ≥ ν,
2ν − 2a ≥ μ,

which imply the desired estimate (20) in an obvious way.

However, we assumed that μ > 2k. The contradiction completes our proof of Proposition 3.1 and Theorem 1 as well.

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