Lower deformations of isolated hypersurface singularities

E. Shustin ^{∗}

Abstract

Given a smooth real or complex algebraic variety V and a hypersurface W ⊂ V with isolated singular points, we study deformations of these singular points which can be realized when varying W in the linear system |W |. For isolated singularities of hypersurfaces we define lower deformations which have an explicit topological-analytic description, and we show that such deformations can be independently realized under some conditions even if the linear system |W | is not a joint versal deformation of all singular point of W. We give few applications of lower deformations, among them construction of divisors with prescribed isolated singularities and construction of real algebraic curves with prescribed topology.

1 Introduction

Let V be a smooth real or complex algebraic variety of dimension n ≥ 2, and W be a divisor on V with isolated singular points. Assuming that dim |W | > 0, we ask two natural questions:

• What deformations of the singular points of W are possible when varying W in |W | ?

• Given a collection of isolated hypersurface singularities, does there exist a divisor W′ ∈ |W | with the given singularities ?

The questions are related in the following way: knowing deformations of a divisor with complicated singularities, one can construct divisors with prescribed simpler singularities. The classical example is Severi’s theorem [22] claiming that singular points of a plane nodal algebraic curve can be independently smoothed out or preserved when varying the curve in the space of curves of the same degree, and this is the basic construction of plane curves of a given degree with a given number of nodes.

∗This work was started during the author’s stay at the University of Toronto and was completed during the author’s visit to the Max-Plank-Institut für Mathematik. I would like to thank these institutions for for hospitality, financial support and excellent work conditions. The author also was partially supported by the Hermann Minkowski – Minerva Center for Geometry at Tel Aviv University and Grant No. G 039-304.01/95 from the German Israeli Foundation for Research and Development.

If the linear system |W | is a joint versal deformation of the multisingularity of W then all possible local deformations of singular points of W are independently realizable when varying W in |W |. However, often we do not know what are “possible local deformations” of singular points. On the other hand, the above versality condition for |W | is quite restrictive. It can be expressed in the following form [1, 2, 10, 11]

h^{1}(V, J_{X^{ea}/V} ⊗ O_{V} (W)) = 0 ,  (1)

where J_{X^{ea}/V} is the ideal sheaf of the zero-dimensional scheme X^{ea}, concentrated at the singular locus Sing(W) of W and defined at each point z ∈ Sing(W) by the ideal I_{z_{i}}^{ea} =< f, \frac{∂f}{∂x_{1}} , ..., \frac{∂f}{∂x_{n}} > in the local ring O_{V,z}, with f(x_{1}, ..., x_{n}) = 0 being an equation of the germ (W, z) in some coordinates x_{1}, ..., x_{n} in a neighborhood of z on V. If (1) fails, one looses control on the simultaneous deformation of singular points of W.

The aim of this paper is to introduce a class of deformations of singular points (called below “lower”), which have an explicit topological analytic description and are independently realizable in variation of W in |W | under a condition weaker than (1). In section 2 we give a geometric description of lower deformations of semiquasihomogeneous and Newton nondegenerate singular points, in section 3 we establish a general sufficient cohomological criterion for the independence of lower deformations (Theorem 1), in sections 4, 5 we express the above cohomological independence criterion through numerical invariants of the divisor W and its singularities (Lemmas 2, 3, Corollary 1 and Lemmas 6, 7, 8, Corollary 2). As application, we give a criterion for the existence of hypersurfaces with prescribed isolated singularities and consider some special deformations of real plane algebraic curves (Lemma 9, Corollary 3 in section 6, Lemma 12 in section 7).

We were motivated by the following:

• The Viro method of constructing real nonsingular hypersurfaces with prescribed topology [30, 31, 32] (see also [20]) appeared to be very efficient in the topology of real algebraic varieties. One procedure in this method, the gluing of polynomials whose Newton polyhedra form a subdivision of a convex polyhedron, has been extended on polynomials with given singular and critical points [25, 26] (for other generalizations see [17]). In the present paper we give similar generalizations for another procedure in the Viro method, a nonsingular deformation of Newton nondegenerate singularities of a given hypersurface which are located at the fundamental points of the coordinate system. Namely, we allow any number on singular points in an arbitrary position and do not require the Newton nondegeneracy. Moreover, we allow also singular deformation and describe both the real and complex parts of the deformation.

• The existence problem for divisors with prescribed isolated singularities has an asymptotically proper solution in the case of plane curves [13]. In the present paper we develop the techniques of [13] to be applicable to any algebraic varieties of any dimension.

