Birationally rigid iterated Fano double covers

A.V.Pukhlikov^{1}

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

November 17, 2002

Abstract

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higherdimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted projective spaces. A generic variety in these families is proved to be birationally superrigid; in particular, it admits no nontrivial structures of a fibration into rationally connected (or uniruled) varieties, it is non-rational and its groups of birational and biregular self-maps coincide.

1The author was financially supported by Russian Foundation of Basic Research, grants 02-15-99258 and 02-01-00441, by a Science Support Foundation grant for young researchers and by INTAS-OPEN, grant 2000-269.

CONTENTS

0. Introduction 0.1. Birationally rigid varieties 0.2. The known results and natural conjectures 0.3. The iterated Fano double covers 0.4. The structure of the paper 0.5. Historical remarks and acknowledgements

1. The method of maximal singularities 1.1. A criterion of birational superrigidity 1.2. The first proof: counting multiplicities 1.3. The second proof: the connectedness principle 1.4. The Lefschetz theorem 1.5. Proof of Proposition 0.1

2. Iterated double covers 2.1. Coordinate presentations 2.2. The regularity condition 2.3. Non-regular sets of polynomials 2.4. Start of the proof of Proposition 2.1 2.5. The polynomials h_{i,j} depend on each other 2.6. Estimates for the codimension

3. Hypertangent divisors 3.1. How to obtain a bound for the multiplicity 3.2. Construction of hypertangent divisors 3.3. The regularity condition for hypertangent divisors 3.4. The Lefschetz theorem once again 3.5. The points of class e = 0 3.6. The points of class e ≥ 1

References

Introduction

0.1 Birationally rigid varieties

Birational rigidity is one of the most striking phenomena of higher-dimensional algebraic geometry. Speaking informally, birational rigidity means that certain algebraic varieties, on which there are no non-zero global regular differential forms (that is, rationally connected varieties), behave as if they were of general type. Starting from the pioneer paper of V.A.Iskovskikh and Yu.I.Manin on the three-dimensional quartic [IM] of 1970 it has been gradually understood that in higher dimensions birationally rigid varieties and fibrations not only do not form an exceptional exotic class, but on the contrary, are quite typical. Sarkisov’s theorem [S1,S2] means that in a sense “almost all” conic bundles of dimension three and higher are birationally rigid. In [P1,P7,P8,P10,P12] it was proved that Fano hypersurfaces and more generally certain complete intersections of index 1 are birationally rigid. In [P6,P9] it was proved that “almost all” del Pezzo fibrations and fibrations into Fano hypersurfaces over ℙ^{1} are birationally rigid. Nowadays it is clear that, on the contrary, non-rigid Fano varieties and fibrations are less typical.

In the present paper we use the more traditional definition of birational rigidity. We work over the field ℂ of complex numbers. A smooth variety X of dimension ≥ 3 is said to be birationally superrigid, if for each birational map χ: X − − → X′ onto a variety X′ of the same dimension, smooth in codimension one, and each linear system Σ′ on X′, free in codimension 1 (that is, codim Bs Σ′ ≥ 2), the inequality

c(Σ, X) ≤ c(Σ′, X′)  (1)

holds, where Σ = (χ^{−1})_{∗}Σ′ is the proper inverse image of Σ′ on X with respect to χ, and c(Σ, X) = c(D, X) stands for the threshold of canonical adjunction

c(D, X) = sup{b/a|b, a ∈ ℤ_{+} \ {0}, |aD + bK_{X}| ≠ ∅}

D ∈ Σ, and similarly for Σ′, X′.

A smooth variety X of dimension ≥ 3 is said to be birationally rigid, if for each X′, χ and Σ′ as above there exists a birational self-map

χ^{∗} = χ_{(X′,χ,Σ′)}^{∗} ∈ Bir X,

depending on the triple (X′, χ, Σ′), such that

c(Σ^{∗}, X) ≤ c(Σ′, X′),

where Σ^{∗} is the strict transform of Σ with respect to (χ^{∗})^{−1}, or, equivalently, the strict transform of Σ with respect to the composition

χ ∘ χ^{∗}: X − − → X′.

The difference between the rigid and superrigid cases is not too big. Roughly speaking, superrigidity is rigidity combined with the property that the groups of birational and biregular maps coincide (indeed, if this is the case, then twisting by a birational=biregular self-map χ^{∗} does not change the threshold of canonical adjunction, hence rigidity implies the inequality (1), that is, superrigidity).

The most natural object for the rigidity theory is formed by Fano varieties. Assume that X is a smooth Fano variety of dimension ≥ 3 with Pic X = ℤK_{X}. The important geometric properties of birationally rigid and superrigid varieties from this class (the properties that motivate the very choice of the word “rigidity”) are summarized in the following

Proposition 0.1. Assume that X is rigid. Then:

(i) X can not be fibered into uniruled varieties by a non-trivial rational map, (ii) if χ: X − − → X′ is a birational map onto a Fano variety X′ with ℚ-

factorial terminal singularities such that Pic X′ ⊗ℚ = ℚK_{X′}, then X is (biregularly) isomorphic to X′ (although χ itself is not necessarily an isomorphism),

(iii) X is non-rational. Assume moreover that X is superrigid. Then any birational map onto a Fano variety X′ with ℚ-factorial terminal singularities such that Pic X′ ⊗ ℚ = ℚK_{X′} is a (biregular) isomorphism. In particular, the groups of birational and biregular self-maps coincide:

Bir X = Aut X.

These implications of birational rigidity are well known. For convenience of the reader, we give a (very easy) proof of Proposition 0.1 below in Sec. 1.5.

Note that Corti and Reid [C2,CR] take the properties (i) and (ii) (property (iii) is an immediate implication of (i)) as a definition of birational rigidity.

Although in this paper we study only Fano varieties, to make the picture complete, let us give the definition of birational rigidity for Fano fibrations, too. The relative version is completely analogous to the absolute one (see the definition above), with the only difference: the group of birational self-maps is replaced by the group of fiber-wise birational self-maps.

Assume that X/S is a rationally connected fibration:

X
π ↓ generic fiber F_{η}
S

The fiber of general position is assumed to be rationally connected, so that X itself is uniruled and the thresholds of canonical adjunction are finite. We define the group of proper birational self-maps of the fibration X/S, setting

Bir(X/S) = Bir F_{η},

that is, Bir(X/S) ⊂ Bir X is the subgroup consisting of all maps χ: X − − → X such that χ transforms each fiber into itself. Now the fibration is said to be birationally rigid (as a fibration!), if for any variety X′ of the same dimension, smooth in codimension one, any birational map χ: X − − → X′ and any linear system Σ′ on X′ there exists a self-map

χ^{∗} ∈ Bir(X/S)

such that for the strict transform Σ^{∗} of the linear system Σ′ with respect to the composition (χ ∘ χ^{∗})^{−1} the inequality

c(Σ^{∗}, X) ≤ c(Σ′, X′)

is satisfied.

Proposition 0.2. Assume that X/S is a Fano fibration with X, S smooth such that

Pic X = ℤK_{X} ⊕ π^{∗} Pic S

and for any effective class D = mK_{X} + π^{∗}T the class NT is effective on S for some N ≥ 1. Assume furthermore that X/S is birationally rigid. Then for any rationally connected fibration X′/S′ and any birational map

χ: X − − → X′

(provided that such maps exist) there is a rational dominant map

α: S − − → S′

making the following diagram commutative:

X − − →^{χ} X′
π ↓  ↓ π′
S − − →^{α} S′.

0.2 The known results and natural conjectures

At the moment birational superrigidity is proved for the following classes of higherdimensional Fano varieties:

• smooth hypersurfaces V_{M} ⊂ ℙ^{M} of degree M, M ≥ 4 (for M = 4 it is the classical theorem of V.A.Iskovskikh and Yu.I.Manin [IM], the case M = 5 was proved in [P1], for generic hypersurfaces of degree M ≥ 6 superrigidity was proved in [P7], finally, for arbitrary smooth hypersurfaces it was proved in [P12]); • generic (in the sense of Zariski topology) Fano complete intersections

V =  V_{d_{1}·d_{2}·...·d_{k}}  ⊂ ℙ^{M+k}
∥
F_{1} ∩ F_{2} ∩ … ∩ F_{k},

where deg F_{i} = d_{i}, d_{1} + … + d_{k} = M + k, d_{i} ≥ 2 and the inequality 2k < M holds [P10];

• generic (in the sense of Zariski topology) Fano double covers

σ: V → Q = Q_{m} ⊂ ℙ^{M+1},

where M ≥ 4, Q is a hypersurface of degree m ≥ 3, σ is ramified over the divisor

W = W_{m·2l} = W^{∗} ∩ Q,

where W^{∗} = W_{2l}^{∗} ⊂ ℙ^{M+1} is a hypersurface of degree 2l, m+l = M+1 [P8]. For m = 1, 2 birational superrigidity was earlier proved in [P2] for smooth double spaces and double quadrics without the assumption of them being generic;

• the known examples [P3,P11] show that mild singularities do not change the picture: the varieties remain rigid, and usually even superrigid.

Conjecture 1 (the absolute case). A smooth Fano variety V of dimension dim V ≥ 5 with the Picard group Pic V = ℤK_{V} is birationally superrigid.

The assumption of V being smooth seems to be unnecessarily strong.

Conjecture 2 (the absolute case). A Fano variety V of dimension dim V ≥ 4 with factorial terminal singularities and the Picard group Pic V = ℤK_{V} is birationally rigid.

One can make Conjecture 2 stronger in its turn, replacing the factorial singularities by the ℚ-factorial ones and modifying the condition on the Picard group in an appropriate way. However, Fano varieties with terminal quotient singularities have not yet been studied in higher (≥ 4) dimensions from the viewpoint of their birational rigidity. Since there are no completely studied examples it seems it is yet too early to formulate general conjectures.

To complete the picture, we give a relative version of the conjectures on birational rigidity.

Let V → S be a fibration into Fano varieties satisfying the following standard conditions:

(i) V is smooth, dim V ≥ 4, (ii) dim S ≥ 1, the anticanonical class −K_{V} is relatively ample, (iii) Pic V = ℤK_{V} ⊕ π^{∗} Pic S. (In dimension dim V = 3 this means that V/S is a Mori fiber space.)

Conjecture 3 (the relative case). If the fibration V/S is sufficiently twisted over the base S, then it is birationally superrigid.

If dim S = 1, that is, S = ℙ^{1}, then the twistedness of the fibration V/S is characterized by the properties of the numerical class

K_{V}^{2} ∈ A^{2}(V),

more precisely, in the known cases it is sufficient to assume that the following “K_{V}^{2} - condition” is satisfied: K_{V}^{2} does not lie in the interior of the cone of the effective cycles of codimension two [P6,P9]. However, this K_{V}^{2} -condition can be somewhat weakened, see [G1,G2,Sob1,Sob2].

In the general case the “twistedness” of the fibration V/S can be imagined in the following way. Let the fiber of general position F_{s}, s ∈ S, belong to a fixed family of Fano varieties F. For simplicity we assume that for a general member F ∈ F the anticanonical class −K_{V} is very ample and determines an embedding

F ↪ ℙ^{N}.

Let H = Hilb(F) be the Hilbert scheme of embedded varieties of the family F. Then the fibrations V/S with the general fiber in F correspond to the maps

S → H.

The degree of the image of S with respect to a fixed class on H can be considered as a characteristic of twistedness of V/S. However, for particular fibrations the twistedness condition takes a simple form and is easy to check, see [P6,P9].

0.3 Iterated Fano double covers

The aim of the present paper is to prove birational superrigidity of a large class of higher-dimensional Fano varieties, generalizing Fano double hypersurfaces [P8]. The varieties under consideration can be realized as complete intersections in weighted projective spaces. Let us give their construction.

The ground field is assumed to be the field of complex numbers ℂ. The symbol ℙ stands for the projective space ℙ^{M+k}, where M ≥ 4, 2k ≤ M − 1. Let us choose a system of homogeneous coordinates on ℙ, say (x_{0} : x_{1} : …: x_{M+k}). Consider a set of homogeneous polynomials

f_{1}, …, f_{k}, g_{1}, …, g_{m}

in the variables (x_{∗}) of degrees, respectively,

d_{1}, …, d_{k}, 2l_{1}, …, 2l_{m},

where m ≥ 1, l_{i} ≥ 2 and the following equality holds:

∑_{i=1}^{k} d_{i} + ∑_{i=1}^{m} l_{i} = M + k.

Set

Q(f_{∗}) = Q(f_{1}, …, f_{k}) = ⋂_{i=1}^{k}{f_{i} = 0}

to be the complete intersection of hypersurfaces F_{i} = {f_{i} = 0} in ℙ. Set also

W_{i} = {g_{i} = 0} ⊂ ℙ.

