SINGULARITY WHICH HAS NO M-SMOOTHING

V.M.Kharlamov, S.Yu.Orevkov, E.I.Shustin

Abstract. The Harnack bound on the number of real components of a plane real algebraic curve has a natural local version which states that the number of closed real components obtained by a perturbation of a real isolated plane curve singularity having at least one real branch is bounded by the genus of the singularity (perturbations attending this extremal value are called M-smoothings). We show that the latter bound is not sharp for some, explicitly given, singularities.

Introduction

Topologically extremal real algebraic varieties reveal spectacular topological properties. Such a phenomenon for plane projective curves was discovered by D. Hilbert, and is stated in the first part of his 16th problem in the form of conjecture for plane curves of degree 6 with maximal number of real components. D.A. Gudkov [7] corrected the conjecture, generalized it to curves of arbitrary degree and proved it for degree 6 by the meticulous study of degenerations of curves of degree 6 (improving the so-called Hilbert-Rohn method). Another, in a sense, purely topological approach was found by V.I. Arnold; it allowed V.I. Arnold [1], V.A. Rokhlin [ 19] and their followers (see, for example, the surveys [8], [ 21], and [31]) to understand the phenomenon and to obtain a series of different related results which acquired the common name of Gudkov-Arnold-Rokhlin congruences.

It is worth to say that the notion of M-variety (M for maximal) plays a crucial role here. In the case of curves it is the usual Harnack maximality: a real plane projective curve of degree m is called maximal if it has \frac{1}{2}(m − 1)(m − 2) + 1 real components. In general, in accordance with the Smith-Thom bound, which states that any real algebraic variety X satisfies β_{∗}(X_{R}) ≤ β_{∗}(X_{C}) where β_{∗} stands for

Typeset by AMS-TEX 2 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

1991 Mathematics Subject Classification. Primary 14P25; Secondary 57M25, 57R91, 57R95.

The first and third authors were supported in part by the Minerva Center for Geometry at the Tel Aviv University. The paper was finished during the stay of the first author in Stanford university and Max-Planck-Institut für Mathematik.

the sum of the Betti numbers over Z/2 and X_{k} is the set of k-points of X, a real variety X is called maximal if β_{∗}(X_{R}) = β_{∗}(X_{C}).

It happens that the problem of existence of M-varieties is not easy and yet not much is known about it. For plane curves the answer is given by Harnack’s theorem [9]: maximal plane curves exist for any degree. Harnack’s recursive construction of M-curves was generalized by O. Viro [27] to complete intersections in projective spaces of any dimension. Viro’s recursion involves dimension of the ambient space, dimension of the complete intersection and its multidegree. For small dimensions it produces M-varieties and it is expected that this is always the case. For hypersurfaces the existence of M-varieties in any degree is proven by I. Itenberg and O. Viro [10] by means of a toric geometry version of the Viro algorithm.

This problem has local analogues (moreover, according to the Thom-Arnold principle, they are inseparable). The first nontrivial local objects are the smoothings of real isolated plane curve singularities.

Here, by a real isolated plane curve singularity, we mean a germ at 0 ∈ R^{2} of a real analytic function f in 2 variables with finite Milnor number μ. Commonly speaking, we identify it with the germ of a curve C defined by f = 0, refer to a curve C in a real Milnor disc B_{R} ⊂ R^{2} and, also, in the complex Milnor ball B_{C} ⊂ C^{2}, and denote by C_{R} and C_{C} the real and complex point sets of C.

A real analytic curve C′ in a Milnor disc B_{R} of C is called smoothing of C if there exists a real analytic 1-parameter family {C_{t}} of curves in B_{R} such that C_{0} = C, C′ = C_{t} for some t > 0, and each C_{t} with t ≠ 0 is nonsingular and transversal to the boundary of B_{R}. We call such a family smoothing out deformation of C.

The real part C_{R} of a smoothing C of an isolated real plane curve singularity C consists of a finite number of smooth circles (called ovals) and non-closed arcs. The number of arcs does not depend on smoothing and is equal to the number r_{R} of real branches of C. The principal local Harnack bound, which is completely similar to the Harnack bound for projective curves, reads as follows: if C has at least one real branch, the number v of ovals is bounded from above by g = \frac{1}{2}(μ − r + 1), where r is the number of all, real and imaginary, branches of C; otherwise, v ≤ g+1 (similarly to the global one, it can be seen as the Smith-Thom inequality). Yet in the middle 70th, V.I. Arnold [3] posed the question: whether this local Harnack bound is sharp?

SINGULARITY WHICH HAS NO M-SMOOTHING 3

The existence of M-smoothings (i.e., the smoothings which achieve this bound) was proven for many classes of singularities, see, e.g., [11], [15], [17], [18], [28], and [24]. In [18] it is erroneously stated that the construction given there provides M-smoothings for any real plane singularity having no imaginary branches and branches with common tangent. However, Risler’s construction [18] proves the following fact: M-smoothings do exist for any unibranch singularity.

The purpose of this paper is to show that there are singularities for which the local Harnack bound is not sharp.

We treat one real equisingular (i.e., μ-constant) family of singularities, namely, Sirler cusp singularities (introduced by S. Carlson at Arnold’s anniversary conference, Moscow, 1988; unpublished). A Sirler cusp singularity is a bouquet of any three real ordinary cusps, having distinct tangents and spread out as shown in Figure 1. By a real ordinary cusp we mean a germ u^{2} + v^{3} = 0, where u and v are real function germs vanishing at 0 ∈ R^{2} with du ∧ dv ≠ 0. Thus, a Sirler cusp singularity is a product of three real ordinary cusps (u_{1}^{2} + v_{1}^{3})(u_{2}^{2} + v_{2}^{3})(u_{3}^{2} + v_{3}^{3}) = 0 with v_{1} + v_{2} + v_{3} = 0, dv_{1} ∧ dv_{2} ≠ 0, dv_{1} ∧ dv_{3} ≠ 0, and dv_{2} ∧ dv_{3} ≠ 0.

We prove that for all Sirler cusps singularities the local Harnack bound, which gives v ≤ g = 13, is not sharp:

Theorem 1. Any Sirler cusp singularity has no M-smoothings (i.e., it has no smoothing with g = 13 ovals).

Figure 1  Figure 2  Figure 3

It is worth noting that the germ (y^{2} −x^{3})(x^{2} −y^{3})((x−y)^{2} −(x+y)^{3}) = 0 in B_{R}, which is topologically equivalent in B_{R} to Sirler cusp singularities, has, contrary to them, an M-smoothing. The same is true for a bouquet of any three ordinary

4 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

cusps contained in a half plane and having distinct tangents. Indeed, one can either apply the local version of Harnack construction (see [18]) or simplify the singular point into an ordinary 6-fold point and then replace the latter singular point by the Harnack affine sextic (see [14]).

Note also, that for the germs (y^{2} − x^{3})(x^{2} − y^{3})(x^{2} + ky^{3}) = 0 , k > 0 which are degenerations of both the Sirler cusp singularities and the germ in the above remark, there is an M-smoothing at least for some values of k. It would be interesting to study the space of singularities which have M-smoothings.

The paper is organized as follows. In Section 1 we formulate and prove some general geometric properties of smoothings (similar to properties of real algebraic curves), and derive some preliminary prohibitions on M-smoothings C′ of a Sirler cusp singularity C. We start from an application of an improved local Harnack bound which takes into account how the boundary of C_{R} is filled by the nonclosed arcs of C′_{R}. It implies, in particular, that the non-closed arcs in C′_{R} should be arranged as is shown in Figure 2; it is this arrangement, which is then considered throughout the paper. Afterwards, in Section 1, we associate with C′ a 4-dimensional manifold Y with an involution Conj : Y → Y. This manifold is glued from two double coverings of B_{C}: one ramified in C, the other in C′; Conj is induced by the ordinary complex conjugation. Applying traditional tools (like Bézout’s theorem, complex orientations, and congruences and bounds for the inertia indices of Conj_{∗} acting on the quadratic form of Y) to (Y, Conj), we restrict the number of topological types of (B_{R}, C′_{R}) and (B_{R} ∪ C′_{C}, C′_{R}).

In Section 2 we present some methods of constructing sublattices of H_{2}(Y). Such sublattices are used in Section 3 to prohibit all the remaining topological types of (B_{R}, C′_{R}) except two. These prohibitions are based on calculations of the inertia indices and discriminant for full and overfull sublattices of Ker{(1+Conj_{∗}) : H_{2}(Y) → H_{2}(Y)}, cf. [14, 30].

The two exceptional topological types of (B_{R}, C′_{R}) are prohibited in Section 4 by using the methods of [16]; i.e. by applying the Murasugi-Tristram inequality to the links associated with C′ and some, attached to C′, special real pencil of lines.

All the statements containing concrete prohibitions on smoothings of Sirler cusp singularities are called Lemmas; the others are called Propositions.

Note that the proof of Theorem 1 contains many case-by-case prohibitions based

SINGULARITY WHICH HAS NO M-SMOOTHING 5

on various tools, and often the same result can be obtained by different methods simultaneously, and we used this occasion to demonstrate better the variety of tools available (specially, when it allows to diminish the complexity of the proof).

Acknowledgments. Our special gratitude to J.-J. Risler for numerous useful discussions of related problems.

§1. Local analogues of basic facts in the geometry of real plane algebraic curves

1.1 Improved local Harnack bound. This bound takes into account the relative position of the non-closed arcs and the complex boundary of a smoothing:

Proposition 1. (see [11] and [13]) If C is a real isolated plane curve singularity with at least one real branch, the number of ovals of any smoothing C′ of C is bounded by g − a, where a + 1 is the number of topological circles obtained from the union of non-closed arcs of C′_{R} by identification of boundary points coming from a common real branch of C (in other words, a + 1 is the number of connected components of (C′_{R} ∪ ∂C′_{C})/Conj minus the number of ovals), g = \frac{1}{2}(μ − r + 1), μ is the Milnor number and r the number of all, real and imaginary, branches of C.

Proof. Glue a disc on each hole of C′_{C} and apply the global Harnack bound. □

Corollary 1. The non-closed arcs of any M-smoothing of a Sirler cusp singularity are arranged as in Figure 2. □

Note that the above improved local Harnack bound can be sharp when the usual one is not. For example, a local version of Harnack’s construction of M-curves applied to a Sirler cusp singularity gives a smoothing with v = 11 and a = 2; the non-closed arc arrangement for this smoothing is shown in Figure 3 (certainly, Theorem 1 implies that a smoothing with the non-closed arcs arrangement shown in Figure 2 never turns the improved bound into an equality). It would be interesting to know whether the improved local Harnack bound is sharp for any singularity (see [KR] for some related information).

1.2 Elementary consequences of Beźout’s theorem. With an isolated plane curve singularity C one can associate two numeric characteristics: N_{1}(C) and N_{2}(C), which are, respectively, the maximum and minimum of the intersection

6 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

numbers of C′ with the straight lines going through the singularity (the maximum is achieved on one of the tangents). For the Sirler cusp singularities, N_{1} = 7 and N_{2} = 6. These characteristics have the following evident property: Proposition 2. Let C be an isolated plane curve singularity. Then, for any sufficiently small Milnor ball B, and any smoothing C′ sufficiently close to C,

(1) C′ meets any straight line at most in N_{1}(C) points, counting multiplicities;

(2) there exists a ball B_{0} ⊂ B, centered at the singular point, such that smoothings close to C meet any straight line crossing (B_{0})_{C} at least in N_{2}(C) points, counting multiplicities.

Let C be a Sirler cusp singularity and the non-closed arcs of its smoothing C′ be arranged as in Figure 2 (recall that due to Corollary 1 any M-smoothing must be of this kind). Denote the four connected components of B_{R}\C′_{R} by A, B_{1}, B_{2}, B_{3}, and the number of ovals in them by a, b_{1}, b_{2}, b_{3}, respectively (see Figure 2). Corollary 2. An M-smoothing of a Sirler cusp singularity has at most one nonempty oval, and if it does exist then it lies in A. Proof. If there are two non-empty ovals or a non-empty oval in B_{i}, trace a real straight line through two interior ovals, in the first case, and through the interior oval and a boundary point of B_{j}, j ≠ i, otherwise (see Figure 4). Such a line intersects C′_{R} at ≥ 8 points, counting the multiplicities; due to Proposition 2 it is impossible. □

Further on we encode the arrangement of the ovals of C′ by (a; b_{1}, b_{2}, b_{3}), if all the ovals are empty, and by (1⟨a_{1}⟩ ⊔ a_{2}; b_{1}, b_{2}b_{3}), if there is a non-empty oval embracing a_{1} ovals.

In the sequel we use intersections with real conics as well. Lemma 1. Let C be a Sirler cusp singularity, {C′_{t}}_{0≤t≤t^{∗}} its smoothing and {K_{t}}_{0≤t≤t^{∗}} a continuous 1-parameter family of real conics. Assume that for each t the intersection of K_{t} with C′_{t} consists of at least 8 points counting multiplicities. Then there exists a Milnor ball B such that, up to taking smaller t^{∗}:

(1) Each K_{t} meets C′_{t} in at most 14 points counting multiplicities.

(2) Either each (K_{t})_{R}, t ≠ 0, is an ellipse, contained in B_{R}, or each (K_{t})_{R}, t ≠ 0, has two connected components in B_{R}. In the latter case, K_{t} converges

SINGULARITY WHICH HAS NO M-SMOOTHING 7

to the union of two real straight lines (may be, coinciding), as t → 0. These straight lines L_{1}, L_{2} pass through the singular point 0 of C, and K_{t} intersects ∂B_{R} at four points close to the intersection points of L_{1}, L_{2} with ∂B_{R}.

Proof. The statements follow from the fact that the intersection multiplicity of a conic with C at 0 is bounded by 7 if the conic is non-singular at 0 and by 13 if the conic is not a pair of straight lines tangent to C at 0. □

The following classical statement is of major importance in further applications.

The Cayley lemma. If D ∈ R^{2} lies inside the triangle generated by A, B, C ∈ R^{2}, each real conic passing through A, B, C, and D crosses any compact real conic embracing A, B, C, and D at four real points. If five points in R^{2} are the vertices of a convex polygon, then they lie on the same connected real component of the conic passing through them.