• There are few isotopy types of real curves of degrees 6 in ℝ^{2} and of degree 8 in ℝP^{2} which cannot be realized by the original Viro method [6, 19, 23], but require different statements on deformation of singular curves. Our generalization includes all these statements as very special cases.

• Biran [5] proposed a construction of ample divisors on algebraic varieties which seems to be an analogue of the Viro construction for the problem considered. One of our goals has been to generalize the Viro construction to divisors on any smooth algebraic variety.

• In the above notation, the germ at W of the family of hypersurfaces F ∈ |W | with singular points of the same analytic type as for W, is smooth and has “expected” dimension if and only if |W | is a joint versal deformation of all singular points of W, i.e. arbitrary local deformations of all singular points of W can be independently realized by variation of W in |W |. The meaning of our result is that if the germ at W of the family of hypersurfaces F ∈ |W | with singular points of the same topological type is smooth and has an expected dimension then the lower deformations of all singular points of W can be independently realized by variation of W in |W |.

Our approach to the problem studied in based on the Viro method of gluing polynomials [30, 31, 32] and its modifications given in [23, 26, 13].

I am grateful to G.-M. Greuel, who explained to me few basic facts on semiquasihomogeneous and Newton nondegenerate singularities.

2 Singular points and their deformations

For the sake of notation simplicity we shall denote x = (x_{1}, ..., x_{n}) a point in ℂ^{n}, i = (i_{1}, ..., i_{n}) a point in ℤ_{+}^{n}, ℤ_{+} = {n ∈ ℤ : n ≥ 0}, |i| = i_{1} + ... + i_{n}, and x^{i} = x_{1}^{i_{1}}...x_{n}^{i_{n}} a monomial.

2.1 Equivalence of singular points and related zerodimensional schemes

Let V be a smooth algebraic projective variety of dimension n ≥ 2. Isolated singular points z_{1}, z_{2} of hypersurfaces W_{1}, W_{2} ⊂ V, respectively, are called analytically equivalent if the complete local rings Ô_{W_{1},z_{1}} and Ô_{W_{2},z_{2}} are isomorphic. This means that there exist neighborhoods U_{1}, U_{2} of z_{1}, z_{2} in V (in metric topology) and an analytic diffeomorphism of U_{1} onto U_{2} which takes W_{1} ∩ U_{1} to W_{2} ∩ U_{2}. In the above notation, the points z_{1} ∈ W_{1}, z_{2} ∈ W_{2} are called topologically equivalent if there exists a homeomorphism of U_{1} onto U_{2} which takes W_{1} ∩ U_{1} to W_{2} ∩ U_{2}. If V, W_{1}, W_{2}, z_{1}, z_{2} are real, i.e. invariant with respect to an antiholomorphic involution (complex conjugation), then we suppose the above isomorphism, diffeomorphism, or homeomorphism to be equivariant.

Let z be an isolated singular point of a hypersurface W ⊂ V defined in a neighborhood of z by an equation F (x) = 0 in some local coordinates on V. For any sufficiently large integer k the germ at F of the stratum in O_{V,z}/𝔪_{z}^{k}, where 𝔪_{z} ⊂ O_{V,z} is the maximal ideal, which consists of elements G ∈ O_{V,z}/𝔪_{z}^{k} defining hypersurface germ on V with a singular point analytically equivalent to z ∈ W, is smooth and its tangent space can be identified with I_{z}^{ea}/𝔪_{z}^{k}, where I_{z}^{ea} ⊂ O_{V,z} is the ideal generated by F, \frac{∂F}{∂x_{1}} , ..., \frac{∂F}{∂x_{n}}.

If n = 2, then similarly for any sufficiently large integer k the germ at F of the stratum in O_{V,z}/𝔪_{z}^{k} which consists of elements G ∈ O_{V,z}/𝔪_{z}^{k} defining a curve germ with a singular point topologically equivalent to z ∈ W, is smooth and its tangent space can be identified with I_{z}^{es}/𝔪_{z}^{k}, where I_{z}^{es} ⊂ O_{V,z} is the so-called equisingular ideal [8, 33].

In the sequel we will be given an analytic or topological (for n = 2) equivalence of isolated singular points, which will be denoted by S. This data, for a singular point z ∈ W ⊂ V, defines an ideal I_{z}^{S} ⊂ O_{V,z} equal to I_{z}^{ea} or I_{z}^{es}, and thereby a zero-dimensional scheme X_{z}^{S} concentrated at z.