Define a sequence of double covers

σ_{i}: ℙ^{(i)} → ℙ^{(i−1)},

where ℙ^{(0)} = ℙ, i = 1, …, m, in the following way. The cover σ_{1} is branched over W_{1}. Assume that σ_{1}, …, σ_{i} are already constructed. For an arbitrary j ∈ {0, …, i − 1} set

σ_{i,j} = σ_{j+1} ∘ … ∘ σ_{i}: ℙ^{(i)} → ℙ^{(j)}.

In particular, σ_{i} = σ_{i,i−1}. Obviously σ_{i,j} is a finite morphism of degree 2^{i−j}. Now the double cover

σ_{i+1}: ℙ^{(i+1)} → ℙ^{(i)}

is determined by the branch divisor

W̃_{i+1} = σ_{i,0}^{−1}(W_{i+1}) ⊂ ℙ^{(i)}.

As a result we get a sequence of double covers

ℙ^{(m)} →^{σ_{m}} ℙ^{(m−1)} →^{σ_{m−1}} … →^{σ_{1}} ℙ^{(0)} = ℙ.

Finally, set

σ = σ_{m,0}: ℙ^{(m)} → ℙ.

It is a finite morphism of degree 2^{m}.

The collections of homogeneous polynomials (f_{∗}; g_{∗}) are parametrized by the space

H = ∏_{i=1}^{k} [H^{0}(ℙ, O_{ℙ}(d_{i})) \ {0})] × ∏_{i=1}^{m} [H^{0}(ℙ, O_{ℙ}(2l_{i})) \ {0})] .

For a general collection (f_{∗}; g_{∗}) ∈ H all the varieties

Q = Q(f_{∗}), ℙ^{(i)}, W̃_{i}, W̃_{i} ∩ σ_{i−1,0}^{−1}(Q(f_{∗})),

are obviously smooth. Set

V = V (f_{∗}; g_{∗}) = σ^{−1}(Q) ⊂ ℙ^{(m)}.

This is a smooth subvariety of codimension k in ℙ^{(m)}. Obviously

V = ⋂_{i=1}^{k} σ^{−1}(F_{i})

is a smooth complete intersection in ℙ^{(m)}. It is easy to see that

K_{V} = [−(M + k + 1) + ∑_{i=1}^{k} d_{i} + ∑_{i=1}^{m} l_{i}] σ^{∗}H

∥
−σ^{∗}H,

where H ∈ Pic ℙ is the class of a hyperplane. By the Lefschetz theorem we get Pic V = ℤσ^{∗}H, so that V is a smooth Fano variety of index 1 and dimension M.

Here is the main result of the present paper.

Theorem. There is a non-empty open Zariski subset U ⊂ H such that for any collection of polynomials (f_{∗}; g_{∗}) ∈ U the variety V = V (f_{∗}; g_{∗}) is a (smooth) birationally superrigid variety.

The open subset U is defined below by explicit regularity conditions. There is no doubt that any smooth variety V = V (f_{∗}; g_{∗}) is birationally superrigid. However, both the classical technique of hypertangent divisors, which is used in this paper, and the new technique based on the connectedness principle of Shokurov and Kollár, are not strong enough to prove this fact.

0.4 The structure of the paper

The paper is organized in the following way. Section 1 contains the “general theory” of the method of maximal singularities, which for convenience of the reader is given here with the necessary details. In Sec. 1.1 we give a criterion of birational superrigidity, reducing the proof of birational superrigidity to checking certain explicit conditions for subvarieties of codimension two. In Sec. 1.2 we give the first proof of the main claim of this criterion. This proof makes use of the technique of counting multiplicities. Sec. 1.3 contains another proof, which makes use of Corti’s idea [C2] of reduction to a non-log-canonical singularity of a linear system on a smooth surface. The corresponding two-dimensional fact is proved by means of an elementary technique. Actually, it is a simple implication of one fact which was proved already in [P3]. In Sec. 1.4 the proof of the basic criterion of Sec. 1.1 is completed. Finally, in Sec. 1.5 we give an elementary proof of Proposition 0.1, describing geometry of birationally rigid varieties.

Section 2 is of more technical character. Its aim is to describe the regularity condition that defines the open set U ⊂ H in the main theorem and to prove that this set is non-empty. In order to do it, we use the method developed in [P10]. However, in the present case this method should be modified. In this section we also give certain convenient coordinate presentations, which are used later.

Section 3 contains the heart of the proof — there we check the criterion of Section 1.1 for the regular iterated double covers. In Sec. 3.1 the technique of estimating the multiplicity of a subvariety at a given point is developed. It is based on the concept of a hypertangent divisor. The method is presented in a general form. After that in Sec. 3.2 we construct a family of hypertangent divisors for an arbitrary point x ∈ V of the iterated double cover V. Sec. 3.3 studies some properties of hypertangent divisors. These properties follow from the regularity condition. It is here that this condition is made use of. Finally, in Sec. 3.4 – 3.5 we complete the proof of the main theorem.

0.5 Historical remarks and acknowledgements

Not pretending to be complete in any sense, we just point out the principal landmarks in the development of the theory of birational rigidity. The very phenomenon of rigidity was guessed by Fano [F1-F3] when he tried to extend Noether’s method [N] to dimension three. However, the techniques of algebraic geometry of his time were not strong enough (intersection theory, sheaves and cohomology, resolution of singularities either were non-existent at all or just made its first steps) to enable him to obtain complete results. The first outlines of the theory of birational rigidity can be seen in the papers of Yu.I.Manin of 60ies on surfaces over non-closed fields, see, for instance, [M1-M3], where, using B.Segre’s earlier results, it is proved what is now formulated as birational superrigidity of del Pezzo surfaces of degree 1 with the Picard group ℤ. In [M2,M3] a graph is associated to a finite sequence of blow ups. Its combinatorial invariants are very important for the classical technique of counting multiplicities.

The decisive step was made by V.A.Iskovskikh and Yu.I.Manin in [IM] where birational superrigidity of a smooth three-dimensional quartic was established (the authors prove the coincidence of birational and biregular isomorphisms of threedimensional quartics, however the arguments of the paper do not need any modification to produce birational superrigidity, so that essentially it is superrigidity that was proved there). After that the technique that was developed in [IM] was applied in [I1] to a few families of Fano 3-folds, which resulted in proving their birational rigidity.

The next step was made by V.G.Sarkisov in [S1,S2]. Starting with V.A.Iskovskikh’s results on the surfaces with a pencil of rational curves, V.G.Sarkisov proved birational rigidity of conic bundles with a “sufficiently big” discriminant divisor. The concept of “birational rigidity” was in 1980 yet non-existent, so in [S1,S2] the main result is formulated as uniqueness of the conic bundle structure on a given variety. The breakthrough that was made in [S1,S2] was the more impressing that embraced conic bundles in arbitrary dimension.

In a few years after Sarkisov’s theorem the first attempts were made to extend the three-dimensional technique of V.A.Iskovskikh and Yu.I.Manin to the field of higher-dimensional Fano varieties [P1,P2]. Besides, birational geometry of a threedimensional quartic with a non-degenerate double point was described [P3]. However, the technical side of this work was getting more and more complicated. The methods needed to be improved.

In [P4] and especially [P5] (the latter paper was written in 1995 and has been distributed among the experts since that time, although published only in 2000) the classical technique of the method of maximal singularities was essentially simplified and clarified, which in particular made it possible to prove birational rigidity of del Pezzo fibrations over ℙ^{1} [P6] — that is, the only class of three-dimensional Mori fiber spaces, which stubbornly refused all attempts to study its birational geometry for about 15 years (there were quite a few well-studied examples of Fano varieties, starting from the quartic, and for the conic bundles there was Sarkisov’s theorem).

The further development of the theory went in two directions. Already in the late 80ies V.G.Sarkisov suggested a general program of factorization of birational maps between three-dimensional Mori fiber spaces into a composition of elementary links [S3]. M.Reid did a lot of work to popularize Sarkisov’s ideas among the experts in Mori theory [R]. In [C1] Corti gave a complete proof of the main claim of Sarkisov’s program and thus brought the construction of this theory to an end. Combining the classical methods with the Sarkisov program, it was proved in [CPR] that three-dimensional ℚ-Fano hypersurfaces of index 1 in weighted projective spaces are birationally rigid. Generators of their groups of birational self-maps were described. Besides, in [C2] Corti suggested to use the connectedness principle of Shokurov and Kollár (based, in its turn, on the Kawamata-Viehweg vanishing theorem) in the investigation of maximal singularities. This idea turned out to be very fruitful and has already been used several times, both in dimension three [CM] and in arbitrary dimension [P12]. This technique was discussed and further developed in [Ch,ChPk], see also [I2,I3].

The classical technique was developing parallel to the ideas coming from the Mori theory and the log minimal model program. In [P7] the construction of hypertangent divisors was introduced. This construction proved extremely fruitful and made it possible to prove birational rigidity of generic Fano varieties and Fano fibrations for several important families [P8-P11]. This construction makes the basis of the present paper, either.

This paper was started by the author during his work at the University of Bayreuth as a Humboldt Research Fellow, and completed at Max-Planck-Institut für Mathematik in Bonn. The author is grateful to Alexander von Humboldt Foundation and to Max-Planck-Institut für Mathematik in Bonn for the financial support of his research. I would like to thank Mathematisches Institut von Universität Bayreuth and, in the first place, Prof. Th. Peternell for the warm friendly atmosphere which surrounded me in Bayreuth and for the constant interest to my work. I am very grateful to Max-Planck-Institut für Mathematik in Bonn for the stimulating creative atmosphere, general hospitality and for exceptionally good conditions of work.

1 The method of maximal singularities

1.1 A criterion of birational superrigidity

We will prove the main theorem checking the following convenient sufficient condition of birational superrigidity.

Proposition 1.1. Let X be a smooth Fano variety with Pic X = ℤK_{X}. Assume that for any irreducible subvariety Y ⊂ X of codimension two the following two properties are satisfied:

(i) mult_{Y} Σ ≤ n for any linear system Σ ⊂ | − nK_{X}| without fixed components; (ii) the inequality

mult_{x} Y ≤ \frac{4}{deg X} deg Y  (2)

holds for any point x ∈ Y, where

deg X = (−K_{X})^{dim X}, deg Y = (Y · (−K_{X})^{dim Y})

and mult_{Y} Σ means multiplicity of a general divisor D ∈ Σ along Y.

Then the variety X is birationally superrigid.

It is in this way that birational superrigidity was proved for the majority of known classes of Fano varieties [P7-P11].

Proof of Proposition 1.1. For convenience of the reader we give it here with all the significant details. For the other details and comments see [P5,P8]. Assume that X is not superrigid. Then, by the definition of superrigidity, we get a birational map χ: X− − → X′ and a pair of linear systems Σ, Σ′, transformed one into another by χ, such that inequality (1) is not true. The next step in the arguments is given by

Proposition 1.2. There exists a geometric discrete valuation ν on X such that the Noether-Fano inequality

ν(Σ) > n · discrepancy(ν)  (3)

holds, where n ∈ ℤ_{+} is defined by the inclusion Σ ⊂ | − nK_{X}| and ν(Σ) = ν(D) for a general divisor D ∈ Σ. The discrete valuations ν, satisfying (3), are called maximal singularities of the linear system Σ.

Recall [C2,CPR,P5,P8] that a discerete valuation is said to be geometric, if it is realizable by a prime Weil divisor on some model of the field of rational function ℂ(X).

Let B ⊂ X be the centre of ν on X. If codim_{X} B = 2, then it is easy to see that mult_{B} Σ > n, which contradicts (i). Therefore codim_{X} B ≥ 3. Here we come to the crucial point of the proof.

Let D_{1}, D_{2} ∈ Σ be general divisors. They have no common components and therefore their intersection is of codimension 2. Denote by (D_{1} ∘ D_{2}) the effective cycle of their scheme-theoretic intersection. Obviously,

deg(D_{1} ∘ D_{2}) = n^{2} deg X.  (4)

Now the crucial fact is given by

Proposition 1.3. The following inequality holds

mult_{B}(D_{1} ∘ D_{2}) > 4n^{2}.  (5)

Proof of the proposition is given below. We give here both known arguments: the classical one, based on the technique of counting multiplicities, and the recent argument of Corti, based on his idea of using the connectedness principle of Shokurov and Kollár.

Now, comparing (4) and (5), we find an irreducible subvariety Y ⊂ X of codimension 2 (a component of the effective cycle (D_{1} ∘ D_{2})) such that

mult_{B} Y > \frac{4}{deg X} deg Y,

which contradicts the assumption (ii) of Proposition 1.1.

Therefore, out initial assumption is false and X is superrigid. Q.E.D. for Proposition 1.1.

1.2 The first proof: counting multiplicities

First of all, we take a resolution of the discrete valuation ν. Consider the sequence of blow ups

φ_{i,i−1} : X_{i} → X_{i−1}
⋃  ⋃
E_{i} → B_{i−1}

i ≥ 1, where X_{0} = X, φ_{i,i−1} blows up the cycle B_{i−1} = Z(X_{i−1}, ν) of codimension ≥ 2, E_{i} = φ_{i,i−1}^{−1}(B_{i−1}) ⊂ X_{i}. Set also for i > j

φ_{i,j} = φ_{j+1,j} ∘ … ∘ φ_{i,i−1} : X_{i} → X_{j},
φ_{i,i} = id_{X_{i}} .