1.3 Complex orientations. Let us introduce some invariants of the boundary of an oriented real isolated plane curve singularity involved in the local versions of Rokhlin’s complex orientations formula. Consider in S^{3} = ∂B_{C} a 1dimensional smooth submanifold Z which is invariant under the complex conjugation and nowhere tangent to ∂B_{R}. Suppose that Z is oriented and decomposed into two halves, Z^{+} and Z^{−}, with ∂Z^{+} = ∂Z^{−} = Z ∩ ∂B_{R}. Given an orientation ω of B_{R} and ξ ∈ H_{1}(∂B_{R}, Z ∩∂B_{R}; Z) such that ∂ξ = −∂[Z^{−}] = ∂[Z^{+}] (Z^{±} inherit the orientation from Z), there are well defined linking numbers link([Z^{−}] + ξ, [Z^{+}] − ξ) and link([Z^{−}] + ξ, ∂[B_{R}]).

To make sense of these linking numbers (depending on ω) we push Z^{+} ∪ ∂B_{R} along a special vector field V \sqrt{−1}. Namely, on ∂B_{R} oriented as the boundary of B_{R} we take V tangent to ∂B_{R}, directing its orientation, and then extend V in an arbitrary way to Z^{+}. Clearly, the linking numbers obtained do not depend on the choice of the extension and are invariant under auto-homeomorphisms of ∂B_{C} commuting with complex conjugation and preserving orientation of ∂B_{R}.

Indeterminacy in the choice of ξ is avoided, if we pick up a point p ∈ ∂B_{R} \ Z and take ξ in H_{1}(∂B_{R} \ {p}, Z ∩ ∂B_{R}; Z).

Let C be a real isolated plane curve singularity, B its Milnor ball and Z = C ∩ ∂B_{C}. Orient Z as the boundary of C, fix an orientation ω of B_{R} and pick a

8 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

point p ∈ ∂B_{R} \ C_{R}. Suppose, for simplicity, that C has no imaginary branches. Then C_{R} divides each branch in two halves. The choice of orientation of real branches of C_{R} is equivalent to the selection of an imaginary half on each branch: the complex orientation of an imaginary half determines the boundary orientation of the real branch. Thus, one speaks of a complex orientation of C_{R} and the associated half C^{+}, for which ∂[C^{+}] = [C_{R}]; the conjugated half is denoted by C^{−}. We put Z^{±} = C^{±} ∩ ∂B_{C} and

M_{1} = link([Z^{−}] + ξ, [Z^{+}] − ξ), M_{2} = link([Z^{−}] + ξ, [∂B_{R}]).

Consider a smoothing C′ of C included in a smoothing out deformation {C_{t}} which does not cross p. A smoothing C′ is called of type I if C′_{C} \ C′_{R} is not connected. In this case C′_{C} \ C′_{R} has two connected components, C′^{+} and C′^{−}, the curve C′_{R} is the common part of the boundary of C′^{+} and C′^{−}, and the complex orientation of C′^{+} and C′^{−} defines on C′_{R} the two opposite orientations, called complex orientations of C′_{R}.

Let us pick a complex orientation of C′_{R}. Given a non-closed arc l of C′_{R}, there is one and only one connected component D_{l,p} of B_{R} \ l not containing p. Orient D_{l,p} in a way that its boundary orientation is the complex orientation of l ⊂ C′_{R} and pose ε(D_{l,p}) = 1, if the orientation of D_{l,p} coincides with ω and −1, if it does not.

Two ovals of C′_{R}, bounding an annulus in B_{R}, form a positive injective pair if their complex orientation coincides with a boundary orientation of the annulus, otherwise the injective pair is called negative. Each oval bounds a disc D in B_{R} and the oval is called positive (with respect to ω) if its complex orientation coincides with the boundary orientation defined by ω_{|D}, conversely, it is called negative. Denote by v the number of ovals; by Π^{−} and Π^{+}, the number of negative and positive injective pairs of ovals, respectively; by v^{−} and v^{+}, the number of negative and positive ovals, respectively; by v^{−}(l) and v^{+}(l), the number of negative and positive ovals contained in D_{l,p}, respectively.

If C′ is a type I smoothing of C, the smoothing out deformation transports the complex orientations of C′_{R} and gives in the limit the complex orientations of C_{R}; we call them coherent complex orientations of C_{R} and C′_{R}.

SINGULARITY WHICH HAS NO M-SMOOTHING 9

Proposition 4. (cf. [23]) In the previous notation for any type I smoothing C′ of C and coherent complex orientations of C_{R} and C′_{R},

(1.1)  \begin{aligned}v + 2(Π^{−} − Π^{+}) & + ∑_{l}(r(D_{l,p}) + 2(v^{+}(l) − v^{−}(l))ε(D_{l,p})) \\ & + ∑_{D_{l,p}⊆D_{m,p}} ε(D_{l,p})ε(D_{m,p})r(D_{l,p}) = M_{1} , \end{aligned}

(1.2)  v^{−} − v^{+} − ∑_{l} ε(D_{l,p})r(D_{l,p}) = M_{2} ,

where l, m run over all non-closed arcs of C′_{R} and r(D_{l,p}) = \frac{1}{2}(1 − ε(D_{l,p})).

Proof. Follow Rokhlin’s proof of his global complex orientations formula, see [20]. Glue into C′^{−} the discs D_{l,p} and the discs bounded in B_{R} by the ovals of C′_{R}. This gives an integral relative 2-cycle Σ^{−} in (B_{C}, ∂B_{C}). Its boundary is [Z^{−}] + ξ, where Z^{−} = C′^{−} ∩ ∂B_{C}, ξ ∈ H_{1}(∂B_{R} \ {p}, Z ∩ ∂B_{R}; Z), and Z^{−} inherits the orientation from ∂C′^{−}. By definition, M_{1} = link([Z^{−}] + ξ, [Z^{+}] − ξ) and M_{2} = link([Z^{−}] + ξ, [∂B_{R}]), where Z^{+} = C′^{+} ∩ ∂B_{C}. Take on ∂B_{R} the field V which directs the orientation of ∂B_{R}. We extend it, first, to B_{R} to obtain a field tangent to B_{R} and such that nowhere on C′_{R} it is tangent to C′_{R} with the direction of the complex orientation of C′_{R} (the opposite direction is allowed). Then extend V arbitrarily to Σ^{−} \ B_{R} and shift Σ^{−} along V \sqrt{−1}. The identities (1.1) and (1.2) follow now from counting the intersection number of the shifted cycle with, respectively, −Conj(Σ^{−}) (which is, indeed, Σ^{+}) and B_{R}. □

Proposition 5. If a real isolated plane curve singularity C has only real branches then for any complex orientation of C there exists a smoothing Ĉ of type I whose complex orientation is coherent to the complex orientation of C.

Proof. As in [18, 11] we use a recursion by the number of blowing-ups making the strict transform of C nonsingular.

At each step contracting back an exceptional divisor E we deform the strict transform (and, thus, C) and keep the complex orientation as follows: (1) if E which we are going to contract meets the strict transform C^{∗} (C is the current deformation) at a non-singular branch with multiplicity > 1, then we deform this branch so that it intersects E at distinct and only real points; (2) if C^{∗} has an

10 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

ordinary singularity, a transverse intersection of several real branches, then we move these branches to a general position and smooth out all nodes according to the complex orientation of intersecting branches. □

Figure 4  Figure 5  Figure 6

To apply Proposition 4 to the Sirler cusp singularities we fix the usual counterclock-wise orientation of B_{R} and, in accordance with Proposition 4 and respective notion, call an oval positive, if it is also oriented counter-clockwise, and negative otherwise. Lemma 2. For any M-smoothing of a Sirler cusp singularity the non-closed arcs are arranged as in Figure 2 and have a complex orientation as shown there. The respective complex orientation of the ovals satisfies the following relations:

(1.3)  a^{+} − a^{−} = 1, ∑_{j}(b_{j}^{+} − b_{j}^{−}) = 0

if the smoothing is of type (a; b_{1}, b_{2}, b_{3});

(1.4)  a_{2}^{+} − a_{2}^{−} = 0, a_{1}^{+} − a_{1}^{−} + ∑_{j}(b_{j}^{+} − b_{j}^{−}) = 0

if the smoothing is of type (1⟨a_{1}⟩ ⊔ a_{2}; b_{1}, b_{2}, b_{3}) with positive non-empty oval; and

(1.5)  2(a_{1}^{+} − a_{1}^{−}) + a_{2}^{+} − a_{2}^{−} = 2, a_{1}^{−} − a_{1}^{+} + ∑_{j}(b_{j}^{+} − b_{j}^{−}) = 0

if the smoothing is of type (1⟨a_{1}⟩⊔a_{2}; b_{1}, b_{2}, b_{3}) with negative non-empty oval. Here, by + and − we specify the number of positive and negative ovals in the respective domains. Proof. The first statement follows from Proposition 1. To get the relations (1.3)- (1.5) we apply Proposition 4. The values of M_{1} and M_{2} are calculated using the

SINGULARITY WHICH HAS NO M-SMOOTHING 11

type I smoothing shown in Figure 5; this smoothing is obtained by construction from Proposition 5. Finally, (1.1) and (1.2) give Π^{−} − Π^{+} + ∑(v^{+}(l) − v^{−}(l)) = 0 and v^{+} − v^{−} = 1, and the result follows. □

1.4 Congruence modulo 8.

Lemma 3. An M-smoothing of a Sirler cusp singularity satisfies the congruence

b_{1} + b_{2} + b_{3} − a ≡ 3 mod 8

if the smoothing is of type (a; b_{1}, b_{2}, b_{3}), and the congruence

a_{1} + b_{1} + b_{2} + b_{3} − a_{2} ≡ 4 mod 8

if it is of type (1⟨a_{1}⟩ ⊔ a_{2}; b_{1}, b_{2}, b_{3}).

This is a particular case of the local Gudkov-Arnold-Rokhlin congruences given in [13]. There exist two approaches leading to such congruences. One of them puts into action the double coverings. Since we use further the same coverings to obtain additional prohibitions, we introduce them here and fabricate from them a closed manifold which is the main object of the next section. The construction below is quite general. For brevity, we apply it directly to a Sirler cusp singularity.

Let C be a Sirler cusp singularity and C′ its smoothing. Consider a Milnor ball B, blow up its center, and denote by B̂ the blown-up ball and by Ĉ the strict transform of C (see Figure 6, where the annulus represents B̂_{R} cut along E_{R} and the internal circle represents the double covering of E_{R}). The pairs (B_{C}, C_{C}) and (B̂_{C}, Ĉ_{C}) have the same boundary and there is a diffeotopy {φ_{t}}_{t≥0} of the identity map ∂B_{C} → ∂B_{C} which transforms ∂(C′_{t})_{C} into ∂C_{C} = ∂Ĉ_{C}. Glue B_{C} and B̂_{C} along the boundary by φ_{t}, t > 0. The resulting space, which we denote by X, is a smooth orientable 4-manifold and S = C′_{C} ∪ Ĉ_{C}, where C′ = C′_{t} can be viewed as its smooth orientable 2-submanifold. We equip X (and S) with the orientation which is the usual, complex orientation on B_{C} ⊂ X (and on C′_{C} ⊂ S). Note that the genus of S is 13; indeed, the singularity genus of C, i.e., the genus of S ∩ B_{C}, is 13 and Ĉ_{C} is the union of three discs.

There is a diffeomorphism between X and CP^{2} which transforms the orientation of X into the usual orientation of CP^{2}, and the exceptional curve E ⊂ B̂ into a straight line. We keep on E its original complex orientation. Then [S] ∘ [E] = 6

12 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

and, thus [S] = 6[E] ∈ H_{2}(X) = Z. Since [S] is divisible by 2, there exists a double covering π : Y → X branched along S. It is unique and splits into two coverings π^{+} : Y^{+} → B_{C} and π^{−} : Y^{−} → B̂_{C}. Proposition 6. Y is a simply connected Spin-manifold with β_{2} = 28, χ = 30, and sign = −16, where β_{2}, χ, and sign stand for second Betti number, Euler characteristic and signature, respectively. Proof. To check that π_{1} = 1 apply Van-Kampen’s theorem to Y = Y^{+} ∪ Y^{−}. Since Y^{+} is a Milnor fiber of a space surface singularity z^{2} = f(x, y), it is simply connected, and it remains to observe that π_{1}(Y^{+} ∩ Y^{−}) → π_{1}(Y^{−}) is surjective. The Wu class of Y and the other invariants are calculated by the usual projection formulas for double coverings: w_{2}(Y) = π^{∗}(w_{2}(X)) + [S] = π^{∗}[E] + [S] = 0 ∈ H_{2}(X; Z/2), χ(Y) = 2χ(X) − χ(S) = 6 + 24, and sign(Y) = 2 sign(X) − S ∘_{Y} S = 2 − 18(E ∘_{X} E) = −16. □

The germ C and its smoothing C′ are real. Thus, the complex conjugation Conj : B_{C} → B_{C} induces an involution Conj : X → X and lifts into two commuting involutions Conj_{1}, Conj_{2} : Y → Y. Denote their fixed points sets by X_{R}, Y_{R}^{1}, and Y_{R}^{2}. It is clear that X_{R} = π(Y_{R}^{1}) ∪ π(Y_{R}^{2}) and π(Y_{R}^{1}), π(Y_{R}^{2}) are surfaces with common boundary Ĉ_{R} ∪ C′_{R}. All the involutions preserve orientation of their respective 4-manifolds. Note that these real structures, Conj_{1} and Conj_{2}, considered up to equivariant diffeomorphisms depend on the smoothing C′ chosen. However, for a chosen smoothing they do not depend on t > 0.

Let us set

Y_{R}^{k+} = Y_{R}^{k} ∩ Y^{+}, Y_{R}^{k−} = Y_{R}^{k} ∩ Y^{−}, X_{R}^{+} = X_{R} ∩ B_{C}, X_{R}^{−} = X_{R} ∩ B̂_{C}, and choose the numeration of Conj_{1}, Conj_{2} in a way that π(Y_{R}^{1+}) contains the point p (see Figure 2), and π(Y_{R}^{2+}) does not. Proposition 7. If C′ is an M-smoothing with non-closed arcs arranged as shown in Figure 2, the involution Conj_{2} : Y → Y is Smith-maximal, i.e., β_{∗}(Y_{R}^{2}) = β_{∗}(Y). Proof. The half Y_{R}^{2+} is the trivial double of the half of B_{R} bounded by C′_{R} and not containing p. The half Y_{R}^{2−} covers (non trivially) twice the half of B̂_{R} bounded by Ĉ_{R} and containing E_{R}. It remains to count the number of components of Y_{R}^{2} and calculate χ(E_{R}). □

SINGULARITY WHICH HAS NO M-SMOOTHING 13

Proposition 8. For any k = 1, 2 and any component F of Y_{R}^{k},

F ∘ F = χ(F ∩ Y^{−}) − χ(F ∩ Y^{+}) .