2.2 Newton nondegenerate and semiquasihomogeneous singular points

Let a holomorphic function germ

F : (ℂ^{n}, 0) → (ℂ, 0), F (x) = ∑_{i} A_{i}x^{i}  (2)

define an isolated singularity at the origin. Let Δ(F) be the convex hull of all points i + j such that A_{i} ≠ 0 and j = (j_{1}, ..., j_{n}) ∈ ℤ_{+}^{n}. For a face σ of Δ(F) denote by K(σ) the convex hull of σ ∪ {0}. By the Newton diagram Γ(F) of F (at the origin) we mean the union of those faces σ of Δ(F) which satisfy

dim K(σ) = dim σ + 1, K(σ) ∩ Δ(F) = σ .

Denote by K(Γ(F)) the union of the closures of K(σ) taken over all faces σ ⊂ Γ(F).

Define the truncation F^{σ} of F on the face σ ⊂ Γ(F) by F^{σ}(x) = ∑_{i∈σ} A_{i}x^{i}. Any truncation F^{σ} is a quasihomogeneous polynomial or power series. A polynomial truncation F^{σ} is called nondegenerate if the hypersurface F^{σ} = 0 has no singular points in (ℂ^{∗})^{n}.

A singular point at the origin defined by F : (ℂ^{n}, 0) → (ℂ, 0) is called Newton nondegenerate if Γ(F) is compact and intersect all the coordinate axes, and all truncations F^{σ}, σ ⊂ Γ(F), are nondegenerate.

Let z be an isolated singular point of a hypersurface W in a smooth complex projective algebraic variety V of dimension n ≥ 2. We say that z is generalized Newton nondegenerate (briefly, GNND) if there exists a holomorphic germ F : (ℂ^{n}, 0) → (ℂ, 0) defining a Newton nondegenerate singularity at the origin such that the compete local rings Ô_{V,z} and Ô_{ℂ^{n},0}/(F) are isomorphic (or, equivalently, there is an isomorphism φ : O_{V,z} → O_{ℂ^{n},0} such that F = 0 is an equation of the germ (W, z) in the coordinates induced by φ. We call F a representative of the GNND singular point z ∈ W.

We say that a Newton nondegenerate singular point defined by a germ F : (ℂ^{n}, 0) → (ℂ, 0) is semiquasihomogeneous if there is a face σ ⊂ Γ(F) of dimension n − 1 (called main face) such that F^{σ} has no critical points in ℂ^{n}\{0}. Correspondingly, we call a GNND point z of a hypersurface W ⊂ V generalized semiquasihomogeneous (briefly, GSQH), if it has a semiquasihomogeneous representative. For a given GSQH point, the main face does not depend on its representative [21]. We call it the main face of a GSQH singular point. If the main face σ meets all the coordinate axes at the points with coordinate m, then such a semiquasihomogeneous singular point is called an ordinary singular point of multiplicity m.

2.3 Zero-dimensional schemes associated with GSQH and GNND singular points

Let z ∈ W ⊂ V be a GSQH singular point with a representative F : (ℂ^{n}, 0) → (ℂ, 0), F (x) = ∑ A_{i}x^{i}. Let σ ⊂ Γ(F) be the main face which lies on the hyperplane given by l(i) = 0 with l(0) > 0, then we define two ideals I(F) and Ĩ(F) in O_{ℂ^{n},0}:

I(F) = {G ∈ O_{ℂ^{n},0} : G = ∑_{l(i)<0} α_{i}x^{i}} ,

Ĩ(F) = {G ∈ O_{ℂ^{n},0} : G = ∑_{l(i)≤0} α_{i}x^{i}} .

Here

dim O_{ℂ^{n},0}/I(F) = #{i : l(i) ≥ 0}, dim O_{ℂ^{n},0}/Ĩ(F) = #{i : l(i) > 0}.

If φ : O_{V,z} → O_{ℂ^{n},0} is an isomorphism such that F = 0 is an equation of the germ (W, z) on V in the coordinates induced by φ, then we obtain ideals

I(W, z) = φ^{∗}(I(F)) + I_{z}^{ea}, Ĩ(W, z) = φ^{∗}(Ĩ(F)) + I_{z}^{ea} .

They define in V zero-dimensional subschemes X_{z}^{sqh}, X̃_{z}^{sqh}, respectively, concentrated at the point z.