For any cycle (…) we denote its proper inverse image on X_{i} by adding the upper index i: (…)^{i}.

Remark. (i) Note that φ_{i,j}(B_{i}) = B_{j} for i ≥ j. (ii) Note also that although all the X’s are possibly singular, B_{i} ⊄ Sing X_{i} for all i.

For some K ∈ ℤ_{+} the divisor E_{K} ⊂ X_{K} realizes the discrete valuation ν.

Let us define the structure of an oriented graph on the set of exceptional divisors or, equivalently, on the set of indices {1, …, K}. We draw an arrow

i → j

if i > j and B_{i−1} ⊂ E_{j}^{i−1}. Set p_{ij} to be the number of paths from i to j if i ≠ j, and p_{ii} = 1. Finally, set p_{i} = p_{Ki} for all i = 1, …, K.

Now let Σ^{j} be the strict transform of the linear system Σ on V_{j}. Set

ν_{j} = mult_{B_{j−1}} Σ^{j−1}.

Obviously, the multiplicity of the linear system Σ with respect to the valuation E_{j} is equal to

ν_{E_{j}}(Σ) = ∑_{i=1}^{j} p_{ji}ν_{i}.

Setting δ_{i} = codim B_{i−1} − 1, we get the well-known expression for the discrepancy

a(X, ν_{E_{j}}) = ∑_{i=1}^{j} p_{ji}δ_{i}.

The Noether-Fano inequality takes the form

∑_{i=1}^{K} p_{i}ν_{i} > ∑_{i=1}^{K} p_{i}δ_{i}n.

Now let us consider the following general situation. Let B ⊂ X, B ⊄ Sing X be an irreducible cycle of codimension ≥ 2, σ_{B}: X(B) → X be its blowing up, E(B) = σ_{B}^{−1}(B) the exceptional divisor. Let

Z = ∑ m_{i}Z_{i}, Z_{i} ⊂ E(B)

be a k-cycle, k ≥ dim B. We define the degree of Z as

deg Z = ∑_{i} m_{i} deg (Z_{i} ⋂ σ_{B}^{−1}(b)) ,

where b ∈ B is a generic point, σ_{B}^{−1}(b) ≅ ℙ^{codim B−1} and the right-hand side degree is the ordinary degree in the projective space.

Note that deg Z_{i} = 0 if and only if σ_{B}(Z_{i}) is a proper closed subset of B.

Our computations will be based upon the following statement.

Let D and Q be two different prime Weil divisors on X, D^{B} and Q^{B} be their proper inverse images on X(B).

Lemma 1.1. (i) Assume that codim B ≥ 3. Then

D^{B} ∘ Q^{B} = (D ∘ Q)^{B} + Z,

where ∘ stands for the cycle of the scheme-theoretic intersection, Supp Z ⊂ E(B), and

mult_{B}(D ∘ Q) = (mult_{B} D)(mult_{B} Q) + deg Z.

(ii) Assume that codim B = 2. Then

D^{B} ∘ Q^{B} = Z + Z_{1},

where Supp Z ⊂ E(B), Supp σ_{B}(Z_{1}) does not contain B, and

D ∘ Q = [(mult_{B} D)(mult_{B} Q) + deg Z] B + (σ_{B})_{∗}Z_{1}.

Proof follows easily from the standard intersection theory [Ful].

Now let us come back to our discrete valuation ν.

We divide the resolution φ_{i,i−1} : X_{i} → X_{i−1} into the lower part, i = 1, …, L ≤ K, when codim B_{i−1} ≥ 3, and the upper part, i = L + 1, …, K, when codim B_{i−1} = 2. It may occur that L = K and the upper part is empty.

Let D_{1}, D_{2} ∈ Σ be two different general divisors. We define a sequence of codimension 2 cycles on X_{i}’s setting

D_{1} ∘ D_{2} = Z_{0},
D_{1}^{1} ∘ D_{2}^{2} = Z_{0}^{1} + Z_{1},
⋮
D_{1}^{i} ∘ D_{2}^{i} = (D_{1}^{i−1} ∘ D_{2}^{i−1})^{i} + Z_{i},
⋮

where Z_{i} ⊂ E_{i}. Thus for any i ≤ L we get

D_{1}^{i} ∘ D_{2}^{i} = Z_{0}^{i} + Z_{1}^{i} + … + Z_{i−1}^{i} + Z_{i}.

For any j > i, j ≤ L set

m_{i,j} = mult_{B_{j−1}}(Z_{i}^{j−1})

(the multiplicity of an irreducible cycle along a smaller cycle is understood in the usual sense; for an arbitrary cycle we extend the multiplicity by linearity).

Set d_{i} = deg Z_{i}.

We get the following system of equalities:

ν_{1}^{2} + d_{1} = m_{0,1},
ν_{2}^{2} + d_{2} = m_{0,2} + m_{1,2},
⋮
ν_{i}^{2} + d_{i} = m_{0,i} + … + m_{i−1,i},
⋮
ν_{L}^{2} + d_{L} = m_{0,L} + … + m_{L−1,L}.

Now

d_{L} ≥ ∑_{i=L+1}^{K} ν_{i}^{2} deg(φ_{i−1,L})_{∗}B_{i−1} ≥ ∑_{i=L+1}^{K} ν_{i}^{2}.

Definition. A function a : {1, …, L} → ℝ_{+} is said to be compatible with the graph structure, if

a(i) ≥ ∑_{j→i} a(j)

for any i = 1, …, L.

We will actually use only one compatible function, namely a(i) = p_{Ki} = p_{i}.

Proposition 6. Let a(·) be any compatible function. Then

∑_{i=1}^{L} a(i)m_{0,i} ≥ ∑_{i=1}^{L} a(i)ν_{i}^{2} + a(L) ∑_{i=L+1}^{K} ν_{i}^{2}.

Proof. Multiply the i-th equality by a(i) and put them all together: in the right-hand part for any i ≥ 1 we get the expression

∑_{j≥i+1} a(j)m_{i,j}

In the left-hand part for any i ≥ 1 we get

a(i)d_{i}.

Lemma 1.2. If m_{i,j} > 0, then i → j.

Proof. If m_{i,j} > 0, then some component of Z_{i}^{j−1} contains B_{j−1}. But Z_{i}^{j−1} ⊂ E_{i}^{j−1}. Q.E.D.

Besides, we can compare the multiplicities m_{ij} with the corresponding degrees. Lemma 1.3. For any i ≥ 1, j ≤ L we have

m_{i,j} ≤ d_{i}.

Proof. The cycles B_{a} are non-singular at their generic points. But since

φ_{a,b} : B_{a} → B_{b}

is surjective, we can count multiplicities at generic points. Now the multiplicities are non-increasing with respect to blowing up of a non-singular cycle, so we are reduced to the obvious case of a hypersurface in a projective space. Q.E.D.

Consequently, we have the following estimate:

∑_{j≥i+1} a(j)m_{i,j} = ∑_{\begin{matrix}j≥i+1 \\ m_{i,j}≠0 \end{matrix}} a(j)m_{i,j} ≤ d_{i} ∑_{j→i} a(j) ≤ a(i)d_{i}.

So we can throw away all the m_{i,∗}, i ≥ 1, from the right-hand part, and all the d_{i}, i ≥ 1, from the left-hand part, replacing = by ≤. Q.E.D.

Corollary 1.1. Set m = m_{0,1} = mult_{B}(D_{1} ∘ D_{2}), D_{i} ∈ Σ. Then

m (∑_{i=1}^{L} a(i)) ≥ ∑_{i=1}^{L} a(i)ν_{i}^{2} + a(L) ∑_{i=L+1}^{K} ν_{i}^{2}.

Corollary 2. The following inequality holds:

m (∑_{i=1}^{L} p_{i}) ≥ ∑_{i=1}^{K} p_{i}ν_{i}^{2}.

Proof: for i ≥ L + 1 obviously p_{i} ≤ p_{L}. Q.E.D.

Taking into account the Noether-Fano inequality for ν, we see that the right-

hand part here is strictly greater than the value of the quadratic form ∑_{i=1}^{K} p_{i}ν_{i}^{2} at the

point

ν_{1} = … = ν_{K} = \frac{∑_{i=1}^{K} p_{i}δ_{i}n}{∑_{i=1}^{K} p_{i}}.

Now set

Σ_{l} = ∑_{δ_{j}≥2} p_{j},

Σ_{u} = ∑_{δ_{j}=1} p_{j}.

Then

mult_{x} Z > \frac{(2Σ_{l} + Σ_{u})^{2}}{Σ_{l}(Σ_{l} + Σ_{u})}n^{2}.

Now easy computations show that the right-hand side is not smaller than 4n^{2}. Q.E.D. for Proposition 2.

1.3 The second proof: the connectedness principle of Shokurov and Kollár

Here we give an alternative proof of Proposition 2, suggested by Corti [C2]. Our version is slightly different from that of [C2], we reduce the proof to one simple fact about oriented graphs which was proved originally in [P3].

It is sufficient to consider the case when B = o is a point. Let S ∋ o be a general germ of a smooth surface on the variety X. Obiovuly, Λ = Σ|_{S} is a germ of a linear system of curves on S with the point o as an isolated base point. Since the pair

(X, \frac{1}{n}Σ)

is not canonical at the point o (it is a reformulation of the Noether-Fano inequality (3)), according to the inversion of adjunction (which is a direct consequence of the connectedness principle of Shokurov and Kollár [K,Sh]), the pair

(S, \frac{1}{n}Λ)

is not log-canonical at the point o. In other words, for a certain birational morphism

φ: S̃ → S

of smooth surfaces there exists a prime divisor E ⊂ S̃, satisfying the log-version of the Noether-Fano inequality

ν_{E}(Λ) > n(a(E) + 1),  (6)

where a(·) is the discrepancy, ν_{E}(·) is the multiplicity of a general divisor of the system at E. Let D_{1}, D_{2} ∈ Λ be generic curves,

Z = (D_{1} ∘ D_{2})

be a zero-dimensional subscheme on S. One may assume that it is supported at the point o.

Proposition 1.4. The following inequality holds

mult_{o} Z(= deg Z) > 4n^{2}.

Since our considerations are local, mult_{o} Z = deg Z is just the degree of the zero-dimensional scheme Z. Since S ∋ o is a general germ of a surface, we obtain immediately the claim of Proposition 1.3.

Proof. We give an elementary argument based on explicit computations. The original argument of Corti see in [C2]. Let

\begin{aligned}& S_{i} & ⊃ & E_{i} \\ φ_{i,i−1} & ↓ & & ↓ \\ & S_{i−1} & ∋ & x_{i−1} \end{aligned}

be the resolution of the discrete valuation ν_{E}, i = 1, …, N, x_{0} = o, x_{1}, …, x_{N−1} points on S_{1}, …, S_{N−1}, respectively, where

x_{i} ∈ E_{i} and ν_{E_{N}} = ν_{E}.

Set Γ to be the graph of this resolution:

{1, …, N}

is the set of vertices, and the vertices i and j, i > j, are connected by an oriented edge (notation:

i → j

always implies that i > j), if and only if

x_{i−1} ∈ E_{j}^{i−1},

where E_{j}^{i−1} is the strict transform of the exceptional line E_{j} on S_{i−1}. Set also

p_{j} = (the number of paths from N to j)

for j ≤ N − 1, p_{N} = 1. Set

ν_{i} = mult_{x_{i−1}} Λ^{i−1},

where Λ^{i−1} is the strict transform of the linear system Λ on S_{i−1}. It is easy to see that in terms of the resolution the log-inequality (6) takes the form

∑_{i=1}^{N} p_{i}ν_{i} > n (∑_{i=1}^{N} p_{i} + 1) .

Besides, the following estimate is true

mult_{o} Z ≥ ∑_{i=1}^{N} ν_{i}^{2}.

Lemma 1.4. The following inequality holds

mult_{o} Z > \frac{(∑_{i=1}^{N} p_{i} + 1)^{2}}{∑_{i=1}^{N} p_{i}^{2}} n^{2}.

Proof is obtained by elementary computations: the minimum of the quadratic form

∑_{i=1}^{N} s_{i}^{2}

under the restriction

∑_{i=1}^{N} p_{i}s_{i} = c  (7)

is attained at s_{i} = p_{i}a, where the common constant a is found from (7). Q.E.D. for the lemma.

In view of this lemma Proposition 1.4 is an implication of the following fact. Lemma 1.5. The following inequality holds

(∑_{i=1}^{N} p_{i} + 1)^{2} ≥ 4 ∑_{i=1}^{N} p_{i}^{2}.  (8)

Proof. Note first of all that in (8) the equality can be attained, for instance when N = 1. Assume that N ≥ 2. Set

{2, …, k ≤ N} = {i | i → 1}.