Proof. The smooth surfaces Y_{R}^{k±} are totally real in Y^{±}, as soon as Y^{±} are equipped with their natural complex structure. Our orientation of Y is the complex one on Y^{+} and the opposite one on Y^{−}. □

Remark. As it follows, f.e., from Proposition 8, there is no almost complex structure on the whole Y (and, similarly, on the whole X) for which Y_{R} (and X_{R}) is totally real. It is why, in particular, we should adjust properly many of traditional calculations, though, in many respects, S looks as a “flexible” real curve of degree 6 in P^{2}.

Proof of Lemma 3. Due to Proposition 7, the Gudkov-Arnold-Rokhlin congruence (see [19]) applies. According to this congruence, Y_{R}^{2} ∘ Y_{R}^{2} ≡ sign(Y) mod 16. In our case, due to Proposition 8, Y_{R}^{2} ∘ Y_{R}^{2} = χ(Y_{R}^{2−}) − χ(Y_{R}^{2+}). This gives Y_{R}^{2} ∘ Y_{R}^{2} = 2(b_{1} +b_{2} +b_{3} −a−3) for type (a; b_{1}, b_{2}, b_{3}) and Y_{R}^{2} ∘Y_{R}^{2} = 2(a_{1} +b_{1} +b_{2} +b_{3} −a_{2} −4) for (1⟨a_{1}⟩ ⊔ a_{2}; b_{1}, b_{2}, b_{3}). □

1.5 Arnold inequalities.

Lemma 4. A Sirler cusp singularity has no M-smoothing of type (1⟨a_{1}⟩⊔a_{2}; b_{1}, b_{2}, b_{3}) with b_{1} + b_{2} + b_{3} > 2.

In fact, this is a particular case of a local version of Arnold inequalities (cf. [2]). To prove it we need to know some numerical characteristics of the eigenlattices H′_{ε} = {x ∈ H_{2}(Y)|Conj_{1}x = εx}, H′′_{ε} = {x ∈ H_{2}(Y)|Conj_{2}x = εx}, and H^{ε_{1},ε_{2}} = H′_{ε_{1}} ∩ H′′_{ε_{2}}. Note, right away, that the lattices H^{ε_{1},ε_{2}} are pairwise orthogonal, since the involutions Conj_{1} and Conj_{2}, as any diffeomorphisms preserving orientation, preserve the intersection form in Y.

In what follows the inertia indices of a lattice (i.e., the numbers of positive and negative entries in diagonalizations over R) are denoted by ind_{+}, ind_{−}.

Proposition 9. For any M-smoothing of a Sirler cusp singularity the inertia indices and discriminants of the lattices H′_{ε}, H′′_{ε}, and H^{ε_{1},ε_{2}}, are the following:

H^{−,+} = H′′_{+}, ind_{+}(H′′_{+}) = 1, ind_{−}(H′′_{+}) = 13 − χ_{0}, discr H′′_{+} = −1 ,

14 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

where χ_{0} is the Euler characteristic of the half π(Y_{R}^{j+}) of B_{R} (j = 1, 2) which contains p;

H^{−,−} = Ze, e^{2} = 2,

where e ∈ H′′_{−} is realized by the pull-back of the exceptional curve E ⊂ B̂_{C};

H^{+,−} = H′′_{−} ∩ e^{⊥} = H′_{+}, ind_{+}(H′_{+}) = 4, ind_{−}(H′_{+}) = 9 + χ_{0}, discr H′_{+} = −2;

the other bi-eigenspaces are zero.

Note that χ_{0} = a_{1} − a_{2} + b_{1} + b_{2} + b_{3} for smoothings of type (1 < a_{1} > ⊔a_{2}; b_{1}, b_{2}, b_{3}) and χ_{0} = 1 − a + b_{1} + b_{2} + b_{3} for those of type (a; b_{1}, b_{2}, b_{3}).

Proof. Due to Corollary 1 the non-closed arcs of an M-smoothing should be arranged as in Figure 2. So, according to Proposition 7, Conj_{2} is a Smith-maximal involution, and, hence, H_{2}(Y) = H′′_{+} + H′′_{−}. In particular, the lattices H′′_{+} and H′′_{−} are unimodular. To get their signatures it remains to apply the Lefschetz and Hirzebruch formulas to Conj_{2}. All the other statements follow from the fact that Conj_{1} ∘Conj_{2} is the deck transformation of the covering Y → X and the deck transformation of this covering induces in H_{2}(Y) a reflection with respect to e, which is the generator of the pull-back of H_{2}(X). □

Corollary 3. For any M-smoothing of a Sirler cusp singularity one has −9 ≤ χ_{0} ≤ 13.

Proof. The inertia indices of H′′_{+} and H′_{+} are nonnegative. □

Proposition 10. All the components of Y_{R}^{1} and Y_{R}^{2} are orientable. The components of Y_{R}^{1} realize elements in H′_{+} and that of Y_{R}^{2} elements in H′′_{+}.

Proof. The characteristic class of the double coverings Y_{R}^{1} → π(Y_{R}^{1}) and Y_{R}^{2} → π(Y_{R}^{2}) is induced by w_{1}(X) = [E_{R}]. □

Note that an orientation of Y_{R}^{1} and Y_{R}^{2} can be chosen in the following way. Pick an orientation ω of B_{R}. Represent Y^{+} by z^{2} = f(x, y) where f > 0 in Y_{R}^{1+}. Lift ω to the leafs of Y_{R}^{1+} with Rez > 0 and to the leafs of Y_{R}^{2+} with ℑz > 0. This orientation extends to an orientation of the whole Y_{R}^{1} and Y_{R}^{2}. We call this orientation the upper leaf orientation induced by ω.

SINGULARITY WHICH HAS NO M-SMOOTHING 15

Proof of Lemma 4. Assume that such a smoothing exists. Consider two elements in H′′_{+}: one, [y_{1}], is realized by the component y_{1} of Y_{R}^{2} covering the region of X_{R} containing E_{R}, the second, y_{2}, by the component y_{2} covering the region between the nonempty oval and the ovals embraced by it. They are orthogonal and have nonnegative squares: due to Proposition 8, [y_{1}]^{2} = 2(∑ b_{j} − 3) and [y_{2}]^{2} = 2a_{1} − 2. Then, from Proposition 9, it follows that [y_{1}] and [y_{2}] are dependent over Q. On the other hand, it follows from the exact Smith sequence applied to the deck transformation of Y → X that the images of [y_{1}] and [y_{2}] in H_{2}(Y; Z/2) are independent over Z/2 (the latter follows from the exactness of the Smith sequence and the fact that the boundary of the regions of X_{R} defining [y_{1}] and [y_{2}] does not divide S). □

Summary. Due to Corollary 1, Lemma 3, Corollary 3, and Lemma 4, an Msmoothing of a Sirler cusp singularity has the non-closed arc arrangement shown in Figure 2 and the distribution of its ovals between the regions A, B_{1}, B_{2}, and B_{3}, is one of the following:

(2.1a)  (1; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, b_{1} + b_{2} + b_{3} = 12,
(2.1b)  (5; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, b_{1} + b_{2} + b_{3} = 8,
(2.1c)  (9; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, b_{1} + b_{2} + b_{3} = 4,
(2.1d) (1⟨a_{1}⟩; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, a_{1} + b_{1} + b_{2} + b_{3} = 12, a_{1} ≥ 10,
(2.1e) (1⟨a_{1}⟩ ⊔ 4; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, a_{1} + b_{1} + b_{2} + b_{3} = 8, a_{1} ≥ 6,
(2.1f) (1⟨a_{1}⟩ ⊔ 8; b_{1}, b_{2}, b_{3}), 0 ≤ b_{1} ≤ b_{2} ≤ b_{3}, a_{1} + b_{1} + b_{2} + b_{3} = 4, a_{1} ≥ 2.

§2. Construction of sublattices in H_{2}(Y)

In this section we construct several series of auxiliary 2-cycles in Y and calculate their matrices of intersection numbers. They are used in Section 3 to prohibit most of the remaining schemes of M-smoothings (see Summary in Section 1).

16 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

2.1 Purely real cycles. Recall that, due to our notation, Y_{R}^{1} = FixConj_{1}, Y_{R}^{2} = FixConj_{2}, π(Y_{R}^{1}) contains the point p, and π(Y_{R}^{2}) does not. According to Proposition 10 all the components y_{i}^{1} and y_{j}^{2} of Y_{R}^{1} and, respectively, Y_{R}^{2} are orientable. Let us denote by [y_{i}^{1}] and [y_{j}^{2}] their fundamental cycles corresponding to some (not fixed) orientation.

Let us number the components in such a way that: π(y_{1}^{1}) contains p; π(y_{2}^{1}), …, π(y_{s}^{1}) (s ≥ 2) are the discs bounded by the empty ovals in B_{1}, B_{2}, B_{3} and empty ovals inside the non-empty oval (if it exists); π(y_{1}^{2}) contains the exceptional curve E_{R} ⊂ X_{R}^{−}. If the non-empty oval exists, assume that the component whose projection is bounded by it is y_{2}^{2}.

Note that, according to Summary in Section 1, s = 5, or 9, or 13 and the number of components of Y_{R}^{2} is 14 − s. Proposition 11.

(1) [y_{1}^{1}]^{2} = 6, or 14, or 22 according to whether C′ is of type (2.1a), (2.1d), or (2.1b), (2.1e), or (2.1c), (2.1f);

(2) [y_{1}^{2}]^{2} = 2(b_{1} + b_{2} + b_{3} − 3);

(3) [y_{i}^{1}]^{2} = [y_{j}^{2}]^{2} = −2 for s ≥ i ≥ 2, j ≥ 3;

(4) [y_{2}^{2}]^{2} = −2, or 2a_{1} − 2 according to if π(y_{2}^{2}) is bounded by an empty or non-empty oval;

(5) [y_{i}^{k}] ∘ [y_{j}^{l}] = 0 for (k, i) ≠ (l, j).

Proof. The relations (1) - (4) follow from Proposition 8. The remaining one, (5), is trivial. □ 2.2 Almost real cycles. Pick a point O ∈ B_{R} \ C′ and two real straight lines l_{0} = 0 and l_{1} = 0 passing through this point. Denote by L the part of B_{R} covered by the lines l_{t} = 0, where l_{t} = tl_{0} + (1 − t)l_{1} and 0 ≤ t ≤ 1.

Let us fix k = 1 or 2 and suppose that:

(i) the number m of intersection points of the line l_{t} with C′_{C} is constant for 0 ≤ t ≤ 1, except for a finite number of values of t;

(ii) each of l_{0} and l_{1} intersects C′ transversally in m − 2 real points and with simple tangency in one real point, where the curve C′_{R} comes up from the outside of L (and thus a line from the other half of the pencil, l_{t} = tl_{0} − (1 − t)l_{1}, intersects C′ in m real points, if this line is close to l_{0} or l_{1});

SINGULARITY WHICH HAS NO M-SMOOTHING 17

(iii) the germs of l_{0} and l_{1} at the points of tangency are contained both in the

same component of π(Y_{R}^{k}); (iv) for each 0 ≤ t ≤ 1 the line l_{t} intersects C′_{R} in at least m − 2 points. Under these conditions a natural Conj_{k}-invariant surface with boundary lying

inside ⋃_{t}(l_{t})_{C} appears. The details of the construction (introduced by Viro [29]) can be found in [14]. Here we summarize the results needed for prohibitions.

To construct the desired surface we pick up an one-dimensional submanifold Λ of π(Y_{R}^{k}) ∩ L such that: ∂Λ is the set of the points of tangency of the half pencil l_{t}, 0 ≤ t ≤ 1, with C′_{R}; each l_{t} which is not tangent to C′_{R} intersects Λ in 2 − j points, where 2j = 0 or 2 is the number of imaginary points in the intersection of l_{t} with C′_{C}.

Then there exists a compact orientable surface M ⊂ B_{C} such that

(1) M is contained in ∪_{t}(l_{t})_{C};

(2) ∂M ⊂ C′_{C}, M ∩ B_{R} = Λ;

(3) M = M^{+} ∪ M^{−}, M^{−} = Conj M^{+}, M^{+} ∩ M^{−} = ∂M^{+} ∩ ∂M^{−} = Λ, where M^{+} and M^{−} are homeomorphic to a disc;

(4) the covering Y → B_{C} is trivial over the interior of M.

Denote by Φ the pull-back of M in Y. Due to (2)–(4), Φ is a closed orientable surface. In particular, the homology class ϕ of Φ in H_{2}(Y) is well defined, up to a sign depending on the choice of orientation of Φ. The self-intersection number ϕ∘ϕ does not depend on this choice.

Proposition 12. For any line sector L = ∪_{t}l_{t} satisfying the conditions (i) – (iv) above, one has ϕ ∈ H′_{+}, if k = 2 and ϕ ∈ H′′_{+} otherwise, and in the both cases

ϕ ∘ ϕ = ∑_{q∈∂Λ} v(q),

where the sum is taken over q ∈ ∂Λ and v(q) = −1 if in its neighborhood the germ of l_{t}, which is tangent at q to C′_{R}, is contained in π(Y_{R}^{k}) and 1 otherwise (see Figure 7, where the dotted lines represent Λ).

This fact follows from Lemmas 5.2, 5.3, 5.4 in [14].

Remark. Note that the surface Φ ⊂ Y is smooth and totally real at all points except some of the points of tangency of Λ to C′_{R}. The above formula can be deduced

Figure 7

from the counting the weights of the complex points of a smoothing of Φ and the adjunction formula χ(Φ) + Φ ∘ Φ = c, where c is the total weight of the complex points. The surface in question, Φ, depends on the choice of three objects: the center O, the half pencil l_{t} = tl_{0} + (1 − t)l_{1}, and the one-dimensional manifold Λ. They can be changed with conservation of the homology class of the surface. For example, any small shift of O can be accompanied by a continuous variation of l_{t} and Λ with preserving all conditions imposed on them. If Φ′ is the surface constructed in such a way for a point O′, a variation of O, then each orientation of Φ is transported naturally to Φ′ and Φ′ thus oriented realizes the same homology class ϕ as Φ. The formula for ϕ ∘ ϕ from Proposition 12 was originally obtained by Viro through counting Φ ∘ Φ′.