Let z ∈ W ⊂ V be a GNND singular point with a representative F : (ℂ^{n}, 0) → (ℂ, 0). Let Γ(F) be the Newton diagram of F, and Δ(F), K(Γ(F)) be the unbounded and bounded closed domains in the nonnegative orthant of ℝ^{n} bounded by Γ(F) and the coordinate hyperplanes. Define two ideals I(F) and Ĩ(F) in O_{ℂ^{n},0}:

I(F) = {G ∈ O_{ℂ^{n},0} : G = ∑_{i∈Δ(F)} α_{i}x^{i}} ,

Ĩ(F) = {G ∈ O_{ℂ^{n},0} : G = ∑_{i∈Δ(F)\Γ(F)} α_{i}x^{i}} .

Here

dim O_{ℂ^{n},0}/I(F) = #(ℤ^{n}∩(K(Γ(F))\Γ(F))), dim O_{ℂ^{n},0}/Ĩ(F) = #(ℤ^{n}∩K(Γ(F))) .

If φ : O_{V,z} → O_{ℂ^{n},0} is an isomorphism such that F = 0 is an equation of the germ (W, z) on V in the coordinates induced by φ, then we obtain ideals

I(W, z) = φ^{∗}(I(F)) + I_{z}^{ea}, Ĩ(W, z) = φ^{∗}(Ĩ(F)) + I_{z}^{ea} .

They define in V zero-dimensional subschemes X_{z}^{nnd}, X̃_{z}^{nnd}, respectively, concentrated at the point z.

2.4 Deformations of singular points

In the above notation, let B ⊂ V be a sufficiently small compact neighborhood of z in V, diffeomorphic to a closed 2n-dimensional ball, and such that W_{0} = W ∩ B is a compact variety with an isolated singular point z and a boundary ∂W_{0} ⊂ ∂B which is transversal to ∂B along ∂W_{0} and is homeomorphic to a cone over ∂W_{0}.

Under a (one-parametric) deformation of a singular point z ∈ W ⊂ V we mean an analytic hypersurface W ⊂ B × D_{ε}, D_{ε} = {t ∈ ℂ : |t| < ε}, such that

• (π|_{W})^{−1}(0) = W_{0}, where π : B × D_{ε} → D_{ε} is the projection,

• for 0 < |t| < ε, (π|_{W})^{−1}(t) := W_{t} is a compact variety in B with isolated singularities in Int(B) and boundary ∂W_{t} ⊂ ∂B, so that W_{t} is transversal to ∂B along ∂W_{t}.

Given an equivalence S of hypersurface isolated singular points, we say that a deformation W_{t}, t ∈ (ℂ, 0) (or t ∈ [0, ε) in the real case), of (W, z) is S-compatible, if for any t_{1}, t_{2} ∈ (ℂ, 0), t_{1}t_{2} ≠ 0, there exists a bijection between Sing(W_{t_{1}}) and Sing(W_{t_{2}}) such that the corresponding points in Sing(W_{t_{1}}), Sing(W_{t_{2}}) are S-equivalent.

If z, W, V, W are real, i.e. invariant with respect to the complex conjugation Conj, then we take only real nonnegative values of the parameter t, and consider X ⊂ B × [0, ε).

2.5 Models for deformations of GSQH singular points

Let z ∈ W ⊂ V be a GSQH singular point. Accepting the notation of section 2.2, introduce the space of polynomials

P(σ) = { ∑_{l(i)≥0} α_{i}x^{i}}

and an affine subspace

P(σ, F) = {P ∈ P(σ) : P^{σ} = F^{σ}} ,

where F^{σ}, P^{σ} are the truncations of F (x), P (x), respectively, on σ. Let a polynomial P ∈ P(σ, F) define a hypersurface Φ = {P = 0} ⊂ ℂ^{n} having only isolated singular points. There is R_{0} > 0 such that any (2n − 1)-sphere S_{R} of radius R ≤ R_{0} centered at the origin intersects Φ transversally. Denote by B^{2n} ⊂ ℂ^{n} the 2n-dimensional closed ball of radius R_{0} centered at the origin. The pair (B^{2n}, Φ ∩ B^{2n}) is said to be a model for an S-compatible deformation W_{t}, t ∈ D_{ε}, of the GSQH singular point z ∈ W if, for any t ∈ D_{ε}\{0}, there exists a homeomorphism (B, W_{t}) → (B^{2n}, Y) which takes Sing(W_{t}) to Sing(Y) so that the corresponding singular points are Sequivalent (here B is a small (closed) neighborhood of z in V).