By the definition of the integers p_{i} we get the equality

p_{1} = ∑_{i→1} p_{i} = ∑_{i=2}^{k} p_{i}.

Consequently, (8) can be rewritten as

(2p_{1} + ∑_{i=k+1}^{N} p_{i} + 1)^{2} ≥ 4 ∑_{i=1}^{N} p_{i}^{2}

or, after an easy transformation,

4 (∑_{i=2}^{k} p_{i}) ( ∑_{i=k+1}^{N} p_{i} + 1) + ( ∑_{i=k+1}^{N} p_{i} + 1)^{2} ≥ 4 ∑_{i=2}^{k} p_{i}^{2} + 4 ∑_{i=k+1}^{N} p_{i}^{2}.

It is easy to see that if k = N, then the subgraph of the graph Γ with the vertices {2, …, N} is of the form

2 ← 3 ← … ← N

(since on any surface S_{i} the curve

⋃_{j≤i} E_{j}^{i}

is by smoothness a normal crossing divisor). Hence p_{2} = … = p_{N} = 1 and the inequality (8) holds in an obvious way. So let us assume that N ≥ k + 1. Arguing by induction on the number of vertices of the graph Γ we may assume that the inequality

( ∑_{i=k+1}^{N} p_{i} + 1)^{2} ≥ 4 ∑_{i=k+1}^{N} p_{i}^{2}

is true. Therefore, it is enough to show that the following estimate is true:

(∑_{i=2}^{k} p_{i}) ( ∑_{i=k+1}^{N} p_{i} + 1) ≥ ∑_{i=2}^{k} p_{i}^{2}.  (9)

If k = 2, then by construction we get

p_{2} ≤ ∑_{i=3}^{N} p_{i},

which immediately implies (9). If k ≥ 3, then the subgraph of the graph Γ with the vertices {2, …, } is a chain:

2 ← 3 ← … ← k.

Since k → (k − 1) and k → 1, the vertices k and i, i ∈ {2, …, k − 2}, are not joined by an arrow (oriented edge). Consequently,

j ↛ i

for j ≥ k + 1, i ∈ {2, …, k − 2}. Thus each path from the vertex N to the vertex i ∈ {2, …, k − 2} must go through the vertex k − 1. Therefore

p_{2} = … = p_{k−1} = p_{k} + ∑_{\begin{matrix}i→k−1 \\ i≥k+1 \end{matrix}} p_{i}.  (10)

Lemma 1.6. For any i ∈ {1, …, N} the following inequality holds:

p_{i} ≤ ∑_{j≥i+2} p_{j} + 1  (11)

(if the set {j ≥ i + 2} is empty, the sum is assumed to be equal to zero).

Proof is obtained by decreasing induction on i. If i = N or i = N − 1, then the inequality (11) is true. Now we get

p_{i} − ∑_{j≥i+2} p_{j} = ∑_{j→i} p_{j} − ∑_{j≥i+2} p_{j} =

= p_{i+1} − ∑_{\begin{matrix}j≥i+2 \\ j↛i \end{matrix}} p_{j}.

Write down the set {j | j → i} as {i + 1, …, i + l}. If l = 1, then applying the induction hypothesis, we obtain (11). If l ≥ 2, then similarly to (10) we get

p_{i+1} = … = p_{i+l−1} = p_{i+l} + ∑_{\begin{matrix}j→i+l−1 \\ j≥i+l+1 \end{matrix}} p_{j}.

Therefore

p_{i+1} − ∑_{\begin{matrix}j≥i+2 \\ j↛i \end{matrix}} p_{j} = p_{i+l−1} − ∑_{j=i+l+1}^{N} p_{j}.

Applying the induction hypothesis to i + l − 1, we complete the proof.

Let us come back to the proof of Lemma 1.5. We get

p_{2} = … = p_{k−1} ≤ ∑_{i=k+1}^{N} p_{i} + 1.

But p_{k} ≤ p_{k−1}, so that

(∑_{i=2}^{k} p_{i}) ( ∑_{i=k+1}^{N} p_{i} + 1) ≥ p_{k−1} ∑_{i=2}^{k} p_{i} ≥ ∑_{i=2}^{k} p_{i}^{2},

which is what we need.

The proof of Proposition 1.4 is complete.

Remark. The inequality (11) first appeared in [P3] as an auxiliary estimate. Here we have reproduced the inductive proof of this fact given in [P3] for convenience of the reader.

1.4 The Lefschetz theorem

Let us prove that the condition (i) of Proposition 2 is true for any smooth iterated double cover V. Indeed, let Σ ⊂ | − nK_{V} | be a linear system of divisor without fixed components. Let Y ⊂ Bs Σ be an irreducible subvariety of codimension 2. Take two general divisors, D_{1}, D_{2} ∈ Σ and consider the algebraic cycle of their scheme-theoretic intersection:

Z = (D_{1} ∘ D_{2}) = aY + ∑ a_{i}Y_{i},

where a_{i} ≥ 1, Y_{i} ≠ Y are some irreducible subvarieties of codimension 2. Obviously, a ≥ (mult_{Y} Σ)^{2}. Compute the degrees:

n^{2} deg V = deg Z = a deg Y + ∑ a_{i} deg Y_{i},

where deg V = 2^{m}d_{1} … d_{k}. By the Lefschetz theorem Y is numerically equivalent to mK_{V}^{2} , m ≥ 1, so that deg Y = m deg V, whence

(mult_{Y} Σ)^{2} ≤ a ≤ n^{2}/m ≤ n^{2},

which is what we need. Q.E.D. for the condition (i).

1.5 Proof of Proposition 0.1

Part (i) of Proposition 0.1 is almost obvious. If

χ: X − − → X′
↓ π′
S′

is a birational map onto X′, where dim S′ ≥ 1 and fibers of π′ are uniruled, take Σ′ to be a pull back of a moving linear system on S′. Then c(Σ′, X′) = 0 and therefore by superrigidity c(Σ, X) = 0. But Σ ⊂ | − nK_{X}| is a moving linear system, hence c(Σ, X) = n ≥ 1. Contradiction.

Let us prove part (ii) of Proposition 0.1. Let χ: X − − → X′ be a birational map, φ: Y → X be its Hironaka resolution, so that ψ = χ∘φ: Y → X′ is a birational morphism. The variety Y is non-singular and

Pic Y = ℤφ^{∗}K_{X} ⊕ ⨁_{i∈I} ℤE_{i},

where {E_{i}|i ∈ I} is the set of all the φ-exceptional divisors. By assumption,

Pic Y ⊗ ℚ = ℚψ^{∗}K_{X′} ⊕ ⨁_{j∈J} ℚE′_{j},

where {E′_{j}|j ∈ J} is the set of all the ψ-exceptional divisors. For simplicity of notations set K = φ^{∗}K_{X}, K′ = ψ^{∗}K_{X′}. We get

K_{Y} = K + ∑_{i∈I} a_{i}E_{i} = K′ + ∑_{j∈J} a′_{j}E′_{j},  (12)

where a_{i} ∈ ℤ, a_{i} ≥ 1, and a′_{j} ∈ ℚ, a′_{j} > 0. Let Σ′ = | − mK_{X′}|, m ≫ 0, be a very ample system. Obviously, c(Σ′, X′) = m. Take Σ = χ_{∗}^{−1}Σ′ ⊂ | − nK_{X}|; obviously, c(Σ, X) = n. Twisting by a suitable birational self-map, we may assume that the inequality (1) is already satisfied for χ. Hence n ≤ m. The proper inverse image of the linear system Σ on Y coincides with the inverse image of the linear system Σ′ with respect to ψ. Therefore, there exist positive integers b_{i}, i ∈ I, such that

−mK′ = −nK − ∑_{i∈I} b_{i}E_{i}.

Dividing by m and substituting into (12), we get

(1 − \frac{n}{m}) K = ∑_{i∈I} (\frac{b_{i}}{m} − a_{i}) E_{i} + ∑_{j∈J} a′_{j}E′_{j}.

Since the divisors E_{i} are φ-exceptional and a′_{j} > 0, we get the equality n = m: otherwise we get a contradiction with the ampleness of (−K_{X}). Furthermore, all the divisors E′_{j} turn out to be φ-exceptional and, moreover, {E_{i}|i ∈ I} = {E′_{j}|j ∈ J}, otherwise rk Pic X′ ≥ 2. Thus χ is an isomorphism in codimension one: set

U = X \ ⋃_{i∈I} φ(E_{i}), U′ = X′ \ ⋃_{j∈J} ψ(E′_{j}),

then χ: U → U′ is an isomorphism. Therefore Σ = | − nK_{X}| and χ induces an isomorphism of the linear systems Σ and Σ′. Consequently, χ: X → X′ is an isomorphism. (Strictly speaking, we have proved that for an arbitrary birational map χ: X − − → X′ there exists χ^{∗} ∈ Bir X such that χ ∘ χ^{∗} is an isomorphism.) The rest is obvious. Proof of Proposition 0.1 is complete.

Proof of Proposition 0.2 is similar to the elementary arguments for the part (i) of Proposition 0.1 above.

2 Iterated double covers

2.1 Coordinate presentations

Let ℙ^{♯} be the weighted projective space

ℙ(\underbrace{1, 1, …, 1}_{M+k+1}, l_{1}, …, l_{m}),

where to the weights l_{i} ≥ 2 correspond the new homogeneous coordinates y_{i}, i = 1, …, m. The variety ℙ^{(m)} can be realized as a complete intersection of the type 2l_{1} · . . . · 2l_{m} in ℙ^{♯}:

ℙ^{(m)} = ⋂_{i=1}^{m}{y_{i}^{2} = g_{i}} ⊂ ℙ^{♯}.

In a similar way, the variety V ⊂ ℙ^{(m)} ⊂ ℙ^{♯} is a complete intersection of the type d_{1} · . . . · d_{k} · 2l_{1} · . . . · 2l_{m}.

Let p ∈ Q be an arbitrary point, (z_{1}, …, z_{M+k}) a system of affine coordinates with the origin at the point p. Set

\begin{matrix}f_{i} = g_{i,1} + … + g_{i,d_{i}}, \\ g_{i} = w_{i,0} + w_{i,1} + … + w_{i,2l_{i}} \end{matrix}

to be the Taylor decompositions of the (non-homogeneous) polynomials f_{i}, g_{i} in the coordinates z_{∗}. Here

deg q_{i,j} = j, deg w_{i,j} = j.

If w_{i,0} ≠ 0, that is, p ∉ W_{i}, then for the convenience of computations we assume always that w_{i,0} = 1.

Definition 2.1. The point p ∈ Q is of class e ∈ {0, 1, …, m}, if

e = ♯{i | p ∈ W_{i}}.

We write the set L = L(p) = {i | p ∈ W_{i}} as

{i_{1} < … < i_{e}}.

Let us define a convenient coordinate system at a point p ∈ Q of class e. For simplicity assume that z_{i} = x_{i}/x_{0}, i = 1, …, M + k. Set

u_{i} = y_{i}/x_{0}^{l_{i}}

for i = 1, …, m. The set of regular functions (z_{∗}, u_{∗}) is a system of affine coordinates on an open affine subset U ⊂ ℙ^{♯}, U ≅ ℂ^{M+k+m}. With respect to these coordinates the variety V is given by the system of equations

\{ \begin{matrix}f_{i}(1, z_{1}, …, z_{M+k}) = 0, & i = 1, …, k, \\ u_{i}^{2} = g_{i}(1, z_{1}, …, z_{M+k}), & i = 1, …, m, \end{matrix}  (13)

and from now on we will identify the homogeneous polynomials f_{i}, g_{i} with their non-homogeneous presentations of the type f(1, z_{∗}).

If the point p ∈ Q is of class 0, then for all i = 1, …, m we have g_{i}(p) ≠ 0, that is, w_{i,0} = 1. In this case for any point q ∈ σ^{−1}(p) the linear maps

σ_{∗}: T_{q}ℙ^{(m)} → T_{p}ℙ,
σ_{∗}: T_{q}V → T_{p}Q

are isomorphisms, so that the σ-preimage of any system of local coordinates on ℙ and Q makes a system of local coordinates on ℙ^{(m)} (for instance, (z_{1}, …, z_{M+k})) and V, respectively.

If the point p ∈ Q is of class e ≥ 1, assume that L(p) = {1, …, e}. In this case a natural system of local coordinates on ℙ^{(m)} is given by the set of functions

(z_{j_{1}}, …, z_{j_{M+k−e}}, u_{1}, …, u_{e}),

where z_{j_{1}}, …, z_{j_{M+k−e}} make a system of local coordinates on the complete intersection W_{1} ∩ … ∩ W_{e} ⊂ ℙ.

Lemma 2.1. For i ∈ {1, …, e} the inverse image of the hyperplane {w_{i,1} = 0} is tangent to V at the point q ∈ σ^{−1}(p). In particular, the following isomorphism holds

T_{q}V ≅ T_{p}(Q ∩ W_{1} ∩ … ∩ W_{e}) ⊕ ⟨u_{1}, …, u_{e}⟩^{∗}.