To calculate the intersection number of ϕ with the real cycles some additional auxiliary numerical characteristics are needed.

Fix an orientation ω of B_{R}. Consider the point q_{0} where Λ meets l_{0}. Orient C′_{R} locally at q_{0} as the boundary of π(Y k̄ ; R), k̄ = 3 − k, where the latter is oriented by ω. Denote by M^{+} that half of M whose cone of tangent rays at q_{0} contains ξ\sqrt{−1} where ξ directs the chosen orientation of C′_{R} at q_{0}.

Represent Y^{+} by z^{2} = f(x, y) with f > 0 in Y_{R}^{1+}, and consider the upper half orientation of Y_{R}^{1} and Y_{R}^{2} induced by ω (see 1.5). Orient in an arbitrary way M̃^{+}, where M̃^{+} is a half of the pull back of M^{+}. Extend this orientation to an orientation of the whole Φ.

SINGULARITY WHICH HAS NO M-SMOOTHING 19

Denote by λ(q) = ±1 the sign of Rez, if k = 1, or ℑz, if k = 2, on the part of Λ̃ = π^{−1}Λ∩M̃^{+} adjacent to q. In other words, this sign is well defined at each point of the interior of Λ, it is constant on each connected component and we extend it to the boundary points by continuity.

Proposition 13. Under conditions of Proposition 12, for any connected component y of Y_{R}^{3−k} equipped with its upper half orientation

[y] ∘ ϕ = ∑_{q∈π(y)∩∂Λ∩L_{1}} λ(q)v(q) − ∑_{q∈π(y)∩∂Λ∩L_{2}} λ(q)v(q);

where L_{1}, L_{2} are the two, properly numbered, halves of L \ O. All the values of λ(q) are determined by λ(q_{0}) and the position of Λ with respect to the branches of C′_{R}: when q jumps from a connected component of Λ to the next one on ∂M^{+}, the variation of λ(q) is equal mod 4 to the number of the components of C′_{R} ∩ L lying in between.

Proof. It follows, for instance, from Lemmas 5.2, 5.3, 5.4 in [14].

Corollary 4. Let L = ∪_{t}l_{t} be a line sector satisfying (i) – (iv) and touched from the outside by two ovals ν_{1} and ν_{2} lying in B_{s}. Suppose that k = 2 and the lines l_{t} with 0 < t < 1 do not intersect any oval of C′. Then ϕ ∈ H′_{+} and

ϕ^{2} = −2, ϕ ∘ [y_{i}^{1}] = λ(q_{0}), ϕ ∘ [y_{j}^{1}] = λ(q_{0}),
ϕ ∘ [y_{m}^{1}] = 0, m ≠ i, j,

where: y_{i}^{1}, y_{j}^{1} are the two cycles defined in section 2.1 and coming from the ovals ν_{1}, ν_{2}; and [y_{·}^{1}] is the fundamental class corresponding to the upper half orientation. In particular, ϕ can be chosen in a way that λ(q_{0}) = 1.

2.3 Imaginate cycles. This construction contains less information but it is more general than the others. It is applied to any Z/2 × Z/2 action. We are applying it to Y^{+}. There, such an action is generated by Conj_{1} and Conj_{2}.

Let y_{1} and y_{2} be two components of Y_{R}^{k}, k = 1 or 2. Assume that C′ is dividing and there is a smooth arc Λ ⊂ X_{R}^{k̄+}, k̄ = 3−k, which connects two points q_{1} ∈ ∂π(y_{1}) and q_{2} ∈ ∂π(y_{2}).

Connect q_{1} and q_{2} by a smooth arc Σ in one half of C′. Lift Λ to a smooth arc Λ̃ connecting in Y the pull backs q̃_{1} and q̃_{2} of q_{1} and q_{2}. Since Y^{+} is simply

20 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

connected (see 1.4), the loop γ formed by Λ̃ and Σ bounds in Y^{+}. Let D be an oriented surface in Y^{+} with ∂D = γ. The combined cycle (1 + Conj_{k})(1 − Conjk̄)D represents an element s(D) ∈ H_{2}(Y). Proposition 14. Let C′ be dividing. In the above notation, s(D) ∈ H′_{+} if k = 1 and s(D) ∈ H′′_{+} otherwise. For a component y of Y_{R}^{k}, s(D) ∘ [y] = 0 mod 4 if y does not contain any of q_{j} and ±1 mod 4 if y contains one and only one of them. It holds

s(D) ∘ s(D) = 0 or 2 mod 4

if, respectively, a complex orientation of C′_{R} at q_{1} and q_{2} is directed to the same or opposite sides of Λ. Moreover, for two imaginate cycles s(D) and s(D′) in generic mutual position

s(D) ∘ s(D′) = 2([Λ ∘ Λ′] + [Σ ∘ Σ′]) mod 4.

The proof is by straightforward calculation of the orbits of intersection (or selfintersection) points. The result is contained in Lemmas 2 – 4 in [22]. 2.4 Mixed cycles. Such cycles are built out of Y_{R} and some surfaces in Y invariant simultaneously under Conj_{1} and Conj_{2} (such an invariance implies the invariance under the deck transformation of Y → X).

Let us fix k = 1 or 2. Let F^{j}, 1 ≤ j ≤ s, be smooth oriented surfaces embedded in Y which intersect each other transversally and satisfy the following conditions:

(i) the intersection points lie outside (Y^{+} ∩ Y^{−}) ∪ Y_{R}^{k}, each F^{j} is transversal to ∂Y^{+} = ∂Y^{−} and, thus, F^{j,+} = F^{j} ∩ Y^{+} and F^{j,−} = F^{j} ∩ Y^{−} are oriented surfaces with common boundary ∂F^{j,+} = ∂F^{j,−} = F^{j} ∩ ∂Y^{±};

(ii) for every j the real part F_{R}^{j} = F^{j} ∩ Y_{R}^{k} of F^{j} divides F^{j} in two halves Φ_{+}^{j}

and Φ_{−}^{j} = Conj_{k}Φ_{+}^{j} with F_{R}^{j} as the common boundary; (iii) the sum of the fundamental classes of F_{R}^{j} corresponding to the boundary

orientation determined by Φ_{+}^{j} realizes 0 ∈ H_{1}(Y_{R}^{k}; Z); (iv) for each j the tangent bundle of Y_{R}^{k} restricted to ∂Φ_{±}^{j} contains a field K of

nonempty open convex cones of lines such that all the complex directions generated by the elements of K are transversal to F^{j} (in particular, K does not contain the directions tangent to ∂Φ_{±}^{j}) and at the points of ∂Φ_{±}^{j} ∩ ∂Y^{±} the real directions tangent to ∂Y_{R}^{k} belong to K.

SINGULARITY WHICH HAS NO M-SMOOTHING 21

Note, that the condition (iv) and the orientability of F^{j} are satisfied if F^{j,+} and F^{j,−} are holomorphic.

Equip Φ_{+}^{j}, 1 ≤ j ≤ s, with the orientation inherited from F^{j} and Φ_{−}^{j} with the opposite one. Denote by F and Φ_{±} the oriented surfaces which are the union of F^{j} and, respectively, Φ_{±}^{j} over all 1 ≤ j ≤ s. Then [Φ_{−}] = Conj_{k}[Φ_{+}].

Under conditions (i)–(iii), there exists a 2-dimensional singular cycle [N] which is an integral combination of the fundamental cycles of the closures of the connected components of Y_{R}^{k}\F and which satisfies the relation ∂[N] = −∂[Φ_{+}] − ∂[Φ_{−}]. Let us denote by n_{N} the corresponding integral valued function on Y_{R}^{k} and set

ξ = [Φ_{+}] + [N] + [Φ_{−}].

Proposition 15. For any F = ∪_{j}F^{j} with F^{j} satisfying the conditions (i)–(iv),

ξ^{2} = [F]^{2} + ∫_{Y_{R}^{k−}} n_{N}^{2}(x)dχ(x) − ∫_{Y_{R}^{k+}} n_{N}^{2}(x)dχ(x);

and, for any j,

ξ ∘ [y_{j}^{k}] = ∫_{y_{j}^{k}∩Y_{R}^{k−}} n_{N}(x)dχ(x) − ∫_{y_{j}^{k}∩Y_{R}^{k+}} n_{N}(x)dχ(x).

The integrals above are taken against Euler characteristic (see, for example, [30]).

Proof. Due to Proposition 10, the components of Y_{R}^{k} are orientable. Hence, ∂Φ_{±} is two sided in Y_{R}^{k}. Pick a nonzero section ν_{1} of the field K (see (iv) above) on ∂Φ_{±} which is tangent to ∂Y_{R}^{k±} on ∂Φ_{±} ∩ ∂Y^{±}. It is everywhere tangent to Y_{R}^{k} and transversal to ∂Φ_{±}. Extend ν_{1} to a section ν_{1}^{+} of the tangent vector field of Y_{R}^{+} and a section ν_{1}^{−} of the tangent vector field of Y_{R}^{−} respecting the following conditions: ν_{1}^{±} are nowhere zero on ∂Y_{R}^{±}; ν_{1}^{ϵ} is directed inside Y_{R}^{ϵ} on the components β of ∂Y_{R}^{ϵ} \ ∂Φ_{±} where ν_{1} is directed inside β and ν_{1}^{ϵ} is directed outside Y_{R}^{ϵ} on the other components of ∂Y_{R}^{ϵ} \ ∂Φ_{±}; ν_{1}^{+} and ν_{1}^{−} restricted to ∂Y_{R}^{±} have equal components tangent to ∂Y_{R}^{±} and their normal components are opposite.

Due to these conditions and (iv), multiplying ν_{1}^{+} and ν_{1}^{−} by \sqrt{−1} (in respective natural complex structures on Y^{±}) we get a continuous vector field ν_{2} on Y_{R} which is transversal to N^{±} and F on ∂Φ_{±}. The numbers ξ^{2} and ξ ∘ [y_{j}^{k}] are degree 2 or 1 polynomials in the index of ν_{2} and obstruction for extending ν_{2} to Φ_{±}. It remains

22 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

to use the fact that Y_{R}^{k±} are totally real with respect to the complex structure of Y^{±} (cf. Proposition 8). □

Consider a smooth Conj-invariant oriented surface G in X which meets S = C′_{C} ∩ Ĉ_{C} transversally. Let U be a simply connected domain in B_{R} or in B̂_{R} (f.e., U can be the interior of B_{R}). Pick a Conj-invariant function f : U → C such that f = 0 defines S in U and f ≥ 0 in π(Y_{R}^{k}) ∩ U. Then over a regular neighborhood T (U) of U the covering Y → X is given by the standard projection V → T (U), V = {z^{2} = f} ⊂ C × T (U), and Conj_{k} = Conj × Conj on V.

Assume that the covering surface F = π^{−1}(G) is divided by F_{R} = F ∩ Y_{R} into two halves Φ_{+} and Φ_{−} with ∂Φ_{+} = ∂Φ_{−} = F_{R}. An orientation of F induces an orientation on Φ_{+}, which, in its turn, defines an orientation of ∂Φ_{+}. In particular, {z ≥ 0}∩π^{−1}(G_{R})∩Y_{R}^{k} ⊂ ∂Φ_{+} becomes oriented. We descend the latter orientation via projection into an orientation of G_{R} ∩ π(Y_{R}^{k}). Proposition 16. Let G and U be as above. If at each point of G_{R} ∩ S the tangent planes of G and their orientations are complex, then the introduced orientations of the connected components of G_{R} ∩ π(Y_{R}^{k}) alternate when moving along G_{R} in U. Proof. Same as for the alternating rule for the complex orientations of the real part of the Riemann surface of a function z = \sqrt{Π(x − a_{i})}, see [22, Lemma 8]. □ Remark. The condition on the tangent planes and their orientation can be replaced by the hypothesis that G is holomorphic in U.

Let us introduce one auxiliary notation. It is applied to smooth curves g, g′ in X_{R} intersecting transversally and equipped with alternating orientations of g∩U∩π(Y_{R}^{k}) and g′ ∩ U ∩ π(Y_{R}^{k}), where, as above, U is a simply connected domain in B_{R} or B̂_{R}. If q ∈ X_{R}\π(Y_{R}^{k}) is an intersection point of g with g′ and q ∈ U, put w(q) = 1 if the compound orientation of g ∩ U ∩ π(Y_{R}^{k}) and g′ ∩ U ∩ π(Y_{R}^{k}) alternate when moving along g to p and then along g′ out of p, and put w(q) = −1 otherwise. Proposition 17. Suppose that F = ∪F^{j}, 1 ≤ j ≤ s, where the family F^{j} satisfies the conditions (i)–(iv) and each F^{j} is the pull-back of an orientable surface G^{j} ⊂ X satisfying the hypotheses of Proposition 16. If all the singular points of F belong to B_{R} ∪ U, U being a simple connected domain of B̂_{R}, then

[F]^{2} = 2 ∑_{j}[G^{j}]^{2} + 4 ∑_{q∈B_{R}} w(q) − 4 ∑_{q∈U} w(q) ,

SINGULARITY WHICH HAS NO M-SMOOTHING 23

where q runs over all intersection points of G^{j}.

For two mixed cycles ξ, ξ′, constructed by means of the corresponding objects F, G = ∪G^{j}, Φ_{±}, N, n_{N}, U and F′, G′ = ∪G^{j}′, Φ′_{±}, N′, n_{N′}, U′ = U, respectively, assume, in addition, that any two irreducible components of G and G′ meet transversally and only at points belonging to (B_{R} ∪ U)\π(Y_{R}^{k}). Then

ξ ∘ ξ′ = 2 ∑_{q∈B_{R}} w(q) − 2 ∑_{q∈U} w(q) + ∫_{Y_{R}^{k−}} n_{N}n_{N′}(x)dχ(x) − ∫_{Y_{R}^{k+}} n_{N}n_{N′}(x)dχ(x) ,

where q runs over all intersection points of surfaces G, G′.

Proof. Same as in [22, Lemma 10] and similar to the proof of Proposition 15.