Given an integer l_{0} > max{|i| : l(i) ≥ 0}, we consider P, P(σ), P(σ, F) to be included in the space Σ(l_{0}) of polynomials of degree ≤ l_{0}. For any singular point w of Φ ∩ B^{2n}, introduce the germ M_{d}(w, P) at P of the set of polynomials in Σ(d) close to P and having a singular point S-equivalent to (Φ, w) in a neighborhood of w. We say that a model (B^{2n}, Φ ∩ B^{2n}) for deformations of the GSQH singular point z ∈ W satisfies the S-transversality (resp., strong S-transversality) condition if there exists sufficiently large integer d such that

• the intersection M_{d}(P) of the germs M_{d}(w, P), w ∈ Sing(Φ ∩ B^{2n}), is smooth, and

• M_{d}(P) intersects with P(σ) (resp., P(σ, F)) in Σ(d) transversally.

Remark 1 The S-transversality is defined in accordance with the similar notion in [26], section 3.

2.6 Models for deformations of GNND singular points

Let z ∈ W ⊂ V be a GSQH singular point, and F = ∑_{i∈Δ(F)} A_{i}x^{i} be its representative. In the notation of subsection 2.2, let us be given

• a subdivision of K(Γ(F)) into convex lattice polyhedra Δ_{1}, ..., Δ_{N} and a convex piece-wise linear real-valued function ν on the nonnegative orthant of ℝ^{n} such that Δ_{1}, ..., Δ_{N}, Δ(F) are its linearity domains, ν|_{Δ(F)} = 0,

• a set of numbers {A_{i} : i ∈ (K(Γ(F))\Γ(F)) ∩ ℤ_{+}^{n}} (real in the real case, complex in the general case) such that, for any Δ_{k} and any proper face σ ⊂ Δ_{k}, the polynomial F^{σ}(x) = ∑_{i∈σ} A_{i}x^{i} has no singular points in (ℂ^{∗})^{n}.

Note that the coefficients A_{i}, i ∈ Γ(F), are taken from (2), and that all the polynomials F_{k}(x) = ∑_{i∈Δ_{k}} A_{k}x^{k} define hypersurfaces in ℂ^{n} with isolated singular points only.

The following construction is basically introduced in [18]. For any k = 1, ..., N we have the momentum map of the positive orthant ℝ_{+}^{n} into the polyhedron Δ_{k}

μ_{k}(x) = \frac{∑_{i∈Δ_{k}} i · x^{i}}{∑_{i∈Δ_{k}} x^{i}} {def \atop =} (μ_{k}^{(1)}(x), ..., μ_{k}^{(n)}(x)) ,

which is a diffeomorphism of ℝ_{+}^{n} onto Int(Δ_{k}) (see [3, 4]). We introduce the complexification of the polyhedron Δ_{k}

ℂΔ_{k} = {(w_{1}, ..., w_{n}) ∈ ℂ^{n} : (|w_{1}|, ..., |w_{n}|) ∈ Δ_{k}},

and the complexification of the momentum map

ℂμ_{k} : (ℂ^{∗})^{n} → ℂΔ_{k}, ℂμ_{k}(w_{1}, ..., w_{n}) = (μ_{k}^{(1)}(x)v_{1}, ..., μ_{k}^{(n)}(x)v_{n}),

x = (|w_{1}|, ..., |w_{n}|), v_{1} = \frac{w_{1}}{|w_{1}|}, ..., v_{n} = \frac{w_{n}}{|w_{n}|} ,

which is a diffeomorphism of (ℂ^{∗})^{n} onto Int(ℂΔ_{k}) ∩ (ℂ^{∗})^{n}. By the (complex) chart Ch(F_{k}) of the polynomial F_{k} we call the closure of the set ℂμ_{k}({F_{k} = 0} ∩ (ℂ^{∗})^{n}).

Proposition 1 (1) The set ℂK(Γ(F)) = ℂΔ_{1} ∪ ... ∪ ℂΔ_{N} is homeomorphic to a closed ball B^{2n}. The set Φ = Ch(F_{1}) ∪ ... ∪ Ch(F_{N}) is a (topological) (2n − 2)- manifold with the boundary Φ∩∂(ℂK(Γ(F))) and the finite singular locus Sing(Φ) = ℂμ_{1}(Sing(F_{1})) ∪ ... ∪ ℂμ_{N}(Sing(F_{N})).

(2) Let {F = 0} intersect the boundary of a closed ball B^{2n} centered at the origin transversally, and {F = 0} ∩ B^{2n} be a cone over {F = 0} ∩ ∂(B^{2n}). Let functions Ã_{i}(t), t ≥ 0, with Ã_{i}(0) = A_{i}, i ∈ ℤ_{+}^{n}, be such that, for a sufficiently small t > 0, there is a bijection between the singular point set of the hypersurface F̃(x) = ∑ Ã_{i}(t)x^{i}t^{ν(i)} = 0 in Int(B^{2n}) and the disjoint union Sing(F_{1}, ..., F_{N}) of the sets Sing(F_{1})∩(ℂ^{∗})^{n}, ..., Sing(F_{N})∩(ℂ^{∗})^{n}, so that the corresponding singular points are S-equivalent. Then there exists a homeomorphism of (B^{2n}, {F̃ = 0} ∩ B^{2n}) onto (ℂK(Γ(F)), Φ) which extends the above bijection Sing(F̃) ∩ B^{2n} ↔ Sing(F_{1}, ..., F_{N}).