With respect to this isomorphism the tangent cone to the intersection V ∩ {w_{i,1} = 0} is given by the quadratic equation

u_{i}^{2} = w_{i,2}|_{T_{p}(Q∩W_{1}∩...∩W_{e})}.

Proof. It is obvious from the system of equations (13).

2.2 The regularity condition

Let g(z_{∗}) = 1 + w_{1} + … + w_{2l} be a polynomial. Following [P8], consider the formal series

(1 + t)^{1/2} = 1 + ∑_{i=1}^{∞} γ_{i}t^{i} = 1 + \frac{1}{2}t − \frac{1}{8}t^{2} + …,

and make the following formal series in the variables z_{∗}:

\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 polynomials of degree i in z_{∗}. Obviously,

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

For instance, Φ_{1}(w_{∗}) = \frac{1}{2}w_{1}. Furthermore, for j ≥ 1 set

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

and

g^{(j)} = g − [\sqrt{g}]_{j}^{2}.

It is easy to see that the first non-zero homogeneous component of g^{(j)} is of degree j + 1. Denote it by the symbol h_{j+1}[g]. Obviously,

h_{j+1}[g] = w_{j+1} + A_{j}(w_{1}, …, w_{j}),  (14)

where we are not interested in the particular structure of the polynomial A_{j}. Now let us formulate the regularity condition. Set h_{i,j} = h_{j}[g_{i}]. Definition 2.2. (i) A point p ∈ Q of class e = 0 is regular with respect to the

set (f_{∗}; g_{∗}) if the set of homogeneous polynomials

{q_{i,j} | (i, j) ∈ J_{q}} ∪ {h_{i,j} | (i, j) ∈ J_{h}},  (15)

where

\begin{matrix}J_{q} = \{(i, j) | 1 ≤ i ≤ k, 1 ≤ j ≤ d_{i}\}, \\ J_{h} = \{(i, j) | 1 ≤ i ≤ m, l_{i} + 1 ≤ j ≤ 2l_{i}, (i, j) ≠ (m, 2l_{m})\}, \end{matrix}

is regular at p = o = (0, …, 0) ∈ ℂ^{M+k}, that is, makes a regular sequence in O_{p,ℙ} or, in other words, the set of its common zeros is one-dimensional.

(ii) A point p ∈ Q of class e ≥ 1 is regular with respect to the set (f_{∗}; g_{∗}), if the set of homogeneous polynomials

{q_{i,j} | (i, j) ∈ J_{q}} ∪ {w_{i,1} | 1 ≤ i ≤ e} ∪ {h_{i,j} | (i, j) ∈ J_{e}},  (16)

makes a regular sequence in O_{p,ℙ}. Here for simplicity of notations we assume that

L(p) = {1, …, e},

the set J_{q} was defined above and

J_{e} = {(i, j) | e + 1 ≤ i ≤ m, l_{i} + 1 ≤ j ≤ 2l_{i}}.

Proposition 2.1. For a general set (f_{∗}; g_{∗}) ∈ H any point p ∈ Q(f_{∗}) is regular.

Proof. Let us consider, to begin with, the general problem of estimating the codimension of “incorrect” sets of polynomials. Here we follow [P10]. This problem is of an independent interest.

2.3 Non-regular sets of polynomials

Let z_{1}, …, z_{N+1} be a set of variables. The symbol P_{a} stands for the space of homogeneous polynomials of degree a in the variables z_{∗}. Set

L = ∏_{i=1}^{l+1} P_{m_{i}} = {(p_{1}, …, p_{l+1})}

to be the set of all (l + 1)-uples of homogeneous polynomials in the variables z_{∗}, 0 ≤ l ≤ N − 1. With each (l + 1)-uple (p_{∗}) ∈ L we associate the projectivized set of its zeros

Z(p_{∗}) = {p_{1} = … = p_{l+1} = 0} ⊂ ℙ^{N} = ℙ(ℂ^{N+1}) = X.

Here we write Z(p_{∗}) and not V (p_{∗}) in order to make this notation different from our complete intersection V = V (f_{∗}), the principal object of study in this paper. Let

Y = {(p_{∗}) ∈ L| codim_{X} Z(p_{∗}) ≤ l}

be the set of “irregular” (l + 1)-uples. We need an estimate for the codimension of Y. The case l = N − 1, when the “correct” dimension of Z(p_{∗}) is zero, is especially important for applications to Fano varieties. However, for technical reasons, it is more convenient to consider the general case of an arbitrary l ∈ {0, …, N − 1}. Set I = {1, …, l + 1} and

μ_{j} = min_{S⊂I,♯S=j} {∑_{i∈S} m_{i}} ,

b = 1, …, l + 1. Assume that

m = μ_{1} = min{m_{1}, …, m_{l+1}} ≥ 2.

Proposition 2.2. For any l ∈ {1, …, N − 1} the following estimate holds:

codim_{L} Y ≥ min_{j∈{0,...,l}}{(μ_{j+1} − j)(N − j) + 1}.  (17)

Remark. (i) For l = 0 we have the trivial estimate codim_{L} Y = dim P_{m}. (ii) It is not so easy to say whether the estimate (17) is optimal or not. It is obtained below by the method which is completely different from the technique used in [P7] for a similar purpose, that is, to prove existence of regular hypersurfaces V_{M} ⊂ ℙ. The technique of [P7] does not work here: the resulting estimates are too weak for complete intersections; however, it is possible that a combination of the method of this paper and that of [P7] could improve (17).

Proof of Proposition 2.2. Set

L_{a} = ∏_{i=1}^{a} P_{m_{i}} = {(p_{1}, …, p_{a})}.

For each irregular (l + 1)-uple (p_{1}, …, p_{l+1}) fix the first (counting from the left to the right) moment when the codimension of the set of zeros p_{1} = … = p_{a} = 0 fails to take the correct value. Consider the sets

Y_{a} = {(p_{∗}) ∈ L_{a}| codim_{X} Z(p_{1}, …, p_{a}) = codim_{X} Z(p_{1}, …, p_{a−1}) = a − 1}.

Obviously,

Y = ∐_{a=1}^{l+1} (Y_{a} × ∏_{i=a+1}^{l+1} P_{m_{i}}) .

In particular,

codim_{L} Y = min{codim_{L_{a}} Y_{a}|1 ≤ a ≤ l + 1}.

Set I_{a} = {1, …, a} ⊂ I and

μ_{a,j} = min_{S⊂I_{a},♯S=j} {∑_{i∈S} m_{i}} ,

j = 1, …, a. Obviously, μ_{a,j} ≥ μ_{j}. Therefore, it is sufficient to prove the estimate

codim_{L_{a}} Y_{a} ≥ min_{j∈{0,...,a−1}}{(μ_{a,j+1} − j)(N − j) + 1}  (18)

for each a = 2, …, l + 1. We omit the trivial case a = 1, because in this case codim_{L_{1}} Y_{1} = dim P_{m_{1}} ≥ dim P_{m}, which is certainly higher than the right-hand side of (17), just set in (17) j = 0.

The space L_{a}, the set Y_{a} and the inequality (18) do not depend on l. Thus we may simplify our notations, setting a = l + 1 and μ_{a,j} = μ_{j}. In other words, we prove the inequality (17) for Y_{l+1} instead of Y.

Denote Y_{l+1} by Y^{∗}. We have reduced our original problem to a simpler task of estimating codimension of Y^{∗} in L, where Y^{∗} consists of all such (l + 1)-uples of polynomials (p_{1}, …, p_{l+1}) that the set p_{1} = … = p_{l} = 0 has the correct dimension and there exists an irreducible component B ⊂ Z(p_{1}, …, p_{l}), on which p_{l+1} vanishes. Let ⟨B⟩ be the linear span of B, and set codim⟨B⟩ = b ≤ l.

Now set Y^{∗}(b) to be the set of all those (l + 1)-uples (p_{∗}) ∈ Y^{∗}, for which there exists a component B ⊂ Z(p_{1}, …, p_{l}) such that

codim_{X}⟨B⟩ = b, p_{l+1}|_{B} ≡ 0.

Obviously,

Y^{∗} = ⋃_{b=0}^{l} Y^{∗}(b).

Thus it is sufficient to prove that

codim_{L} Y^{∗}(b) ≥ (μ_{b+1} − b)(N − b) + 1.  (19)

for each b = 0, …, l. Let us prove (19).

The case b = 0. Here ⟨B⟩ = ℙ^{N} and therefore each non-zero monomial in the linear forms in z_{1}, …, z_{N+1} of degree d_{l+1} ≥ d does not vanish on B. The space of monomials

{ ∏_{i=1}^{m_{l+1}}(a_{i,1}z_{1} + … + a_{i,N+1}z_{N+1})} ⊂ P_{m_{l+1}}

is closed. Its dimension is equal to

m_{k+1}N + 1 ≥ μ_{1}N + 1.

On the other hand, the set of polynomials p_{l+1} ∈ P_{m_{l+1}} that vanish on B is closed. These two closed sets intersect each other at zero only. Therefore the codimension of Y (0) in L is no smaller than μ_{1}N + 1. This gives (19) for b = 0.

The case b ≥ 1. Here ⟨B⟩ = ℙ^{N−b}. Our strategy is to reduce this case to the previous one (b = 0), restricting the polynomials p_{i} to the linear span P = ⟨B⟩. Although our arguments are rather simple, they are not straightforward and require some extra work.

Definition 3. Let g_{1}, …, g_{e} be homogeneous polynomials on the projective space P, e ≤ dim P −1, deg g_{i} ≥ 2 for i = 1, …, e. An irreducible subvariety C ⊂ P such that ⟨C⟩ = P and codim_{P} C = e is called an associated subvariety of the sequence (g_{∗}), if there exists a chain of irreducible subvarieties R_{j} ⊂ P, j = 0, …, e, satisfying the following properties:

• R_{0} = P;

• for each j = 0, …, e − 1 the subvariety R_{j+1} is an irreducible component of

the closed algebraic set

{p_{j+1} = 0} ∩ R_{j},

where p_{j+1}|_{R_{j}} ≢ 0, so that codim_{P} R_{j} = j for all j;

• R_{e} = C.

If the sequence (g_{∗}) has an associated subvariety, this sequence is said to be good.

Lemma 2.2. (i) The property of being good is an open property.

(ii) A good sequence (g_{∗}) can have at most

[\frac{1}{e + 1} ∏_{j=1}^{e} deg g_{j}]

associated subvarieties.

Proof is easily obtained by induction on e. For g_{1} we have the condition g_{1} ≢ 0, which is clearly an open one. Furthermore, at least one irreducible component of the hypersurface g_{1} = 0 must be of degree ≥ 2, which is also an open condition. There can be at most [deg g_{1}/2] such components.

Assume that Lemma is true for each e = 1, …, j, where j ≤ dim P − 2. Denote by G_{j} the open set of good sequences of length j. By (ii), for each (g_{∗}) ∈ G_{j} there exist at most

[\frac{1}{j + 1} ∏_{α=1}^{j} deg g_{α}]

associated subvarieties. The polynomial g_{j+1} should be non-zero on at least one of them, say R_{j}, and moreover, the intersection

{g_{j+1} = 0} ∩ R_{j}

should contain an irreducible component, the linear span of which is P. Obviously, this determines an open set in

G_{j} × H^{0}(P, O_{P} (deg g_{j+1})).

Each associated subvariety has codimension j + 1 and does not lie in a hyperplane; therefore, its degree is not smaller than j + 2. Q.E.D. for the lemma.

Now let us return to the polynomials p_{∗} and assume that l > b. We claim that we can find (l − b) polynomials among them — after re-numbering we may assume that they are p_{1}, …, p_{l−b} — such that the sequence

p_{1}|_{P} , …, p_{l−b}|_{P}  (20)

is good and B is one of its associated subvarieties.

Proof of the claim. Assume that we have already found j polynomials — let them be p_{1}, …, p_{j} — such that the sequence (p_{1}|_{P} , …, p_{j}|_{P}) is good and one of its associated subvarieties, say R_{j}, contains B. If j < l − b, then R_{j} ≠ B and there exists a polynomial p_{α}, α ∈ {j + 1, …, l}, such that

p_{α}|_{R_{j}} ≢ 0.

Otherwise, R_{j} ⊂ Z(p_{1}, …, p_{l}) and we get a contradiction, since dim R_{j} > dim B. After re-numbering, we may assume that α = j + 1. Now p_{j+1}|_{B} ≡ 0, so that there exists an irreducible component R_{j+1} of the set {p_{j+1} = 0}∩R_{j}, such that R_{j+1} ⊃ B. Proceeding in this way, we obtain our claim.