§3. Prohibitions via lattice calculations

In this section we prohibit most of the remaining schemes of M-smoothings of a Sirler cusp singularity (see Summary in the end of §1). To prohibit them we apply the constructions from Section 2 to get some special sublattices in H_{2}(Y). The prohibitions come from analysis of their inertia indices and discriminants. Often, the lattices obtained are isomorphic to A_{n}, n ∈ N, the standard integral negative definite lattice of rk = n generated by elements e_{i}, 1 ≤ i ≤ n, with e_{i}^{2} = −2 and e_{i} ∘ e_{j} = 1 for |i − j| = 1 and 0 for |i − j| > 1.

Typically, the construction of cycles is preceded by application of the Bézout theorem to specially selected straight lines and conics.

3.1 Prohibitions for smoothing with non-empty oval.

Lemma 6. There is no smoothing of type (2.1d), (2.1e), (2.1f), except, perhaps,

(3.1)  (1⟨a_{1}⟩ ⊔ a_{2}; 0, 1, 1), a_{1} = 10, 6, or 2 .

Proof. Let C′ be a smoothing of type (2.1d), (2.1e), or (2.1f) (recall that b_{1} ≤ b_{2} ≤ b_{3}). Without loss of generality assume that all the ovals of C′ are in a sufficiently small ball B_{0} ⊂ B, each non-closed arc of C′ crosses ∂B_{0} at 2 points, and the real straight lines intersecting B_{0} cross C′_{R} near ∂B_{R} at ≤ 1 point. Fix a point q on the non-closed arcs of C′ as shown in Figure 8, consider the pencil of real straight lines through q, denote by Q the minimal segment of this pencil, containing all lines which intersect empty ovals inside the non-empty one, and denote by R the

Figure 8

minimal segment of this pencil, containing all the lines which intersect ovals in the domain B_{3} (it may be empty if b_{3} = 0). Proposition 18. The lines L ∈ Q do not intersect ovals lying outside the nonempty oval. The lines L ∈ R do not intersect ovals lying outside the domain B_{3}. Proof. If the first statement is violated, one obtains the situation shown in the right drawing in Figure 9. Then the conic passing through q and intersecting the four empty ovals indicated has ≥ 16 common points with C′ in contradiction to Lemma 1.

Assume that the second statement is violated. Then, up to switching q to q′, we have either the situation shown in the left drawing in Figure 9, or that in the left drawing in Figure 12, or that in Figure 11. In the first two cases the conics shown contradict Lemma 1. In the last case the straight line passing through the two ovals meets C′ at ≥ 8 points, which contradicts Proposition 2. □

Now, using Proposition 18, the construction of real and almost real cycles presented in subsections 2.1, 2.2, and Propositions 11 and 12, one obtains a sublattice of type A_{2a_{1}−1} ⊕ A_{2b_{3}−1} if b_{3} > 0, or a sublattice of type A_{2a_{1}−1} if b_{3} = 0 in H′_{+}. According to Proposition 9, this implies

ind_{−}(H′_{+}) = 9 + χ_{0} = 2(a_{1} + b_{1} + b_{2} + b_{3}) − 3 ≥ 2(a_{1} + b_{3}) − 2 if b_{3} > 0,

ind_{−}(H′_{+}) = 9 + χ_{0} = 2(a_{1} + b_{1} + b_{2} + b_{3}) − 3 ≥ 2a_{1} − 1 if b_{3} = 0, which contradicts b_{1} ≤ b_{2} ≤ b_{3}, and thus completes the proof of Lemma. □

Figure 9

Lemma 7. There is no smoothing of type (3.1).

Proof. Let C′ be a smoothing of type (3.1). First, construct a mixed cycle as described in 2.4. Pick a point q_{0} in the disc bounded by an empty oval inside the non-empty one, and consider the pencil P of real straight lines through q_{0}. Let L′_{1}, L′_{2} ∈ P be the tangents to the non-closed arcs bounding the domains B_{1}, B_{2} such that the interval (L′_{1}, L′_{2}) ⊂ P consist of lines which do not meet the nonclosed arcs indicated. Denote by L_{1}, L_{2} ∈ P the lines close to L′_{1}, L′_{2}, respectively, and intersecting the above non-closed arcs in two real points (see Figure 10).

Figure 10 Figure 11

Without loss of generality, one can suppose that q_{0} is the singular point of C and that the gluing diffeomorphism φ : ∂B → ∂B̂ defining X takes ∂(L_{1})_{C} and ∂(L_{2})_{C}

26 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

to ∂(L̂_{1})_{C} and ∂(L̂_{2})_{C}, where L̂_{1}, L̂_{2} are the strict transforms of L_{1}, L_{2} in B̂. Thus, for every j = 1, 2, we get a sphere G^{j} = (L_{j})_{C} ∪ (L̂_{j})_{C} ⊂ X meets S = C′_{C} ∪ C_{C} at eight points which are all real. Hence F^{j} = π^{−1}(G^{j}) ⊂ Y is a surface of genus 3. It is invariant with respect to Conj_{1} and Conj_{2}, F_{R}^{j} = Y_{R}^{2} ∩ F^{j} consists of four ovals and divides F^{j} into two halves, Φ_{+}^{j} and Φ_{−}^{j}, cf. 2.4.

Proposition 19. Under proper orientations of Φ_{+}^{1} and Φ_{+}^{2}

[∂Σ_{1}^{+}] + [∂Σ_{2}^{+}] = 0 ∈ H_{1}(Y_{R}^{2}) .

Proof. From Proposition 16 applied to G = G^{1} ∪G^{2} and U = B_{R} it follows that Φ_{+}^{1} and Φ_{+}^{2} can be oriented in a way that the associated orientations of the connected components of ((L_{1})_{R} ∪ (L_{2})_{R}) ∩ π(Y_{R}^{2}) are as in Figure 10. Under this choice of orientations, there exists an oriented part N ⊂ Y_{R}^{2} such that [∂N] = [∂Φ_{+}^{1}] + [∂Φ_{+}^{2}]. □

To finish the construction of the mixed cycle take N as in the proof of Proposition 19 and lying over

⋃_{L∈[L_{1},L_{2}]} L_{R} ∩ Y_{R}^{2},

where [L_{1}, L_{2}] ⊃ [L′_{1}, L′_{2}]. Then put

ξ = [Φ_{+}^{1}] + [Φ_{−}^{1}] + [Φ_{+}^{2}] + [Φ_{−}^{2}] + 2[N] .

Note that N is contained in Y_{1}^{2} ∪ Y_{2}^{2} and denote Δ_{1} = N ∩ Y_{1}^{2}, Δ_{2} = N ∩ Y_{2}^{2}.

Proposition 20. One has

ξ^{2} = 8s, ξ ∘ [y_{2}^{2}] = ±4s, [y_{2}^{2}]^{2} = 2(a_{1} − 1),

where s is the number of empty ovals bounding from inside the domain π(Δ_{2}).

Proof. The third formula is contained in Proposition 11. The second one follows from Proposition 15: the only nontrivial contribution in the integrals is given by Δ_{2} where n_{N} = 2 and, thus, ξ ∘ [y_{2}^{2}] = −2χ(Δ_{2}) = ±4s. To calculate ξ^{2} we apply Propositions 15, 17: the contribution of 2(Δ_{1} + Δ_{2}) is

4(−χ((Δ_{1} ∩ Y^{+}) ∪ Δ_{2}) + χ(Δ_{1} ∩ Y^{−})) = 8s ,

SINGULARITY WHICH HAS NO M-SMOOTHING 27

the contribution of self intersections of Φ_{+}^{1} + Φ_{−}^{1} and Φ_{+}^{2} + Φ_{−}^{2} is 4, and the contribution of (Φ_{+}^{1} + Φ_{−}^{1}) ∩ (Φ_{+}^{2} + Φ_{−}^{2}) is −4 by the definition of w(q). □

By Proposition 20 the classes ξ, [y_{2}^{2}] generate a sublattice in H′′_{+}, which is positive definite if 0 < s < 2a_{2} − 1, and is positive semidefinite if s = 0 or 2a_{2} − 1. Since ind_{+}(H′′_{+}) = 1 by Proposition 9, to complete the proof of Lemma 7, it remains to show that ξ, [y_{2}^{2}] are linearly independent for the cases s = 0 and s = 2a_{2} − 1.

Assume that s = a_{2} − 1. One should show that [ξ] ≠ ±2[y_{2}^{2}].

Pick a smooth arc λ ⊂ A, which does not intersect L_{1}, L_{2}, and C′ and joins a point p_{1} ∈ ∂B_{2} and a point p_{2} on the non-empty oval such that p_{1}, p_{2} ∉ π^{−1}(Δ_{1} ∪ Δ_{2}). The construction in 2.3 provides us with an imaginate cycle c such that ξ ∘ c ≡ 0 mod 2, while [y_{2}^{2}] ∘ c ≡ 1 mod 2 according to Proposition 14.

Assume that s = 0. Then one has to show that [ξ] ≠ 0.

Consider the pencil segment Q. Proposition 18 and and the rule of orientations in a pencil [6] imply that a_{1}^{+} − a_{1}^{−} = 0 in the notation of Lemma 2. The formulae (1.4), (1.5) imply that one of the ovals in B_{2} ∪ B_{3}, say, that in B_{3}, is oriented negatively with respect to the boundary of B_{3}. Then the same rule of orientations applied to the pencil generated by L_{1}, L_{2} shows that there exist empty ovals in A swept by lines L ∈ [L_{1}, L_{2}]. Now we pick up a point p_{1} on such an oval, a point p_{2} on the boundary of π(Δ_{1} ∪Δ_{2}), join them by an arc λ ⊂ B_{R}\(C′_{R} ∪(L_{1})_{R} ∪(L_{2})_{R}) and construct an imaginate cycle c. By Proposition 14

[ξ] ∘ [π^{−1}(D ∪ Conj(D))] ≡ 2 mod 4 ,

which completes the proof of Lemma 7. □

3.2 Prohibitions of smoothings of type (a; 0, 0, b_{3}).

Lemma 8. There is no smoothing of type (2.1a), (2.1b), (2.1c) with b_{1} = b_{2} = 0.

Proof. Let C′ be a smoothing of type (2.1a), (2.1b), or (2.1c).

As in the proof of Lemma 6, suppose that all the ovals of C′ are in a small ball B_{0} ⊂ B, consider the pencil of straight lines through the point q and the minimal segment R of this pencil containing all the lines which intersect ovals in B_{3}. Denote by R′ the analogous segment of the pencil of lines through the point q′ (see Figure

28 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

Proposition 21. Each L ∈ R ∪ R′ intersects at most one oval of C′.

Proof. Such a line crosses the non-closed arcs of C′ at ≥ 4 points and the result follows from Lemma 2. □

Let us orient R clock-wise, and R′ counter clock-wise. This defines an ordering O (resp. O′) on the set s (resp., s′) of ovals swept by the lines L ∈ R (resp., L ∈ R′).

Proposition 22. (1) (s, O) = (s′, O′).

(2) The ovals in B_{3} form not more than two maximal sequences (with respect to O = O′) non-interrupted by the ovals from A.

Proof. (1) First, note that O = O′ on the ovals in B_{3}, since otherwise there exist two ovals in B_{3}, located as shown in Figure 11, and a line through these ovals intersects C′ at ≥ 8 points, contradicting Proposition 2.

Therefore, if (s, O) ≠ (s′, O′), we have one of the situations shown in Figure 12 (up to exchanging q and q′). Consider a conic K through the points q, q′, intersecting the three ovals indicated. By Cayley’s lemma, K ∩ B_{R} consists of two branches and their intersection with the non-closed arcs of C′ is determined by the configuration of the straight lines indicated. Then (K·C′_{R}) ≥ 16, which contradicts Lemma 1.

Figure 12

(2) Assuming that the sequence of ovals in B_{3} is interrupted at least twice, we have one of the situations shown in Figure 13 (up to exchanging q, q′). By Cayley’s lemma and due to the arrangement of straight lines through the four ovals indicated, the conic K, passing through q and these ovals, must go as shown in Figure 13, which implies (K · C′_{R}) ≥ 15, contradicting Lemma 1. □

Figure 13 Figure 14

By Proposition 22, we have either one non-interrupted sequence of b_{3} ovals in B_{3}, or two non-interrupted sequences of b′_{3} and b′′_{3} ovals in B_{3}, b′_{3} + b′′_{3} = b_{3}. Hence, the construction of real cycles and almost real cycles in a pencil gives us a sublattice of type A_{2b_{3}−1}, or A_{2b′_{3}−1} ⊕ A_{b′′_{3}−1} in H′_{+}, which implies the contradiction

2b_{3}−1 ≤ ind_{−}(H′_{+}) = 2b_{3}−3, or 2b′_{3}−1+2b′′_{3}−1 = 2b_{3}−2 ≤ ind_{−}(H′_{+}) = 2b_{3}−3,

completing the proof of Lemma. □

3.3 Prohibitions of smoothings of type (a; 0, b_{2}, b_{3}), b_{2}, b_{3} > 0.

Lemma 9. There is no smoothing of type (2.1a), (2.1b), (2.1c) with b_{1} = 0, b_{2}b_{3} > 0.

Proof. Assume that C′ is a smoothing of C of type (2.1a), (2.1b), or (2.1c) with b_{1} = 0, b_{2}b_{3} > 0.

Let a straight line L intersect an oval in B_{2} and an oval in B_{3} (let us call these ovals ν_{2}, ν_{3}, respectively), and not separate any two ovals from B_{2} ∪ B_{3} in B_{R} (see Figure 14). Fix points q_{2}, q_{3} ∈ L inside ν_{2}, ν_{3}, respectively, denote by Q_{i} the pencil of real straight lines through q_{i}, i = 2, 3, and introduce the minimal segments P_{i} ⊂ Q_{i}, containing the lines which intersect ovals in the domain B_{5−i}, i = 2, 3. An orientation of P_{i} defines an ordering on the set V_{i} of ovals swept by lines l ∈ P_{i}, i = 2, 3.

Proposition 23. One of the following statements holds.

(1) The set V_{i} contains only ovals from B_{5−i}, i = 2, 3.

(2) The set V_{3} contains only ovals from B_{2}. The set V_{2} contains all the ovals from B_{3} and a non-empty set V_{0} of ovals in A. The sequence of ovals from B_{3} is

30 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

interrupted by the ovals in V_{0} exactly once. The line L separates in B_{R} the ovals in V_{0} from the rest of the ovals in the domain A.