If all the data is defined over the reals and Sing(F_{1}, ..., F_{N}) = ∅, the statement of Proposition 1 for the real parts of ℂK(Γ(F)), Φ, B^{2n} and {F̃ = 0} is a particular case of Viro’s theorem [30]. In fact, the proof of Viro’s theorem [30] provides the same claim with Sing(F_{1}, ..., F_{N}) ≠ ∅. In full generality Proposition 1 is proven in [18] in a similar way. Here we omit the proof.

The pair (ℂK(Γ(F)), Φ) as above is said to be a model for an S-compatible deformation W_{t}, t ∈ D_{ε}, of the GNND singular point z ∈ W if, for any t ∈ D_{ε}\{0}, there exists a homeomorphism (B, W_{t}) → (ℂK(Γ(F)), Φ) which takes Sing(W_{t}) to Sing(F_{1}, ..., F_{N}) so that the corresponding singular points are S-equivalent (here B is a small (closed) neighborhood of z in V).

We shall define the S-transversality for the above model (ℂK(Γ(F)), Φ) (cf. [26], section 3).

Let Gr be the dual graph of the subdivision ℝ_{+}^{n} = Δ(F) ∪ Δ_{1} ∪ ... ∪ Δ_{N}: namely, its vertices are incident to Δ(F), Δ_{1}, ..., Δ_{N}, and its arcs are incident to common facets of these polyhedra. Define G to be the set of oriented graphs G̃ with support G and without oriented cycles. It is clear that G ≠ ∅. For any G ∈ G we denote by Δ_{i,+}(G) the union of facets of Δ_{i}, which correspond to the arcs of G̃ coming in Δ_{i}, i = 1, ..., N.

For a polynomial F_{i}(x) and an d ≥ deg F_{i} denote by M_{d}(F_{i}) the germ at F_{i} of the set of polynomials of degree ≤ d which, in a neighborhood of Sing(F_{i}) ∩ (ℂ^{∗})^{n}, define singular points S-equivalent to the corresponding singular points of {F_{i} = 0}. The triple (Δ_{i}, Δ_{i,+}(G), F_{i}) is called S-transversal if

• M_{d}(F_{i}) is smooth for sufficiently large d,

• if n ≥ 3 then M_{d}(F_{i}) (d >> 0) intersects transversally with the set of polynomials with Newton polyhedron Δ_{i} having the same coefficients at i ∈ Δ_{i,+}(G) as F_{i}; if n = 2 then M_{d}(F_{i}) (d >> 0) intersects transversally with the set of polynomials with Newton polyhedron Δ_{i}, whose truncations on any connected component of Δ_{i,+}(G) is proportional to the corresponding truncation of F_{i}.

Remark 2 The S-transversality condition for n ≥ 3 is stronger than that for n = 2, if Δ_{k,+}(G) is not connected. In [26], the latter one is called weak S-transversality.

The model (ℂK(Γ(F)), Φ) is called S-transversal if there exists a graph G ∈ G such that no arc of G goes out of Δ(F) and all the triples (Δ_{i}, Δ_{i,+}(G), F_{i}), i = 1, ..., N, are S-transversal. The model (ℂK(Γ(F)), Φ) is called strongly Stransversal if there exists a graph G ∈ G such that no arc of G goes into Δ(F) and all the triples (Δ_{i}, Δ_{i,+}(G), F_{i}), i = 1, ..., N, are S-transversal.

Remark 3 The case of GSQH singular point with all notions introduced (models, S-transversality) is, in fact, a particular case of a GNND singular point. Nevertheless, we treat it separately, because it is important by itself, all definitions look much simpler that the general ones. Finally, if the main face σ of a GSQH singular point intersects the coordinate axes not at integral points, then the treatment of this situation as a general GNND singular point would be unnecessarily complicated.