Now fix a projective subspace P ⊂ ℙ^{N} of codimension b. Let Y^{∗}(P) be the set of all (l + 1)-uples (p_{1}, …, p_{l+1}) ∈ Y^{∗} such that there exists a component B ⊂ Z(p_{1}, …, p_{l}), whose linear span is ⟨B⟩ = P and p_{l+1}|_{B} ≡ 0.

By Lemma 2.2, good sequences form an open set. Thus we may estimate the codimension of Y^{∗}(P) in L, assuming that (p_{1}|_{P} , …, p_{l−b}|_{P}) make a good sequence. Let

B_{1}, …, B_{K}

be all its associated subvarieties, whose linear span is P. If (p_{1}, …, p_{l+1}) ∈ Y^{∗}(P), then the polynomials

p_{l−b+1}|_{P} , …, p_{l+1}|_{P}

must all vanish on one of these subvarieties B_{i}. Now arguing as in the case b = 0, we get

N ∑_{j=l−b+1}^{l+1} deg p_{j} + b + 1 ≥ μ_{b+1}(N − b) + b + 1

independent conditions on p_{l−b+1}, …, p_{l+1}. Taking into account that the Grassmanian has dimension dim G(N + 1 − b, N + 1) = b(N + 1 − b), we get finally

codim_{L} Y^{∗}(b) ≥ μ_{b+1}(N − b) + b + 1 − b(N + 1 − b)
∥
(μ_{b+1} − b)(N − b) + 1,

which is what we need.

In our arguments above we assumed that l > b. If l = b, then B ⊂ ℙ^{N} is a line, l = N − 1 and the inequality (19) can be obtained by an easy dimension count: for a fixed line B the condition p|_{B} ≡ 0 for a polynomial p of degree e ≥ 1 defines a closed algebraic set of polynomials of codimension e + 1 in P_{e}. Therefore,

codim_{L} Y^{∗}(N − 1) ≥ ∑_{i=1}^{N}(m_{i} + 1) − 2(N − 1) = μ_{l+1} − l + 1,

since μ_{l+1} = m_{1} + … + m_{N}.

Q.E.D. for Proposition 2.2.

Corollary 2.1. In the notations of Proposition 2.2 for l ≤ N − 2 the following estimate holds:

codim_{L} Y ≥ mN + 1,  (21)

whereas for l = N − 1 the following estimate holds:

codim_{L} Y ≥ min{mN + 1, μ_{l+1} − l + 1}.  (22)

Proof. Obviously μ_{j} ≥ jm for each j = 1, …, l + 1. Thus

(μ_{j+1} − j)(N − j) + 1 ≥ ε(j) + mN + 1,

where ε(t) = −(m−1)t^{2} +(Nm−N −m)t has two roots, t = 0 and t = N −1− \frac{1}{m−1}. Thus ε(0) = 0 and ε(j) ≥ 0 for j = 1, …, N − 2. Therefore we may omit in (17) the values j = 1, …, l − 1. If l ≤ N − 2, we may also omit the value j = l. Q.E.D. for the corollary.

2.4 Start of the proof of Proposition 2.1

Let U_{sm} ⊂ H be a non-empty Zariski open subset, consisting of all collections (f_{∗}; g_{∗}) such that:

(i) Q = Q(f_{∗}) ⊂ ℙ is a smooth complete intersection;

(ii) all the divisors W_{i}|_{Q} are smooth and the divisor

(W_{1} + … + W_{m})|_{Q}

has normal crossings. In particular, the points of class e ≥ 1 make a smooth quasiprojective variety of codimension e (its closure is the set of points of class ≥ e), and the very variety V = V (f_{∗}; g_{∗}) is smooth.

Let e ∈ {0, …, m} be fixed. Consider the closed subset

Y_{e} = {(x, (f_{∗}; g_{∗})) ∈ ℙ × U_{sm} | x ∈ Q(f_{∗}) is non-regular of class e}.

Let π: ℙ × U_{sm} → U_{sm} be the projection onto the second factor. Now Proposition 2.1 follows from

Proposition 2.3. The closure

\overline{π(Y_{e})} ⊂ U_{sm}

is a proper closed subset for any e ∈ {0, 1, …, m}.

Proof. We show that \overline{π(Y_{e})} has a positive codimension in U_{sm}. Set

Y_{e}(x) = Y_{e} ∩ ({x} × U_{sm}) ⊂ ℙ × U_{sm},
I = {(x, (f_{∗}))|x ∈ Q(f_{∗})} ⊂ ℙ × U_{sm},
I_{e} = {(x, (f_{∗}))|x ∈ Q(f_{∗}) is of class e} ⊂ ℙ × U_{sm},
I_{e}(x) = I_{e} ∩ ({x} × U_{sm}) ⊂ ℙ × U_{sm}.

Identifying {x} × U_{sm} ≅ U_{sm}, we think of Y_{e}(x) and I_{e}(x) as subsets in U_{sm}.

In the non-homogeneous presentation with respect to the system of affine coordinates (z_{1}, …, z_{M+k}) the collection (f_{∗}; g_{∗}) can be identified with the set of homogeneous polynomials q_{i,j}, w_{i,j}. Among the polynomials (15,16) that appear in the regularity condition there are precisely k + e linear forms: these are

q_{1,1}, …, q_{k,1}, w_{1,1}, …, w_{e,1}.

Since (f_{∗}; g_{∗}) ∈ U_{sm}, these linear forms are linearly independent. Set

P = P (f_{∗}; g_{∗}) = {v ∈ ℂ^{M+k} | q_{1,1} = … = w_{e,1} = 0},

P ≅ ℂ^{M−e}. If the point x is non-regular, then the set of common zeros of the rest of polynomials in the set (15) or (16) is of a “wrong” codimension. It is easy to compute that there is not more than M − e − 1 polynomials of degree ≥ 2 in the list (15) or (16). Let us apply Corollary 2.1 to them.

2.5 The polynomials h_{i,j} depend on each other

Note that the polynomials h_{i,j} formally depend on each other, so that generally speaking we cannot apply Corollary 2.1. However, the explicit form of the polynomials h_{i,j} (14) makes it possible to circumwent this obstruction. Indeed, let us assume that the polynomials

w_{i,1}, …, w_{i,l_{i}}

for i ≥ e = 1 are fixed.

Let us construct by induction a sequence of homogeneous polynomials

ξ_{i,l_{i}+1}, …, ξ_{i,2l_{i}},

setting

\begin{matrix}ξ_{i,l_{i}+1} = −A_{l_{i}}(w_{i,1}, …, w_{i,l_{i}}), \\ ξ_{i,j+1} = −A_{j}(w_{i,1}, …, w_{i,l_{i}}, ξ_{i,l_{i}+1}, …, ξ_{i,j}), \end{matrix}

j = l_{i} + 1, …, 2l_{i} − 1. It is obvious that the polynomials ξ_{i,j} depend on w_{i,1}, …, w_{i,l_{i}} only and therefore are also fixed.

Lemma 2.3. For any closed subset T ⊂ P the following conditions are equivalent:

(i) all the polynomials h_{i,j} vanish on T, j = l_{i} + 1, …, a ≤ 2l_{i}; (ii) all the polynomials w_{i,j} − ξ_{i,j} vanish on T, j = l_{i} + 1, …, a ≤ 2l_{i}.

Proof. Indeed,

h_{i,l_{i}+1} ≡ w_{i,l_{i}+1} − ξ_{i,l_{i}+1},

so that for a = l_{i} + 1 the claim of the lemma is obvious. Now we argue by induction on a ≥ l_{i} + 2. If the lemma is true for a ≤ c ≤ 2l_{i} − 1, and any of the conditions (i), (ii) holds for a = c + 1, then in any case

(w_{i,j} − ξ_{i,j})|_{T} ≡ 0

for j = l_{i} + 1, …, c. Therefore

w_{i,j}|_{T} ≡ ξ_{i,j})|_{T}

for j = l_{i} + 1, …, c, so that the conditions

[w_{i,c+1} + A_{c}(w_{i,1}, …, w_{i,l_{i}}, w_{i,l_{i}+1}, …, w_{i,c})]|_{T} ≡ 0

and

[ w_{i,c+1} + A_{c}(w_{i,1}, …, w_{i,l_{i}}, w_{i,l_{i}+1}, …, w_{i,c}) ]|_{T} ≡ 0
∥
−ξ_{i,c+1}

are equivalent. Q.E.D. for the lemma.

Thus for any fixed set of polynomials w_{i,1}, …, w_{i,l_{i}} the polynomials h_{i,j} in the regularity condition can be replaced by the polynomials w_{i,j} − ξ_{i,j}. However, the polynomials w_{i,j}, j = l_{i} + 1, …, 2l_{i}, are arbitrary and therefore the polynomials (w_{i,j} − ξ_{i,j}) are also arbitrary: essentially we just shift the origin in the space of homogeneous polynomials of degree j in z_{∗}. Thus when we estimate the codimension codim_{I(x)} Y (x) that comes from the regularity condition being not satisfied we may apply Corollary 2.1 as if all the polynomials q_{i,j}, h_{i,j} were homogeneous polynomials independent of each other. Now let us, at long last, estimate this codimension.

2.6 Estimate for the codimension

Set Y_{1}(x) = Y^{+}(x) ∪ Y^{♯}(x), where (f_{∗}; g_{∗}) ∈ Y^{+}(x) if and only if l_{1} ≥ 3, whereas (f_{∗}; g_{∗}) ∈ Y^{♯}(x) if and only if l_{1} = 2.

Obviously,

π(Y_{e}) = ⋃_{x∈ℙ}Y_{e}(x).  (23)

It suffices to show that

codim_{U_{sm}} π(Y_{e}) ≥ 1.  (24)

By (23) this estimate follows immediately from the inequality

codim_{U_{sm}} Y_{e}(x) ≥ M + k + 1,

which is what we shall actually prove. Let us consider each possible case in turn. Case e ≥ 2. Here by Definition 2.2 (ii) we have

∑_{i=1}^{k}(d_{i} − 1) + ∑_{i=e+1}^{m} l_{i} = M − ∑_{i=1}^{e} l_{i} ≤ M − e − 2

homogeneous polynomials of degree ≥ 2 on P ≅ ℂ^{M−e}. According to Corollary 2.1,

codim_{I_{e}(x)} Y_{e}(x) ≥ 2M − 2e − 1.

Since obviously

codim_{U_{sm}} I_{e}(x) = k + e,

we get finally the estimate

codim_{U_{sm}} Y_{e}(x) ≥ 2M + k − e − 1.

Taking the union over all x ∈ ℙ, we get

\overline{⋃_{x∈ℙ}Y_{e}(x)}

∥
codim_{U_{sm}} \overline{Y_{e}} ≥ M − e − 1.

Since e < M/2, the inequality (24) is proved, so that \overline{Y_{e}} ⊂ U_{sm} is a proper closed subset.

Case e = 1. In the +-subcase we proceed as above for e ≥ 2. Let consider the ♯-subcase. Here we have precisely M − 2 polynomials on ℂ^{M−1}, so that

codim_{I(x)} Y^{♯}(x) ≥ min(2M − 3, β),  (25)

where

β = ∑_{i=1}^{k} ∑_{j=2}^{d_{i}}(j − 1) + ∑_{i=2}^{m} ∑_{j=l_{i}+1}^{2l_{i}}(j − 1) + 2
∥

1/2 [∑_{i=1}^{k} d_{i}(d_{i} − 1) + ∑_{i=2}^{m} l_{i}(3l_{i} − 1)] + 2.

If the minimum in (25) is attained at 2M − 3, then we proceed as in the +-case above. Thus it suffices to consider the case when the minimum is attained at β. Let us estimate β as a function of non-negative real varieties d_{i}, l_{i}, satisfying the constraints

∑_{i=1}^{k} d_{i} + ∑_{i=2}^{m} l_{i} = M + k − 2, d_{i} ≥ 2, l_{i} ≥ 2.

Lemma 2.4. The following inequality holds:

β ≥ M.

First of all, assuming the claim of the lemma, let us complete the ♯-case. We have

codim_{I_{1}(x)} Y^{♯}(x) ≥ M, codim_{U_{sm}} I_{1}(x) = k + 1,

so that

codim_{U_{sm}} Y^{♯}(x) ≥ M + k + 1,

and taking the union over all the points x ∈ ℙ we get

codim_{U_{sm}} \overline{Y^{♯}} ≥ 1,

which is what we need.

Proof of Lemma 2.4. For convenience of computations we start with the following auxiliary claim.

Lemma 2.5. (i) For s_{i} ≥ 2, ∑_{i=1}^{c} s_{i} = B ≥ 2c, where c ∈ ℤ_{+} is a fixed positive

integer the following inequality holds:

∑_{i=1}^{c} s_{i}(3s_{i} − 1) ≥ 5B.

(ii) For s_{i} ≥ 2, ∑_{i=1}^{k} s_{i} = A ≥ 2k, where k ∈ ℤ_{+} is a fixed positive integer the

following inequality holds:

∑_{i=1}^{k} s_{i}(s_{i} − 1) ≥ A(\frac{A}{k} − 1).