(3) The statement (2) with the indices 2 and 3 exchanged.

Proof. First, note that V_{i} contains no oval from B_{i}, i = 2, 3. Indeed, otherwise we have one of the situations shown in Figures 15a, b. In the situation 15a the straight line L′ crossing the two ovals indicated meets C′ at ≥ 9 points contradicting Proposition 1. In the situation 15b the conic K going through the four ovals indicated and the point q meets C′ at ≥ 15 points, which contradicts Lemma 1.

Figure 15

Assuming that both V_{2} and V_{3} contain ovals in A, one should have the situation shown in Figure 15c, where the conic through the five ovals indicated meets C′ at ≥ 15 points which contradicts Lemma 1.

If V_{2} contains a non-empty set V_{0} of ovals in A, and the straight line through ν_{2} and an oval v ∈ V_{0} separates in B_{R} two ovals from B_{3}, then so does the straight line through v and the point q. Indeed, otherwise we have one of the situations shown in Figures 15d, f, e. In each case the conic through the four ovals indicated and the point q meets C′ at ≥ 15 points which contradicts Lemma 1. Then we finish the proof of Proposition 23 as it was done in the proof of Proposition 22 by

SINGULARITY WHICH HAS NO M-SMOOTHING 31

application of Lemma 1 shown in Figure 13. □

If the statement (1) of Proposition 23 holds true, one takes the real 2-cycles in Y generated by the ovals in B_{2} ∪ B_{3}, and the almost real cycles, constructed by means of the pencils P_{2}, P_{3}, and obtains a sublattice in H′_{+} isomorphic to A_{2b_{2}−1} ⊕A_{2b_{3}−1}, according to Proposition 11 and Corollary 4. Hence

2b_{2} − 1 + b_{3} − 1 = 2(b_{2} + b_{3}) − 2 ≤ ind_{−}(H′_{+}) = 2(b_{2} + b_{3}) − 3,

which is absurd, completing the proof of Lemma in the case considered.

Assume that the statement (2) of Proposition 23 holds true, V_{0} contains a_{1} ≥ 1 ovals and V_{2} contains two non-interrupted sequences of ovals of lengths b′_{3}, b′′_{3}, where b′_{3} + b′′_{3} = b_{3}.

We shall construct a full rank sublattice in H′_{+} and compute its discriminant, obtaining contradiction whenever this discriminant is not −2n^{2}, n ∈ Z.

The real 2-cycles in Y generated by the ovals in B_{2} and the almost real cycles constructed by means of the pencil P_{3} give classes ξ_{i}^{2} ∈ H′_{+}, i = 1, ..., 2b_{2} − 1. Assuming that they are properly numbered, starting with the class defined by the oval ν_{2}, these classes form a standard basis of a lattice of type A_{2b_{2}−1}: (3.2a) (ξ_{i}^{2})^{2} = −2, 1 ≤ i ≤ 2b_{2} − 1, ξ_{i}^{2} ∘ ξ_{j}^{2} = 1, |i − j| = 1, ξ_{i}^{2} ∘ ξ_{j}^{2} = 0, |i − j| > 1 .

Similarly, the real 2-cycles in Y generated by the ovals in B_{3} and the almost real cycles constructed by means of the pencil P_{2} give classes ξ_{i}^{3}, i = 1, ..., 2b_{3} − 1, numbered according to the ordering defined by the orientation of P_{2}, where ξ_{1}^{3} is generated by the oval ν_{3}. By Propositions 11, 12, and Corollary 4, we have (after suitable orientation of the cycles considered)

(3.2b)  (ξ_{i}^{3})^{2} = −2, 1 ≤ i ≤ 2b_{3} − 1, i ≠ 2b′_{3}, (ξ_{2b′_{3}}^{3})^{2} = 2a_{1} − 2,
(3.2c)  ξ_{i}^{3} ∘ ξ_{j}^{3} = 1, |i − j| = 1, ξ_{i}^{3} ∘ ξ_{j}^{3} = 0, |i − j| > 1 .

Then we complete the above set of classes with [y_{1}^{1}] and two more classes obtained from the following mixed cycles τ_{0}, τ_{1}.

For τ_{0} we take G = E_{C}, where E is the exceptional divisor in B̂. Let us deform the curve Ĉ slightly in order to obtain six real intersection points with E (see Figure

32 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

16, where the opposite points on E_{R} should be identified). The surfaces G ⊂ X and F = π^{−1}(G) ⊂ Y, clearly, satisfy the conditions (a)-(c) and (i)-(iv) in subsection 2.4, and we can define the cycle

τ_{0} = Φ^{+} + Φ^{−} + 2(π^{−1}(δ_{1}) + π^{−1}(δ_{2}) + π^{−1}(δ_{1})),

where Φ^{+}, Φ^{−} are presented by the components of F \Y_{R}^{1}, and δ_{1}, δ_{2}, δ_{3} are the discs bounded by Ĉ_{R} and E_{R}, indicated in Figure 16.

Figure 16 Figure 17

For τ_{1} we take G = L_{C} ∪ L̂_{C}, where L is the line introduced in step 1, and L̂ ⊂ B̂_{C} is the strict transform of the copy of L in B̂ (see Figure 16). The surfaces G and F = π^{−1}(G) ⊂ Y satisfy the conditions (a)-(c) and (i)-(iv) in subsection 2.4. Indeed, (i) and (iv) follow directly from the construction of G and F, (ii) is satisfied, since G is diffeomorphic to the sphere and all the intersection points of G and S = C′_{C} ∪ Ĉ_{C} are real, (iii) follows from Proposition 16, establishing the orientation of the components of G_{R} ∩π(Y_{R}^{1}) shown by arrows on Figure 16. Hence we can construct the cycle

τ_{1} = Φ^{+} + Φ^{−} + ∑ ±Λ ,

where λ runs over the closures of all the components of Y_{R}^{1}\F adjacent to F.

Proposition 24. Orientations of the cycles τ_{0}, τ_{1} can be chosen so that (1) [τ_{0}]^{2} = 14, [τ_{1}]^{2} = 2 + 2a, [τ_{0}] ∘ [τ_{1}] = 0,

(2) [τ_{0}] ∘ [y_{1}^{1}] = 6, [τ_{1}] ∘ [y_{1}^{1}] = 2a + 2 − 4a_{1}, [y_{1}^{1}] ∘ ξ_{2b′_{3}}^{3} = 2a_{1},

(3) if b′_{3} > 1 then [τ_{1}] ∘ ξ_{2}^{3} = 1 and [τ_{1}] ∘ ξ_{2b′_{3}}^{3} = −2a_{1},

(4) if b′_{3} = 1 then [τ_{1}] ∘ ξ_{2}^{3} = 1 − 2a_{1},

SINGULARITY WHICH HAS NO M-SMOOTHING 33

(5) if b_{2} > 1 then [τ_{1}] ∘ ξ_{2}^{2} = 1.

All the other intersection indices of [τ_{0}], [τ_{1}] with [y_{1}^{1}], ξ_{i}^{2}, i = 1, ..., 2b_{2} − 1, ξ_{i}^{3}, i = 1, ..., 2b_{3}, are zero.

Proof. All the formulae follow from Propositions 12, 13, 15, 16, and 17. Let us show, for example that [τ_{0}] ∘ [τ_{1}] = 0. We fix the orientation of τ_{0} so that it coincides with the orientation of y_{1}^{1} on π^{−1}(δ_{i}), i = 1, 2, 3, and fix the orientation of τ_{1} so that it coincides with the orientation of y_{1}^{1} on π^{−1}(δ_{2}), π^{−1}(δ_{3}) and is the reverse on π^{−1}(δ_{1}). The corresponding orientations of the components of E_{R}\S_{R} and L_{R} ∪ L̂_{R}\S_{R}, defined in Proposition 16, are shown by arrows in Figure 16. The geometric intersection of τ_{0} and τ_{1} consists of π^{−1}(δ_{1}) ∪ π^{−1}(δ_{2}) ∪ π^{−1}(δ_{3}) and two points π^{−1}(E ∩ L̂). The contribution of π^{−1}(δ_{1}) ∪ π^{−1}(δ_{2}) ∪ π^{−1}(δ_{3}) to the intersection index is 2, and the contribution of π^{−1}(E ∩ L̂) is −2 by Proposition 17. □

Now we obtain a total of 2(b_{2} + b_{3}) + 1 = dim H′_{+} classes

[τ_{0}], [τ_{1}], [y_{1}^{1}], ξ_{i}^{2}, i = 1, ..., 2b_{2} − 1, ξ_{i}^{3}, i = 1, ..., 2b_{3} − 1 .

The determinant of the intersection matrix of these classes is a function Δ(b_{2}, b′_{3}, b′′_{3}, a_{1}) of positive integral variables b_{2}, b′_{3}, b′′_{3}, a_{1}, satisfying

b_{2} + b′_{3} + b′′_{3} = 4, or 8, or 12, a_{1} ≤ 13 − b_{2} − b′_{3} − b′′_{3} .

The direct computation, based on Propositions 11, 24 and formulae (3.2a), (3.2b), (3.2c), shows that the function Δ takes value −2n^{2}, n ∈ Z, only in the following cases:

(3.3a)  b_{2} = 1, b′_{3} = 1, b′′_{3} = 2, a_{1} = 1, Δ = −2^{9},
(3.3b)  b_{2} = 1, b′_{3} = 1, b′′_{3} = 2, a_{1} = 5, Δ = −2^{9} · 3^{2},
(3.3c)  b_{2} = 1, b′_{3} = 1, b′′_{3} = 2, a_{1} = 7, Δ = −2^{11},
(3.3d)  b_{2} = 1, b′_{3} = 1, b′′_{3} = 6, a_{1} = 1, Δ = −2^{13},

Figure 18

(3.3e)  b_{2} = 1, b′_{3} = 2, b′′_{3} = 5, a_{1} = 1, Δ = −2^{7} · 11^{2},
(3.3f)  b_{2} = 1, b′_{3} = 1, b′′_{3} = 10, a_{1} = 1, Δ = −2^{9}.

Next we shall prohibit the cases (3.3d), (3.3e), (3.3f) by constructing a mixed cycle which together with y_{1}^{2} generates a positive sublattice in H′′_{+} in contradiction with ind_{+}(H′′_{+}) = 1 (cf. Proposition 9). Let L′ be a straight line through the point q crossing an oval ν_{0} ∈ V_{0}. We will rotate the line L′ around a point on ν_{0} keeping two real intersection points with the boundary of B_{2} and making L′ close to a tangent to C at the singular point (see Figure 18a, b). Then we can assume that the proper transform L̂′ ⊂ B̂ of L′ crosses Ĉ and the exceptional divisor E as shown in Figure 18c. Following the procedure set up in 2.4, we take F^{1} = π^{−1}(L′_{C} ∪ L̂′_{C}), F^{2} = π^{−1}(E_{C}). Clearly, F^{1}, F^{2} satisfy the conditions (i)-(iv) in 2.4 with k = 2. Then we choose halves Φ_{+}^{1}, Φ_{+}^{2} of F^{1}, F^{2}, respectively, so that to obtain the orientations of the components of (L′_{R} ∪L̂′_{R})∩π(Y_{R}^{2}) and E_{R} ∩π(Y_{R}^{2}) as shown in Figure 18a, b, c. We claim that 6[∂Φ_{+}^{1}] + 2[∂Φ_{+}^{2}] = 0 ∈ H_{1}(Y_{R}^{2}). Indeed, the latter 1-cycle is the boundary of the 2-cycle N combined from the closures of the components of Y_{R}^{2}\(F^{1} ∪ F^{2}) with the multiplicities indicated on their projections to X in Figure 18a, b, c (we assume that these components have the orientation induced from Y_{R}^{2}). Now we can define the class

η = [3Φ_{+}^{1} + 3Φ_{−}^{1} + Φ_{+}^{2} + Φ_{−}^{2} − N] ∈ H′′_{+} .

Remark. The ovals inside B_{2} ∪ B_{3} are not shown in Figure 18. It easily follows from Proposition 23 and its proof that these omitted ovals are located as follows:

SINGULARITY WHICH HAS NO M-SMOOTHING 35

(1) in Figure 18a, b_{2} ovals are in the part of B_{2} marked with “−1”, b′_{3} and b′′_{3} ovals are in the parts of B_{3} marked with “−5” and “+1”, respectively, (2) in Figure 18b, b_{2} ovals are in the part of B_{2} marked with “−1”, b′_{3} and b′′_{3} ovals are in the parts of B_{3} marked with “+5” and “+1”, respectively.

A routine computation based on Propositions 15 and 17 gives In the situation shown in Figure 18a

η^{2} = 2b_{2} + 2b′′_{3} + 50b′_{3} − 56, η ∘ [y_{1}^{2}] = −2b_{2} + 2b′′_{3} − 10b′_{3}

in the situation shown in Figure 18a, and

η^{2} = 2b_{2} + 2b′′_{3} + 98b′_{3} − 80, η ∘ [y_{1}^{2}] = −2b_{2} + 2b′′_{3} + 14b′_{3} − 12

in the situation shown in Figure 18b.

These formulae together with Propositions 11 give the values of the discriminant of the sublattice ⟨η, [y_{1}^{2}]⟩ ⊂ H′′_{+} for the cases (3.3a)-(3.3f)

−64, −64, −64, 80, 416, 224,

respectively, in the situation shown in Figure 18a, and

32, 32, 32, 176, 704, 320,

respectively, in the situation shown in Figure 18b.

Since the positivity of that discriminant contradicts ind_{+}(H′′_{+}) = 1, it remains to prohibit only smoothings of type (3.3a)–(3.3c) with ovals in V_{0} located as shown in Figure 18a.