3 Independence of lower deformations of GSQH and GNND singular points

Theorem 1 Let V be a smooth algebraic projective variety of dimension n ≥ 2 and W ⊂ V be a hypersurface with a finite singular set Sing(W). Let Sing(W) be divided into disjoint subsets S, S_{1}, S_{2}, S_{3}, S_{4}, S_{5} (each of them may be empty), such that the singular points z ∈ S_{2} ∪ S_{3} are GSQH, the singular points in S_{4} ∪ S_{5} are GNND. Let us be given, for any point z ∈ S_{2} ∪ S_{3}, a deformation model (B^{2n}, Φ_{z}), for any point z ∈ S_{4} ∪ S_{5}, a deformation model (ℂK(Γ(F_{z})), Φ_{z}), which is strongly S-transversal if z ∈ S_{2} ∪ S_{4} and S-transversal if z ∈ S_{3} ∪ S_{5}. Define a zero-dimensional scheme

X = ⋃_{z∈S} X_{z}^{S} ∪ ⋃_{z∈S_{1}} X_{z}^{ea} ∪ ⋃_{z∈S_{2}} X_{z}^{sqh} ∪ ⋃_{z∈S_{3}} X̃_{z}^{sqh} ∪ ⋃_{z∈S_{4}} X_{z}^{nnd} ∪ ⋃_{z∈S_{5}} X̃_{z}^{nnd}

and its ideal sheaf J_{X/V} on V. If

h^{1}(V, J_{X/V} ⊗ O_{V} (W)) = 0  (3)

then there exists a family W_{t} ∈ |W |, t ∈ ℂ, |t| < ε, such that W_{0} = W and, the family W_{t} realizes

• for any z ∈ S, an S-equisingular deformation of z ∈ W,

• for any z ∈ S_{1}, an arbitrary deformation prescribed up to local diffeomorphism of a neighborhood of z in V,

• for any z ∈ S_{2} ∪ S_{3}, an S-compatible deformation of z ∈ W with the model (B^{2n}, Φ_{z}),

• for any z ∈ S_{4} ∪ S_{5}, an S-compatible deformation of z ∈ W with the model (ℂK(Γ(F_{z})), Φ_{z}).

In addition, for t ≠ 0,

h^{1}(V, J_{X_{t}/V} ⊗ O_{V} (W_{t})) = 0 ,  (4)

where X_{t} = ∪_{z}X_{z}^{S} with z running over all singular points of W_{t} (this means, in particular, that the germ at W_{t} of the S-equisingular stratum in |W | is smooth and has an expected dimension).

If all the data are invariant with respect to Conj, then there exists an equivariant family W_{t}, t ∈ [0, ε), with the above properties.

Remark 4 (1) Note that all simple singularities are GSQH and for them X_{z}^{S} = X_{z}^{ea}. Hence they should be included into the sets S and S_{1}, when applying Theorem 1.

(2) Comparing with Lemma [23], Theorem 1 not only extends the application range of the statement on independence of deformations, but is stronger even in the situation of [23]. Indeed, the condition of Lemma [23] is equivalent to the h^{1}-vanishing for the ideal sheaf of a zero-dimensional scheme bigger than that in Theorem 1, for instance, such a scheme for an ordinary singular point is defined by the ideal I(F) in [23] (cf. [24]), and is defined by the ideal I(F) + I^{ea} ≠ I(F) in Theorem 1.

Proof. For the reader’s convenience we present our argument in few steps.

Step 1. Assume that S_{2} = {z}, S_{1} = S_{3} = S_{4} = S_{5} = ∅.

1.1. From the exact sequence

0 → J_{X/V} ⊗ O_{V} (W) → O_{V} (W) → O_{X} → 0  (5)

and (3) we obtain

H^{0}(V, O_{V} (W)) = H^{0}(V, J_{X/V} ⊗ O_{V} (W)) ⊕ H^{0}(X, O_{X})
= H^{0}(V, J_{X/V} ⊗ O_{V} (W)) ⊕ O_{V,z}/I_{z}^{sqh} .  (6)

For the point z we have an isomorphism φ : O_{V,z} → O_{ℂ^{n},0} and a semiquasihomogeneous representative F : (ℂ^{n}, 0) → (ℂ, 0) such that F (x) = 0 is an equation of the germ (W, z) in the coordinates defined in a neighborhood of z in V by φ. We can write

F (x) = ∑_{l(i)≤0} A_{i}x^{i} ,  (7)

where l(i) = 0 is an equation of the hyperplane containing the main face σ of the Newton diagram of F, l(0) > 0 and l is integral-valued at integral points. On the other hand the model Φ_{z} of deformation for z can be described in B^{2n} by an equation

Φ_{z}(x) = ∑_{l(i)≥0} A_{i}x^{i} .