Proof: elementary computations. It is easy to see that in both cases the minimum is attained at s_{1} = … = s_{c} (or = s_{k}.)

Setting

∑_{i=1}^{k} d_{i} = A, ∑_{i=2}^{m} l_{i} = B,

we obtain by Lemma 2.5 the estimate

β ≥ \frac{1}{2}(A(\frac{A}{k} − 1) + 5B) + 2.

Now to prove Lemma 2.4 it is sufficient to check that the inequality

ε(A, B) = A(\frac{A}{k} − 1) + 5B + 4 ≥ 2M

is true under the constraints

1 ≤ k ≤ \frac{M − 1}{2}, A ≥ 2k, B ≥ 0, A + B = M + k − 2.

Replacing B by M + k − 2 − A, let us consider the function of real variable A ∈ ℝ_{+}

ζ(A) = A(\frac{A}{k} − 6).

Its minimum on the interval I = [2k, M + k − 2] is attained either at A = 3k (if 2k ≤ M − 2), or at A = M + k − 2 (if 2k = M − 1). In the first case we get

ε(A, B) ≥ 2A + 5B + 6 = 2(M + k − 2) + 3B + 6 ≥ 2M + 4.

In the second case B = 0 and by elementary computations we get

ε(A, B) ≥ (M + k − 2)\frac{2k − 1}{k} + 6 ≥ 2M.

This completes the proof of Lemma 2.4.

Case e = 0. Here to prove the estimate (24) it is sufficient to show that

β = \frac{1}{2} [∑_{i=1}^{k} d_{i}(d_{i} − 1) + ∑_{i=1}^{m−1} l_{i}(3l_{i} − 1) + l_{m}(3l_{m} − 5)] + 2 ≥ M + 1,  (26)

since if this is the case then the estimate

codim_{U_{sm}} Y_{0}(x) ≥ M + k + 1

holds, and arguing as above we see that \overline{Y_{0}} ⊂ U_{sm} is a proper closed subset. Consider β as a function of non-negative real variables d_{i}, l_{i}, and set

∑_{i=1}^{k} d_{i} = A, ∑_{i=1}^{m−1} l_{i} = B.

Assuming A, B to be fixed, we get by Lemma 2.5 the estimate

β ≥ \frac{1}{2}(A(\frac{A}{k} − 1) + 5B + l_{m}(3l_{m} − 5)) + 2.

Thus to prove the estimate (26) it is sufficient to check that the inequality

A(\frac{A}{k} − 1) + 5B + l_{m}(3l_{m} − 5) + 2 ≥ 2M.  (27)

First of all let us get rid of l_{m}. Since

A + B + l_{m} = M + k, l_{m} ≥ 2,

we assume A and B + l_{m} to be fixed. Since the derivative

[−5t + t(3t − 5)]′ = 6t − 10

is positive for t ≥ 2, the minimum of the left-hand side of (27) is attained at l_{m} = 2. Therefore it is sufficient to prove the inequality

A(\frac{A}{k} − 1) + 5B + 4 ≥ 2M

for A + B = M + k − 2 and the standard constraints for A, B and k. But this has already been done when the ♯-case was considered.

Proof of Proposition 2.1 is complete. For a general collection (f_{∗}; g_{∗}) ∈ H each point x ∈ Q(f_{∗}) is regular.

3 Hypertangent divisors

3.1 How to obtain a bound for the multiplicity

Let X be a smooth projective variety, H ∈ Pic X an ample class. For an irreducible subvariety Y ⊂ X its H-degree (or, simply, degree, when it is clear what ample class is meant) is the integer

deg_{H} Y = (Y · H^{dim Y}).

By linearity the H-degree is defined for any cycle (which is assumed to be equidimensional).

The symbol

\begin{matrix}mult & x \\ deg & H \end{matrix} Y

means the ratio (mult_{x} Y)/ deg_{H} Y, where x ∈ X is a point. Set

λ_{e}(x) = sup_{\begin{matrix}T ⊂X, \\ codim_{X} T =e \end{matrix}} \begin{cases}mult & x \\ deg & H \end{cases} T \} ,

where the supremum is taken over all irreducible subvarieties of codimension e ≥ 1. Assume that on X there exists a set of effective divisors

D_{i} ∈ |a_{i}H|,

i = 1, …, N, such that the set-theoretic intersection

D_{1} ∩ … ∩ D_{N}

is of codimension N ≤ dim X in a neighborhood of the point x. Set

μ_{i} = mult_{x} D_{i} ≥ 1.

Let T ⊂ X be an irreducible subvariety of codimension e ≥ 1, where N ≥ e + 1 and T ∋ x.

Lemma 3.1.There exists a subset L ⊂ {1, …, N} of cardinality N − e (after re-numbering we assume, to simplify the notations, that L = {1, …, N − e}) and a sequence of irreducible subvarieties T_{i}, i = 0, 1, …, N − e, such that:

(i) codim T_{i} = e + i;

(ii) T_{0} = T, T_{i} ⊄ D_{i} and T_{i+1} is an irreducible component of the effective cycle T_{i} ∩ D_{i};

(iii) T_{i} ∋ x and the following inequality holds:

\begin{matrix}mult & x \\ deg & H \end{matrix} T_{i} ≥ \frac{μ_{i}}{a_{i}} · \begin{matrix}mult & x \\ deg & H \end{matrix} T_{i−1}  (28)

for all i = 1, …, N − e.

Proof. Let us prove the existence of L and the set of subvarieties T_{i} by induction on i ∈ {0, 1, …, N − e}. For i = 0 we have nothing to prove.

Lemma 3.2. There is a divisor D_{i}, 1 ≤ i ≤ N, such that T ⊄ D_{i}.

Proof. Assume the converse: T ⊂ D_{i} for all i = 1, …, N. Then

T ⊂ D_{1} ∩ … ∩ D_{N}

so that

codim_{x}(D_{1} ∩ … ∩ D_{N}) ≤ codim T = e ≤ N − 1

contrary to our assumption about the set D_{1}, …, D_{N}. Q.E.D. for the lemma. To simplify the notations we assume that T ⊄ D_{1}. Obviously,

mult_{x}(T ∘ D_{1}) ≥ mult_{x} T · mult_{x} D_{1},
deg_{H}(T ∘ D_{1}) = a_{1} deg_{H} T.

Therefore, the following estimate holds:

\begin{matrix}mult & x \\ deg & H \end{matrix} (T ∘ D_{1}) ≥ \frac{μ_{1}}{a_{1}} · \begin{matrix}mult & x \\ deg & H \end{matrix} T.

The inequality

\begin{matrix}mult & x \\ deg & H \end{matrix} Y ≥ γ  (29)

is certainly non-linear in Y. However, it is equivalent to the linear inequality

mult_{x} Y ≥ γ deg_{H} Y.

Therefore if (29) holds for an effective cycle Y, then there exists a component Y^{+} of this cycle, that is, an irreducible subvariety in X, such that

\begin{matrix}mult & x \\ deg & H \end{matrix} Y^{+} ≥ γ

(since the H-degree of an irreducible subvariety is always strictly positive). So we get that there is an irreducible component T_{1} of the effective cycle (T ∘ D_{1}), an irreducible subvariety of codimension e + 1, such that

\begin{matrix}mult & x \\ deg & H \end{matrix} T_{1} ≥ \frac{μ_{1}}{a_{1}} · \begin{matrix}mult & x \\ deg & H \end{matrix} T,

which is what we need for i = 1.

Assume that the subset {1, …, j}, j ≤ N − e − 1, and a sequence of irreducible subvarieties T_{1}, …, T_{j} satisfy the conditions (i)-(iii).

Lemma 3.3. There is a divisor D_{i}, j + 1 ≤ j ≤ N, such that T_{j} ⊄ D_{i}.

Proof. Assume the converse: T_{j} ⊂ D_{i} for all i = j + 1, …, N. Then

T_{j} ⊂ D_{j+1} ∩ … ∩ D_{N}.

Taking into consideration that by construction

T_{j} ⊂ D_{1} ∩ … ∩ D_{j},

we get

T_{j} ⊂ D_{1} ∩ … ∩ D_{N}.

However x ∈ T_{j} and the codimension of the subvariety T_{j} is equal to

e + j ≤ N − 1,

which gives again (as in the proof of Lemma 3.2) a contradiction with what we assumed about the collection D_{1}, …, D_{N}. Q.E.D. for the lemma.

After re-numbering we may assume that T_{j} ⊄ D_{j+1}. Now we argue as above:

\begin{matrix}mult & x \\ deg & H \end{matrix} (T_{j} ∘ D_{j+1}) ≥ \frac{μ_{j+1}}{a_{j+1}} · \begin{matrix}mult & x \\ deg & H \end{matrix} T_{j}

and therefore there is an irreducible component T_{j+1} of the effective cycle (T_{j} ∘D_{j+1}) that satisfies the inequality (28). Proof of Lemma 3.3 is complete.

Corollary 3.1. The following inequality holds

λ_{e}(x) · min_{\begin{matrix}L⊂\{1,...,N\} \\ ♯L=N−e \end{matrix}} (∏_{i∈L} \frac{μ_{i}}{a_{i}} ) ≤ λ_{N}(x).  (30)

Proof. In the notations above

λ_{N}(x) ≥ \begin{matrix}mult & x \\ deg & H \end{matrix} T_{N} ≥ (∏_{i∈L} \frac{μ_{i}}{a_{i}} ) \begin{matrix}mult & x \\ deg & H \end{matrix} T.

Here L ⊂ {1, …, N} is a subset of cardinality N − e, which depends, generally speaking, on T. The more so,

λ_{N}(x) ≥ min_{\begin{matrix}L⊂\{1,...,N\} \\ ♯L=N−e \end{matrix}} (∏_{i∈L} \frac{μ_{i}}{a_{i}} ) · \begin{matrix}mult & x \\ deg & H \end{matrix} T.

The first factor in the right-hand side does not depend on T. Since the variety T ⊂ X is absolutely arbitrary, we get the inequality (30). Q.E.D. for the corollary.

Corollary 3.2. Assume that the linear system |H| is free and defines a (finite) morphism φ_{|H|}: X → ℙ^{k}. Then the following estimate holds:

λ_{e}(x) ≤ ( min_{\begin{matrix}L⊂\{1,...,N\} \\ ♯L=N−e \end{matrix}} (∏_{i∈L} \frac{μ_{i}}{a_{i}} ))^{−1} .  (31)

Proof. For any irreducible subvariety T ⊂ X of codimension e ≥ 1 there exist divisors D_{i} ∈ |H|, i = 1, …, dim T, such that:

• D_{i} ∋ x, in particular mult_{x} D_{i} ≥ 1;

• the intersection

T^{♯} = T ∩ D_{1} ∩ … ∩ D_{dim T}

is zero-dimensional.

Obviously,

deg T^{♯} = deg T

and

mult_{x} T^{♯} ≥ mult_{x} T · ∏_{i=1}^{dim T} mult_{x} D_{i} ≥ mult_{x} T.

However, T^{♯} is a zero-dimensional scheme, so that

deg T^{♯} ≥ mult_{x} T^{♯}.

From this inequality we obtain that λ_{e}(x) ≤ 1 for all x and e. Now applying the previous corollary we complete the proof.

3.2 Construction of hypertangent divisors

To realize the method described in Sec. 3.1 for the iterated double covers, let us first of all fix some notations. As above, we have a system of affine coordinates (z_{1}, …, z_{M+k}) with the origin at the point p ∈ Q. For 1 ≤ i ≤ k, 1 ≤ j ≤ d_{i} − 1 set

f_{i,j} = q_{i,1} + … + q_{i,j},
D_{i,j}^{ℙ} = \overline{\{f_{i,j} = 0\}}

(the closure is taken in ℙ),

D_{i,j}^{Q} = D_{i,j}^{ℙ}|_{Q},
D_{i,j}^{f} = σ^{−1}(D_{i,j}^{Q}).

Assume that the point p ∈ Q is of class e ≥ 0, and, moreover, if e ≥ 1, then p ∈ W_{1} ∩ … ∩ W_{e}. Set

D_{i}^{+} = \overline{\{w_{i,1} = 0\}}|_{Q},
D_{i} = σ^{−1}(D_{i}^{+}),

i ∈ {1, …, e}. Finally, for a point x ∈ V such that σ(x) = p, let us define with respect to the coordinates u_{i}, i ≥ e + 1 (see Sec. 2.1), the following divisors:

D_{i,j}^{g} = \overline{\{u_{i} − [\sqrt{g_{i}}]_{j} = 0\}}|_{V} ,
D_{i,j}^{+} = σ(D_{i,j}),

where j = l_{i}, …, 2l_{i} − 1. It is easy to see that

D_{i,j}^{?} ∈ |jH|, D_{i} ∈ |H|

for any i, j, listed above, where ? ∈ {f, g}.

Lemma 3.4. (i) For any i, j the following inequality holds:

mult_{x} D_{i,j} ≥ j + 1.  (32)

(ii) For any i ∈ {1, …, e}, where e ≥ 1, the following inequality holds:

mult_{x} D_{i} ≥ 2.