For this purpose we introduce one more mixed cycle. Fix a point inside an oval ν_{0} ∈ V_{0} and draw through this point a straight line L_{2} crossing the oval ν_{2}, and a straight line L_{3} crossing the oval ν_{3} (see Figure 19a). Without loss of generality we can suppose that the point L_{2} ∩ L_{3} is the singular point of C, and the strict transforms L̂_{2}, L̂_{3} ⊂ B̂ meet E_{R} and Ĉ_{R} as shown in Figure 19b. The surfaces F^{1} = π^{−1}(L_{2,C} ∪ L̂_{2,C}), F^{2} = π^{−1}(L_{2,C} ∪ L̂_{2,C}), clearly, satisfy the conditions (i)-(iv) in 2.4 with k = 1. Then we choose halves Φ_{+}^{1}, Φ_{+}^{2} of F^{1}, F^{2}, respectively, so that to obtain the orientations of the components of (L_{2,R} ∪ L̂_{2,R}) ∩ π(Y_{R}^{1}) and (L_{3,R}∪L̂_{3,R})∩π(Y_{R}^{1}) as shown in Figure 19a, b. The 1-cycle [∂(Φ_{+}^{1}+Φ_{−}^{1}+Φ_{+}^{2}+Φ_{−}^{2}]

Figure 19

is a boundary of the 2-chain N consisting of the closures of connected components of Y_{R}^{1}\(F^{1} ∪F^{2}) taken with multiplicities indicated in Figure 19a, b on the projections of these components on B_{R} ∪B̂_{R}. Denote the mixed cycle [Φ_{+}^{1} +Φ_{−}^{1} +Φ_{+}^{2} +Φ_{−}^{2} −N by η and consider the sublattice in H′_{+} generated by

(3.4)  ξ_{1}^{2}, ξ_{1}^{3}, ξ_{3}^{3}, ξ_{4}^{3}, ξ_{5}^{3}, [y_{1}^{1}], [τ_{0}], [τ_{1}], [η].

Computations along the formulae in Propositions 15, 17 give in the cases (3.3a)– (3.3c)

[η] ∘ ξ_{1}^{2} = [η] ∘ ξ_{1}^{3} = [η] ∘ ξ_{i}^{3} = 0, i = 3, 4, 5,
[η] ∘ [y_{1}^{1}] = 8 + 4a′_{1} + 4a′_{2}, [η] ∘ [τ_{0}] = 4,
[η] ∘ [τ_{1}] = 4a′_{2} − 4a′_{1}, [η]^{2} = 12 + 8a′_{1} + 8a′_{2},

where a′_{1} is the number of ovals in V_{0} different from ν_{0} and contained in the real part of η, a′_{2} is the number of ovals in A which do not belong to V_{0} and are contained in the real part of η. Combining the latter formulae with Proposition 24, one computes the determinant Δ(a′_{1}, a′_{2}) of the intersection matrix of classes (3.4). Since rkH′_{+} = 9 is equal to the number of classes (3.4), and discrH′_{+} = −2 (cf. Proposition 9), Δ(a′_{1}, a′_{2}) must be −2n^{2}, n ∈ Z. Direct computation shows that the function Δ(a′_{1}, a′_{2}) takes no value 2n^{2}, n ∈ Z, for any

a_{1} = 1, 5, 7, 0 ≤ a′_{1} ≤ a_{1} − 1, 0 ≤ a′_{2} ≤ 9 − a_{1},

SINGULARITY WHICH HAS NO M-SMOOTHING 37

which completes the proof of Lemma 9.

3.4 Prohibitions for smoothings of type (a; b_{1}, b_{2}, b_{3}), b_{1}, b_{2}, b_{3} > 0. Assume that there exists a smoothing of type (a; b_{1}, b_{2}, b_{3}), b_{1}b_{2}b_{3} > 0.

We start with the study of the oval arrangement of such a smoothing.

Proposition 25. Let q_{i} be a point inside an oval in B_{i}, i = 1, 2, and let P_{i}, i = 1, 2, be the minimal segment in the pencil of straight lines through q_{i}, which contains all the lines intersecting the ovals in B_{3}. Then,

(1) the lines L ∈ P_{i} intersect no ovals outside B_{3}, except for the oval embracing q_{i}, i = 1, 2;

(2) suitably oriented segments P_{1}, P_{2} define the same order on the ovals in B_{3}.

Proof. This can be shown by application of Lemma 1 as in the proof of Proposition 23. For instance, assuming the contrary to the statement (1), we obtain five ovals of C′ located as shown in Figure 17. The conic K passing through these ovals must intersect C′_{R} in at least 16 points, which is impossible. □

We order the ovals in each domain B_{i}, i = 1, 2, 3, as in Proposition 25Let us choose the first oval ν_{i} in B_{i}, i = 1, 2, 3, and three real straight lines L_{1}, L_{2}, L_{3} in B such that L_{i} crosses ν_{j}, ν_{k} for i ∉ {j, k}, and L_{i}, L_{j} with i ≠ j meet at a point in B_{k} outside ν_{k}, k ∉ {i, j} (see Figure 20).

Figure 20

The lines L_{1}, L_{2}, L_{3} divide B_{R} into seven domains, which we denote by B(ε_{1}, ε_{2}, ε_{3}), ε_{1}, ε_{2}, ε_{3} = ±1, where

ε_{i}L_{i}(z) > 0 if z ∈ B(ε_{1}, ε_{2}, ε_{3}) ,

and L_{1}, L_{2}, L_{3} are positive inside the triangle with vertices at the intersection points (see Figure 20). Let a^{ε_{1}ε_{2}ε_{3}} be the number of ovals in A^{ε_{1},ε_{2},ε_{3}} = B(ε_{1}, ε_{2}, ε_{3}) ∩ A, and b_{i}^{ε_{1}ε_{2}ε_{3}} be the number of ovals in B_{i}^{ε_{1},ε_{2},ε_{3}} = B(ε_{1}, ε_{2}, ε_{3}) ∩ B_{i} different from ν_{i}, i = 1, 2, 3.

Proposition 26. (1) At most one of b_{i}^{ε_{1}ε_{2}ε_{3}} with ε_{1}ε_{2}ε_{3} = −1 and i = 1, 2, 3 is different from zero.

(2) If b_{3}^{−++} > 0 or b_{1}^{+−+} > 0, then

a^{++−} = a^{−−+} = a^{−++}a^{+−−} = a^{−+−}a^{+−+} = 0 .

(3) If b_{3}^{+++} > 0 or b_{1}^{−−+} > 0, then

a^{−+−} = a^{+−−} = a^{−−+}a^{++−} = 0 ,

and if a^{−−+} > 0 then

a^{+−+}a^{−++} = 0 .

(4) The above statements hold for any renumeration of B_{1}, B_{2}, B_{3}.

Proof. This follows from Bézout’s theorem in the form of Lemma 1, because otherwise one can draw a conic intersecting C′ at ≥ 15 points. □

Lemma 10. There do not exist M-smoothings of types (a; b_{1}, b_{2}, b_{3}), b_{1}, b_{2}, b_{3} > 0, except, perhaps, for the cases

(3.5)  \{ \begin{matrix}b_{1} = b_{2} = 1, b_{3} = 2, a = 9, \\ b_{3}^{−++} = 1, a^{+++} = 3, a^{−++} = 1, a^{−+−} = 5 \end{matrix}

(3.6)  \{ \begin{matrix}b_{1} = b_{2} = 1, b_{3} = 6, a = 5, \\ b_{3}^{−++} = 5, a^{+++} = 1, a^{−+−} = 4, \end{matrix}

(up to permutation of indices 1, 2, 3).

Proof. Our plan is to construct dim H′_{+} + 1 classes in H′_{+} and compute the determinant Δ of their intersection matrix. If Δ ≠ 0 then the corresponding smoothing does not exist.

SINGULARITY WHICH HAS NO M-SMOOTHING 39

We start with the class [y_{1}^{1}] ∈ H′_{+}.

Then, as in 3.3, we construct the real and almost real 2-cycles coming from the ovals in B_{1} ∪ B_{2} ∪ B_{3} and the corresponding segments of the pencil of lines centered inside the ovals ν_{1}, ν_{2}, ν_{3}, and which give the classes

(3.7)  ξ_{i}^{1}, i = 1, ..., 2b_{1} − 1, ξ_{i}^{2}, i = 1, ..., 2b_{2} − 1, ξ_{i}^{3}, i = 1, ..., 2b_{3} − 1,

where ξ_{1}^{j} is generated by the oval ν_{j}, j = 1, 2, 3. By Proposition 25, these classes form the standard basis in a sublattice A_{2b_{1}−1} ⊕ A_{2b_{2}−1} ⊕ A_{2b_{3}−1} ⊂ H′_{+} which is orthogonal to [y_{1}^{1}].

Four more classes in H′_{+} are represented by the following mixed cycles τ_{0}, τ_{1}, τ_{2}, and τ_{3}. Taking a smoothing C′ closer to C, we may suppose that the three points where L_{i} meet are arbitrarily close to the singular point q_{0} of C. Then there are three lines L′_{1}, L′_{2}, L′_{3} close to L_{1}, L_{2}, L_{3} and such that they pass through q_{0}, and there is an equivariant diffeomorphism ψ : B_{C} → B_{C} close to identity which is identical on C ∪ (1 − ϵ)B_{C}, preserves the spheres concentric to B_{C}, and transforms L′_{i} ∩∂B_{C} in φ(L_{i} ∩∂B_{C}), where φ is the gluing diffeomorphism defining X. Denote by L̂_{1}, L̂_{2}, L̂_{3} ⊂ B̂ the strict transforms of L′_{1}, L′_{2}, L′_{3}. Now we construct τ_{0}, τ_{1} exactly as in the proof of Lemma 9: the role of L is played here by L_{1}. The cycles τ_{2}, τ_{3} are constructed in the same manner as τ_{1} by substituting the line L_{1} for the lines L_{2}, L_{3}, respectively. The orientations of the cycles τ_{1}, τ_{2}, τ_{3} are chosen in correspondence with the orientations of components of (L_{1,R} ∪L_{2,R} ∪L_{3,R})∩π(Y_{R}^{1}) shown in Figure 20.

Thus, we obtain a total of 2(b_{1} + b_{2} + b_{3}) + 2 = dim H′_{+} + 1 classes (3.7) and

(3.8)  [y_{1}^{1}], [τ_{0}], [τ_{1}], [τ_{2}], [τ_{3}] .

Proposition 27.

[τ_{i}]^{2} = 2 + 2a, i = 1, 2, 3;

[τ_{i}] ∘ [τ_{j}] = 2 ∑_{ε_{1},ε_{2},ε_{3}=±1} ε_{i}ε_{j}a^{ε_{1}ε_{2}ε_{3}}, 1 ≤ i < j ≤ 3;

[τ_{i}] ∘ [y_{1}^{1}] = 2 + 2 ∑_{ε_{1},ε_{2},ε_{3}=±1} ε_{i}a^{ε_{1}ε_{2}ε_{3}}, i = 1, 2, 3 .

If b_{i}^{ε_{1}ε_{2}ε_{3}} > 0 and ε_{1}ε_{2}ε_{3} = −1, then

ξ_{2}^{i} ∘ [τ_{j}] = −ξ_{2}^{i} ∘ [τ_{k}] = ±1,

where {i, j, k} = {1, 2, 3}. If b_{i}^{ε_{1}ε_{2}ε_{3}} > 0 and ε_{1}ε_{2}ε_{3} = 1, then

ξ_{2}^{i} ∘ [τ_{j}] = ξ_{2}^{i} ∘ [τ_{k}] = ±1,

where {i, j, k} = {1, 2, 3}. All the intersection indices of [τ_{1}], [τ_{2}], [τ_{3}] with the other classes (3.7), (3.8) are zero.

Proof. Follows from Propositions 12, 15, and 17, as in Proposition 24. □

Direct computation of the determinant Δ of the intersection matrix of the classes (3.7) and (3.8), based on Propositions 24 and 27, for all the non-negative values of the parameters a^{ε_{1}ε_{2}ε_{3}}, b_{i}^{ε_{1}ε_{2}ε_{3}}, i = 1, 2, 3, ε_{1}, ε_{2}, ε_{3} = ±1, satisfying

∑_{ε_{1},ε_{2},ε_{3}=±1} (a^{ε_{1}ε_{2}ε_{3}} + ∑_{i=1}^{3} b_{i}^{ε_{1}ε_{2}ε_{3}}) = 13 ,

∑_{ε_{1},ε_{2},ε_{3}=±1} a^{ε_{1}ε_{2}ε_{3}} = 1 or 5 or 9 ,

and the conditions of Proposition 26, shows that Δ ≠ 0 except for the cases (3.5), (3.6) (up to renumeration of B_{i}), which completes the proof of Lemma 10. □

The possible arrangement of ovals for smoothings (3.5) and (3.6) left by Lemma 10 is the following:

Proposition 29. In a smoothing of type (3.5) or (3.6) the ovals are located with respect to the lines L_{1}, L_{2}, L_{3} and the non-closed arcs as shown in Figure 21. Moreover, no straight line through L_{1} ∩ L_{2} and an oval in A^{−++} ∪ B_{3}^{−++} separates an oval in A^{−+−} from L_{1}.

SINGULARITY WHICH HAS NO M-SMOOTHING 41

Proof. The region A^{−+−} may consist of two connected components (see Figure 21), but the component of A^{−+−}, adjacent to the line L_{3}, does not contain ovals. Indeed, otherwise one can draw a conic intersecting an oval in that component, ovals ν_{1}, ν_{2}, ν_{3}, and an oval in B_{3}^{−++}, and then show by Cayley’s lemma and Lemma 1 that such a conic intersects C′_{R} in at least 16 points.

Similarly, assuming that a straight line through L_{1} ∩ L_{2} and an oval from A^{−++} ∪ B_{3}^{−++} separates some oval in A^{−+−} from the line L_{1}, one can draw a conic intersecting these two ovals and the ovals ν_{1}, ν_{2}, ν_{3}, and show by Cayley’s lemma and Lemma 1 that such a conic intersects C′_{R} in at least 16 points. □

§4. Application of Seifert forms

According to the results of §1–3, an M-smoothing C′ of a Sirler singularity C can have, if it exists, only the following arrangements of ovals: (9; 1, 1, 2) and (5; 1, 1, 6). Moreover, as it follows from Lemma 10, there exist three lines L_{1}, L_{2}, L_{3} which are located with respect to C′ as is shown in Figure 21.

Without loss of generality we may assume that L_{1} and L_{2} intersect at the origin of B. Then the pencil λ_{1}L_{1}+λ_{2}L_{2} defines on B̂ a fibration by disks, π : B̂ → E = CP^{1} (here, as before, B̂ is B blown up at the origin and E is the exceptional curve). The Euler number of this fibration is −1. The real part B̂_{R} of B̂ is the Möbius band and π|B̂_{R} is a fibration by segments over E_{R} = RP^{1}. We use the same notation for C′ and its pull-back to B̂. Note, that, according to our above convention, C′ does not go through the origin.

We may assume also that: (1) C′_{R} is located in B̂_{R} with respect to the fibers of π as it is depicted in Figures 23 and 24 where the fibers are viewed as vertical lines and ⟨a⟩ denotes a horizontal chain of a ovals; (2) the number of the fibers tangent to C′_{R} is 28 (2 on each oval and 2 on each of two non-closed branches, which is the minimal number of tangent fibers). The part (1) follows from Proposition 29, (2) is achieved by transformation of C′_{R} described in [16; Prop. 3.5.1]. We suppose, in addition, that no vertical line goes through two ovals. For the group of three ovals in Figure 22 we may suppose this by [16; Prop. 3.5.1], and in other cases it follows from Bézout’s theorem. The two ovals ⟨1⟩ in the Figure 22 are placed one above the other to emphasize that their mutual position with respect to the fibers is unknown. Thus, we have to consider three possibilities: two for Figure 23 and

Figure 22 Figure 23

one for Figure 23.

The full transform Ĉ of C in B̂ is the sum of E taken with multiplicity 6 and three smooth real curves having a quadratic tangency with E at three distinct real points. The real parts of these three curves are in alternate position with respect to the sides of E_{R}: in an affine system of coordinates (u, v) in B̂ with v = 0 defining E, these 3 curves are defined by v = a_{i}(u − u_{i})^{2} with u_{1} < u_{2} < u_{3}, a_{1}a_{2} < 0, and a_{2}a_{3} < 0.

The smoothing C′, considered as situated in B̂, is a deformation of Ĉ. Outside a neighborhood of E, it is a small perturbation. Thus, there are three disjoint disks D_{1} ∪ D_{2} ∪ D_{3} ⊂ E, v_{i} ∈ D_{i} such that π|_{C′} is 7-fold over D_{1} ∪ D_{2} ∪ D_{3} and 6-fold outside.

Let γ_{ε}, ε ≥ 0, be a family of simple closed curves lying in one of the components, E^{+} of E \ E_{R} on distance ε from E_{R} (γ_{0} = E_{R}). Denote by H_{ε} the closed disk bounded by γ_{ε} in E^{+}. Clearly, π^{−1}(H_{ε}) are diffeotopic to π^{−1}(H_{0}) inside π^{−1}(H_{0}), at least for small ε, and they are diffeomorphic to a polydisc D^{2} × D^{2}. Denote by S_{ε}^{3} its boundary.

As in [16], we study the surface N_{ε} = C′ ∩π^{−1}(H_{ε}). There exists ε_{0} such that for 0 < ε < ε_{0} the surfaces N_{ε} are identified by a diffeotopy of π^{−1}(H_{ε}) and we denote all N_{ε} with 0 < ε < ε_{0} by N as well as omit ε in the notation of other objects. The boundary K = ∂N is a link in S^{3} whose isotopy type is determined by C′_{R} up to some unknown integer parameters.

In what follows we compute the nullity of the Seifert form of K as a function of these parameters and show that any of their values contradict the Murasugi– Tristram inequality for the Euler characteristic of N.

Proposition 30. χ(N) = 2.

SINGULARITY WHICH HAS NO M-SMOOTHING 43

Proof. As mentioned in the Introduction, the genus of a Sirler singularity is 13, and hence χ(C′) = 2 − 2g − r = −27. Denote by c the number of branching points of π|C′ → E counted with multiplicities. Clearly, c = c_{R} + 2c_{H} where c_{R} (resp. c_{H}) is the number of branching points lying over E_{R} (resp. over H). By Riemann – Hurwitz formula, χ(C′) = 6χ(CP^{1}) + χ(D_{1} ∪ D_{2} ∪ D_{3}) − c, and, hence, c = 42.

Similarly, χ(N) = 6χ(H) + χ(H ∩ (D_{1} ∪ D_{2} ∪ D_{3})) − c_{H} = 6 + 3 − c_{H}. According to the assumptions made in this section in what concerns real fibers tangent to C′_{R}, we have c_{R} = 28, and, hence, c_{H} = (c−c_{R})/2 = 7. Thus, χ(N) = 6+3−7 = 2. □ Proposition 31. The determinant of the symmetrized Seifert matrix of K vanishes. Proof. The Murasugi–Tristram inequality [26] states that

null(K) ≥ χ(N) + |sign(K)|

where null(K) and sign(K) are respectively the nullity and the signature of the link K. By Proposition 30, this implies null(K) ≥ 2. □ End of the proof of Theorem 1. Here, we find explicitly all the links K which can appear for the smoothings shown in Figures 23, 24 and prove that the determinants of their symmetrized Seifert matrices are non-zero.

Since π^{−1}(H) is identified with D^{2} × D^{2}, the sphere S^{3} = ∂π^{−1}(H) is naturally decomposed into two solid tori S^{3} = T_{1} ∪ T_{2}. In this decomposition, T_{1} = π^{−1}(γ) (recall that γ = ∂H) and T_{2} = H × S^{1}.

44 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

The real curve G := C ∩ S_{0}^{3} (where S_{0} is the sphere S_{ε} for ε = 0) consists of the following seven parts. Four of them, namely, the ”core” E_{R} (the real part of the exceptional curve) and the three parabolas lying on the Möbius band B̂_{R}. The other three pieces are the arcs lying on the surfaces of the ”solid half-tori” ((∂D_{i}) ∩ H) × S^{1}, i = 1, 2, 3 (they are subsets of T_{2}). Note that B̂_{R} is embedded into T_{1} = D^{2} × S^{1} as a left Möbius band \{(z, e^{iφ})|e^{− \frac{1}{2} iφ}z ∈ R\}.

In the Figure 24 (left) we have depicted the solid torus T_{1}, the three solid halftori with the three pieces of G carried by them, and the Möbius band B̂_{R}. In the Figure 24 (right) we have depicted the Möbius band B̂_{R} and the four pieces of G carried by it.

Figure 25

The perturbed curve G′ := C′ ∩ S_{0}^{3} is obtained from this picture by replacing the core and the three parabolas with the curves shown in Figures 23, 24. And to obtain the link K we apply the procedure described in [16, 3.4, 3.5]. The result is depicted in Figure 25 where the variable pattern Q corresponds to the pattern Q in Figures 23, 24 and contains one of the following three braids:

α^{(i)} = β^{(i)} · τ_{1,2}σ_{2}^{−1}σ_{1}^{−1}σ_{3}^{−1} · ( ∏_{j=1}^{h^{(i)}} σ_{2}^{−1}σ_{3}^{e_{j}}σ_{1}^{−e_{j}}) · σ_{3}σ_{1}σ_{2}^{−1}, i = 1, 2, 3 where

\begin{aligned}β^{(1)} & = τ_{2,3}σ_{3}^{−1}τ_{3,4}σ_{4}^{−1}τ_{4,1}σ_{1}^{−5} \\ β^{(2)} & = τ_{2,4}σ_{4}^{−1}τ_{4,3}σ_{3}^{−1}τ_{3,1}σ_{1}^{−5} \\ β^{(3)} & = τ_{2,4}σ_{4}^{−5}τ_{4,1}σ_{1}^{−4} \end{aligned}

SINGULARITY WHICH HAS NO M-SMOOTHING 45

and h^{(1)} = h^{(2)} = 4, h^{(3)} = 2. The braids α^{(1)}, α^{(2)}, correspond to different mutual positions of ⟨1⟩’s in Figure 22; the braid α^{(3)} corresponds to Figure 23. In all the cases we use the convention in Figure 26 for the generators σ_{j} of the braid group; Δ is the Garside element (see Figure 26) Δ = Δ_{6} = Π_{5,1}Π_{5,2}Π_{5,3}Π_{5,4} · σ_{5} where

Π_{k,l} = \{ \begin{matrix}σ_{k}σ_{k+1} … σ_{l} & if l > k \\ σ_{k}σ_{k−1} … σ_{l} & if l < k \end{matrix}

and τ_{k,l} is defined as in [16], by

τ_{k,l} = \{ \begin{matrix}Π_{l,k+1}^{−1}Π_{k,l−1} & if l > k \\ Π_{l,k−1}^{−1}Π_{k,l+1} & if l < k \\ 1 & if l = k \end{matrix}

σ_{i}  σ_{i}^{−1}  Δ_{6} Figure 26

In Figure 25 we marked some arcs of K by zigzags. Replacing them by the dashed lines we obtain a link which is a braid on 7 strings

Δ_{6}^{−1} · Π_{5,1}^{2}Π_{6,1} · α^{(i)} · Π_{1,6} · σ_{1} · Π_{1,5}^{2} = (Π_{5,5}^{−1} … Π_{5,1}^{−1}) · Π_{5,1}^{2}Π_{6,1} · α^{(i)}σ_{2} · Π_{1,6}Π_{1,5}^{2}
= δ^{−1}(σ_{1}Π_{6,1} · α^{(i)}σ_{2}^{2} · Π_{1,6} · σ_{2}σ_{1}σ_{3}σ_{2} · σ_{5}^{−1})δ where δ = Π_{5,3}Π_{5,4}σ_{5}

Using the algorithm from [16, 2.6.5], we have computed the determinant of the Seifert form as a function of the unknown integer parameters e_{j}. In the two cases in Figure 22 the determinant is (up to a non-zero constant factor)

d^{(1)} = − 228 + 28e_{1} + 64e_{2} + 100e_{3} + 136e_{4} − 9e_{1}^{2} − 32e_{2}^{2} − 41e_{3}^{2} − 36e_{4}^{2}
− 16e_{1}e_{2} − 14e_{1}e_{3} − 12e_{1}e_{4} − 48e_{2}e_{3} − 32e_{2}e_{4} − 52e_{3}e_{4};
d^{(2)} = − 1236 − 120e_{1} + 36e_{2} + 192e_{3} + 348e_{4} − 85e_{1}^{2} − 324e_{2}^{2} − 381e_{3}^{2} − 256e_{4}^{2}
− 120e_{1}e_{2} − 70e_{1}e_{3} − 20e_{1}e_{4} − 416e_{2}e_{3} − 184e_{2}e_{4} − 348e_{3}e_{4}.

46 V.M.KHARLAMOV, S.YU.OREVKOV, E.I.SHUSTIN

Each d^{(i)}, i = 1, 2 is a quadratic function of e_{j} whose Hessian is negatively definite and whose value at the minimum is also negative. Hence, the determinant of the Seifert form is non-zero.

In the case of Figure 23 the determinant is

d^{(3)} = −180 + 240e_{1} − 60e_{2} + 109e_{1}^{2} + 256e_{2}^{2} + 76e_{1}e_{2}.

The equation d^{(3)} = 0 has no integer solution.

Thus, in all the cases we obtain a contradiction with Proposition 31. □

§4. Concluding remarks

1. Since the local Harnack bound is not sharp for individual singularities, it is natural to ask whether it is sharp in each complex equisingular (i.e., μ-constant) family of singularities. The answer is positive. This result can be obtained by the blowing-up construction like in [18]. Using an additional freedom (one may “reorient” branches) and the induction over the number of blow-ups, one proves the existence of M-smoothing with 2 basis in each complex equisingular class. The proof will appear in [12].

2. The known list of individual singularities admitting M-smoothings (see, f.e., [11, 18]) can be extended by semiquasihomogeneous singularities. As it is shown in [24], a real semiquasihomogeneous singularity admits an M-smoothing, if its quasihomogeneous part is non-degenerate (i.e., defines an isolated singularity) and all the real local branches lie in two quadrants.

3. It is natural also to ask what are the real equisingular families containing a singularity which admits an M-smoothing. According to [24], any real semiquasihomogeneous singularity can be deformed in its real equisingular class to a singularity admitting an M-smoothing, and any Newton non-degenerate plane curve singularity can be deformed in its real equisingular class to a singularity admitting a weak M-smoothing, i.e., providing equality in the improved local Harnack bound (Proposition 1).

4. As noted in Introduction, there is a close relation between smoothings of a Sirler cusp singularity and affine sextics: one can simplify this singularity into an ordinary 6-fold point and then replace the singular point by an affine sextic. Thus, Theorem 1 can be used for prohibiting particular arrangements of affine M-curves.

SINGULARITY WHICH HAS NO M-SMOOTHING 47

In particular, Theorem 1 prohibits the arrangement D(a, b, c) (in the notation of [14]), the result obtained differently in [14], [16].

5. There is an approach for prohibiting M-smoothings of type (a, 0, b_{2}, b_{3}), b_{2}b_{3} ≠ 0, different from the one presented in §3 and based on computations of braids and Alexander polynomials, cf. [16].

6. There is an approach for prohibiting the arrangements (9; 1, 1, 2) and (5; 1, 1, 6) different from the one presented in §4. Similar to the cases (3.3d), (3.3e), (3.3f), it is sufficient to construct a mixed cycle which together with y_{1}^{2} generates a positive rank 2 sublattice in H′′_{+} in contradiction with ind_{+}(H′′_{+}) = 1. The mixed cycle to consider is determined by a line through the boundary point of C′_{R} lying between L_{2} and L_{3} (see Figure 21) and an oval in the group of 5 ovals (see the left hand part of Figure 21), if the arrangement is (9; 1, 1, 2), and in the group of 4 ovals (see the right hand part), if the arrangement is (5; 1, 1, 6).

7. The improved local Harnack bound given by Proposition 1 is, in fact, a particular case of the following general bounds for pairs (X, A) with Z/2-action c : (X, A) → (X, A), which state that dim H_{∗}(X_{R}; Z/2) ≤ dim H^{1}(H_{∗}(X; Z/2)) ≤ dim H_{∗}(X; Z/2) − 2a, where X_{R} = Fix{c : X → X}, 2a = dim H_{∗}(A; Z/2) − dim K −(dim H^{1}(H_{∗}(A; Z/2)/K), K is the kernel of the map H_{∗}(A; Z/2) → H_{∗}(X; Z/2) induced by inclusion A ⊂ X and H^{1} stands for the Galois cohomology of a Z/2vector space with Z/2-action. The first bound is Swan’s inequality, see [25], the second one is an easy consequence of the usual Ker − Coker sequence, cf. [4]. To get the bound from Proposition 1 take X = C′_{C} and A = C′_{R} ∪ ∂C′_{C}.

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Address of the authors:

V.M. Kharlamov, Université Louis Pasteur et CNRS, Strasbourg, France.

S.Yu. Orevkov, Steklov Math. Institute, Moscow, Russia.

E.I. Shustin, Tel Aviv University, Israel.