Proof. (i) To begin with, take 1 ≤ i ≤ k. Obviously,

f_{i,j}|_{Q} = (−q_{i,j+1} − … − q_{i,d_{i}})|_{Q},

since f_{i}|_{Q} ≡ 0, which implies the estimate (32). Now assume that i ≥ e + 1. By the definition of the class of a point, g_{i}(p) ≠ 0. In the open affine subset U ⊂ ℙ^{♯}, U ≅ ℂ^{M+k+m} with coordinates (z_{∗}, u_{∗}) we get

[u_{i}^{2} − g_{i}(1, z_{1}, …, z_{M+k})]|_{V} ≡ 0.

Since obviously

(y_{i} + [\sqrt{g_{i}}]_{j})(x) ≠ 0,

we obtain that locally the divisor D_{i,j} is given by the equation

g_{i}^{(j)}|_{V} = 0.

As we have seen above, the first non-zero homogeneous component in g_{i}^{(j)} is h_{j+1}[g_{i}]; it is of degree j + 1.

This completes the proof of the first part of the lemma.

(ii) Assume that e ≥ 1. The hyperplane {w_{i,1} = 0} is tangent to the hypersurface W_{i} at the point p. Hence the divisor D_{i} is singular at the point x. This is what we need.

3.3 The regularity condition for hypertangent divisors

To apply the techniques of Sec. 3.1 it is not enough just to know the multiplicities of hypertangent divisors at the point x. We need more precise information about the tangent cones to these divisors. Set

E = T_{p}(Q ∩ W_{1} ∩ … ∩ W_{e}), U = ⟨u_{1}, …, u_{e}⟩^{∗}.

As we have seen above (Lemma 2.1), the following isomorphism holds

T_{q}V ≅ E ⊕ U.  (33)

The subspace E ⊂ T_{p}ℙ = ℂ_{(z_{∗})}^{M+k} is given by the system of linear equations

q_{1,1} = … = q_{k,1} = w_{1,1} = … = w_{e,1} = 0.

The local computations performed in the proof of the previous lemma show that by the isomorphism (33) the tangent cones to the hypertangent divisors are given by the following equations:

• to the divisors D_{i,j}^{f} —

q_{i,j+1}|_{E} = 0  (34)

for 1 ≤ i ≤ k;

• to the divisors D_{i,j}^{g} —

h_{j+1}[g_{i}]|_{E} = 0  (35)

for i ≥ e + 1;

• to the divisors D_{i} —

u_{i}^{2} = w_{i,2}|_{E}  (36)

for 1 ≤ i ≤ e, if e ≥ 1.

Indeed, the equations (34) have been obtained above in Sec. 3.2. The equations (35) and (36) follow from the local computations made in Sec. 3.2 if one takes into consideration that any linear form L(z_{∗}) that vanish on E defines an element in the square of the maximal ideal M_{x,V} of the point x in the local ring O_{x,V}:

σ^{∗}(L(z_{∗})|_{Q}) ∈ M_{x,V}^{2} .

Thus if a pair of homogeneous polynomials P^{+}(z_{∗}), P^{−}(z_{∗}) of degree a ≥ 1 coincide on E, that is, (P^{+} − P^{−})|_{E} ≡ 0, then

σ^{∗}(P^{+}(z_{∗})|_{Q}) ≡ σ^{∗}(P^{−}(z_{∗})|_{Q}) mod M_{x,V}^{a+1}.

From here the equations (35) and (36) follow immediately.

Lemma 3.5. The set D = {D_{i,j}^{f}, D_{i}, D_{i,j}^{g}} of all hypertangent divisors satisfies the regularity condition

codim_{x} ⋂_{D∈D} D = ♯D.

Proof. It is sufficient to compute the codimension

codim_{T_{x}V} ⋂_{D∈D} T_{x}D.  (37)

Since T_{x}V = U ⊕ E and the coordinates u_{i} come into the equations (36) only — each in its own, the codimension (37) is equal to

codim_{E}{q_{i,j+1}|_{E} = 0, h_{j+1}[g_{i}] = 0},  (38)

where the indices i, j comprise the sets

{1 ≤ i ≤ k, 1 ≤ j ≤ d_{i} − 1}

and

{e + 1 ≤ i ≤ m, l_{i} + 1 ≤ j ≤ 2l_{i}},

respectively. Adding the equations of the hyperplane E, we see that the codimension (37) is precisely the codimension in ℂ^{M+k} of the set determined by all the polynomials that come into the regularity condition (15) or (16). The claim of Lemma 3.5 follows from this fact immediately.

3.4 The Lefschetz theorem once again

Assume that the point p = σ(x) is of class e = 0.

Lemma 3.6. The set-theoretic intersection

T = D_{1,1} ∩ … ∩ D_{k,1} ⊂ V

is of codimension k, coincides with the scheme-theoretic intersection

T = (D_{1,1} ∘ … ∘ D_{k,1})

and satisfies the equalities

deg T = deg V, mult_{x} T = 2^{k}.

Proof. Let show by induction on i = 1, …, k, that the set-theoretic intersection

T_{i} = ⋂_{j=1}^{i} D_{j,1} ⊂ V

is of codimension i, coincides with the scheme-theoretic intersection:

T_{i} = (D_{1,1} ∘ … ∘ D_{i,1})

and satisfies the equalities

deg T_{i} = deg V, mult_{x} T_{i} = 2^{i}.

Indeed, for i = 1 it is true in an obvious way: the tangent cone

T_{x}D_{1,1} ⊂ T_{x}V ≅ T_{p}Q

is given by the quadratic equation

q_{1,2}|_{{q_{1,1}=...=q_{k,1}=0}} = 0,

which is non-trivial by the regularity condition. To come over from i to i + 1, let us use the Lefschetz theorem: by the regularity condition the set of common zeros of the system of equations

q_{1,1} = … = q_{k,1} = q_{1,2} = … = q_{i,2} = q_{i+1,2} = 0  (39)

is of codimension precisely i + 1 in T_{x}V. Hence

T_{x}T_{i} ⊄ T_{x}D_{i+1,1}

and thus

T_{i} ⊄ D_{i+1,1}.

But T_{i} ⊂ V is an irreducible subvariety of codimension i. Therefore the schemetheoretic intersection

T_{i+1}^{+} = (T_{i} ∘ D_{i+1,1})

is an effective cycle of codimension i + 1. However,

deg T_{i} = deg T_{i+1}^{+} = deg V

and i + 1 < dim V/2, so by the Lefschetz theorem we get that T_{i+1}^{+} = T_{i+1} is an irreducible subvariety, the class of which generates A^{i+1}V. Finally, the system of equations (39) gives precisely the tangent cone T_{x}T_{i+1} (as an effective algebraic cycle, that is, respecting the multiplicities of the components). Thus we get

mult_{x} T_{i+1} = 2^{i+1},

which is what we need.

Definition 3.1. An irreducible subvariety x ∈ Y ⊂ V of codimension 2 is said to be correct at the point x (where p = σ(x) is a point of class 0), if there exists an irreducible subvariety R ⊂ Y of codimension

codim_{V} R = k + 1,

satisfying the inequality

\frac{mult_{x}}{deg} R ≥ 2^{k−1} · \frac{mult_{x}}{deg} Y.  (40)

Lemma 3.7. If the subvariety Y ∋ x is not correct at the point x, then the following estimate holds:

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

Proof. Let us apply Lemma 3.1 to the irreducible subvariety Y and the set of hyperplane sections {D_{i,1}}. We obtain that there exists an irreducible subvariety R^{♯} ⊂ Y of codimension k (with respect to V) such that

\frac{mult_{x}}{deg} R^{♯} ≥ 2^{k−2} · \frac{mult_{x}}{deg} Y.

Two cases are now possible. If

R^{♯} ≠ T = D_{1,1} ∩ … ∩ D_{k,1},

then there is a hyperplane section D_{a,1}, 1 ≤ a ≤ k, which does not contain R^{♯}. Thus

codim_{V} (R^{♯} ∩ D_{a,1}) = k + 1,

so that (R^{♯} ∘D_{a,1}) is an effective cycle of codimension k +1, satisfying the inequality

\frac{mult_{x}}{deg} (R^{♯} ∘ D_{a,1}) ≥ 2^{k−1} · \frac{mult_{x}}{deg} Y.

Therefore, there is an irreducible component R of the effective cycle (R^{♯} ∘ D_{a,1}), satisfying the inequality (40). Therefore, Y is a correct subvariety, which contradicts our assumption.

Thus R^{♯} = T = D_{1,1} ∩ … ∩ D_{k,1}. Consequently, by Lemma 3.6 we get the inequality

\frac{mult_{x}}{deg} Y ≤ 2^{2−k} · \frac{mult_{x}}{deg} T = \frac{4}{deg V} ,

as we have claimed. Q.E.D. for the lemma.

3.5 The points of class e = 0

Let us prove, at long last, the estimate (2) for a point x ∈ V such that its image p = σ(x) ∈ ℙ is of class 0. In order to do this, we apply the construction of Lemma 3.1 to an arbitrary subvariety R ⊂ V of codimension k + 1, R ∋ x, and to the set of M − 1 divisors, which we get putting together the f-collection

{D_{i,j}^{f} | 1 ≤ i ≤ k, 1 ≤ j ≤ d_{i} − 1}

and the g-collection

{D_{i,j}^{g} | 1 ≤ i ≤ m, l_{i} ≤ j ≤ 2l_{i} − 1, (i, j) ≠ (m, 2l_{m} − 1)}.

Since the linear system |H| is by construction free (and defines precisely the cover φ_{|H|} = σ: V → ℙ), we get by Corollary 3.2 (inequality (31))

λ_{k+1}(x) ≤ (∏_{i=1}^{k} ∏_{j=2}^{d_{i}−1} \frac{j + 1}{j})^{−1} · (∏_{i=1}^{m} ∏_{j=l_{i}}^{2l_{i}−1} \frac{j + 1}{j})^{−1} · \frac{2l_{m}}{2l_{m} − 1} · \frac{a + 1}{a},  (41)

where a = 2, if max_{i}{d_{i}} ≥ 3, and a = min_{i}{l_{i}} in the opposite case. Indeed, to apply the operation min in the inequality (31) means in the notations of Corollary 3.2 to delete from the product

∏_{i=1}^{N} \frac{μ_{i}}{a_{i}}

precisely the e highest factors. In our case these factors obviously are k twos (corresponding to the tangent hyperplane sections D_{i,1}^{f}) and the next factor (a + 1)/a. The factor 2l_{m}/(2l_{m} − 1) comes into (41) simply because the divisor D_{m,2l_{m}−1}^{g} is not present in our collection. Making in (41) the obvious cancellations, we obtain

λ_{k+1}(x) ≤ \frac{2^{k}}{deg V} · \frac{2l_{m}}{2l_{m} − 1} · \frac{3}{2},

since in any case a ≥ 2. Finally for the correct subvariety Y we get:

\frac{mult_{x}}{deg} Y ≤ 2^{1−k} · \frac{mult_{x}}{deg} R ≤

≤ \frac{4}{deg V} · \frac{3l_{m}}{4l_{m} − 2} ≤ \frac{4}{deg V} ,

which is what we need. Proof of the crucial inequality (2) for a point x ∈ V such that the point p = σ(x) ∈ Q is of class 0 is complete.

3.6 The points of class e ≥ 1

Now assume that for the point x ∈ V the point p = σ(x) ∈ Q is of class e ≥ 1, that is,

p = σ(x) ∈ Q ∩ W_{1} ∩ … ∩ W_{e}.

By the regularity condition the set-theoretic intersection

(⋂_{i,j} D_{i,j}^{f}) ∩ (⋂_{i} D_{i}) ∩ (⋂_{i,j} D_{i,j}^{g})

is in a neighborhood of the point x of the correct codimension

∑_{i=1}^{k}(d_{i} − 1) + e + ∑_{i=e+1}^{m} l_{i} ≥ 1

(the codimension is taken with respect to V). Now let us apply Corollary 3.2 and obtain an upper bound for λ_{2}(x). Taking into consideration that

min_{\begin{matrix}L⊂\{1,...,N\} \\ ♯L=N−c \end{matrix}} ∏_{i∈L} β_{i} × max_{\begin{matrix}L⊂\{1,...,N\} \\ ♯L=c \end{matrix}} ∏_{i∈L} β_{i} = ∏_{i=1}^{N} β_{i},

we get

λ_{2}(x) ≤ (∏_{i=1}^{k} ∏_{j=1}^{d_{i}−1} \frac{j + 1}{j})^{−1} · 2^{−e} · ( ∏_{i=e+1}^{m} ∏_{j=l_{i}}^{2l_{i}−1} \frac{j + 1}{j})^{−1} · 4

∥

\frac{4}{deg V} ,

which is what we need. Proof of the crucial estimate (2) (and therefore of our theorem) is complete.

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