1.2. Let Λ, Λ_{1}, ..., Λ_{n} ⊂ ℤ_{+}^{n} be (finite) sets such that {x^{i} : i ∈ Λ} is projected to a basis of O_{ℂ^{n},0}/(I(F)+ < ∂F/∂x_{1}, ..., ∂F/∂x_{n} >), and

\{x^{i} : i ∈ Λ\} ∪ ⋃_{j=1}^{n}\{x^{j} \frac{∂F}{∂x_{j}} : j ∈ Λ_{j}\}

is projected to a basis of O_{ℂ^{n},0}/I(F). Denote by [P (x)] the sum of those monomials of a polynomial P (x) which do not belong to I(F).

Relation (6) and the isomorphism φ imply that there exist a basis of H^{0}(V, O_{V} (W)), consisting of elements represented in O_{V,z} = O_{ℂ^{n},0} as

F_{i}(x) = x^{i} + ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj} [x^{j} \frac{∂F}{∂x_{j}} ] + ∑_{l(m)≤0} A_{im}x^{m}, i ∈ Λ,

F_{i}(x) = ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj} [x^{j} \frac{∂F}{∂x_{j}} ] + ∑_{l(m)≤0} A_{im}x^{m}, i = 0, ..., s.

Denote by B_{i}, i ∈ Λ, B_{i}, i = 0, ..., s, the coordinates in H^{0}(V, O_{V} (W)) with respect to this basis. Without loss of generality, assume that F is encoded by B_{i} = 0, i ∈ Λ, B_{0} = 1, B_{i} = 0, i = 1, ..., s.

1.3. Consider a deformation of F in φ_{z}(H^{0}(V, O_{V} (W))) ⊂ O_{ℂ^{n},0} defined by

F_{t}(x) = F (x) + ∑_{i∈Λ} a_{i}F_{i}(x) + ∑_{i=1}^{s} a_{i}F_{i}(x),  (8)

and depending on small parameters a_{i}, i ∈ Λ, a_{i}, i = 1, ..., s, which later will be specified as smooth functions of a small parameter t.

The requirement of Theorem 1 means that the elements of this deformation belong to the germ at W of the S-equisingular stratum Σ in H^{0}(V, O_{V} (W)) corresponding to the points w ∈ S, and, on the other hand, they realize the given model (B^{2n}, Φ_{z}) in a neighborhood of z.

1.4. By (3) Σ is smooth and transversal to the space H^{0}(V, J_{X_{z}^{sqh}}(W)). Hence there is a smooth germ Σ′ ⊂ Σ defined by

B_{0} = 1, B_{i} = A_{i} + ψ_{i}({B_{i} : i ∈ Λ}), i = 1, ..., s,

where ψ_{i}, i = 1, ..., s, are smooth functions vanishing at zero. Therefore the elements of (8) belong to Σ′ if

a_{i} = ψ_{i}({a_{i} : i ∈ Λ}), i = 1, ..., s.  (9)

1.5. Consider a local coordinate change x ↦ T (x) such that

T_{j}(x) = x_{j} + ∑_{j∈Λ_{j}} a_{j,j}x^{j}, j = 1, ..., n,  (10)

where a_{j,j}, j ∈ Λ_{j}, j = 1, ..., n, are small parameters. Then one derives

F̃_{t}(x) = F_{t}(T (x)) = F_{t}(x) + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} a_{j,j}x^{j} \frac{∂F}{∂x_{j}} + O(\{a_{i}^{2}\}, \{a_{j,j}^{2}\})

= ∑_{k∈Λ} c_{k}x^{k} + ∑_{l(k)≤0} c_{k}x^{k} + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} c_{j,j} [x^{j} \frac{∂F}{∂x_{j}} ] ,

where

c_{k} = a_{k} + O({a_{i}^{2}}, {a_{j,j}^{2}}), k ∈ Λ,  (11)
c_{j,j} = ∑_{i∈Λ} a_{i}A_{i,jj} + ∑_{i=1}^{s} a_{i}A_{i,jj} + a_{j,j} + O({a_{i}^{2}}, {a_{j,j}^{2}}),  (12)
j ∈ Λ_{j}, j = 1, ..., n,
c_{k} = A_{k} + ψ_{k}^{(1)}({a_{i}^{2}}, {a_{j,j}^{2}}), l(k) ≤ 0,  (13)

with ψ_{k}^{(1)} being certain smooth functions vanishing at zero. By the implicit function theorem, equations (9), (11), (12) are solvable with respect to a_{i}, i ∈ Λ, a_{i}, i = 1, ..., s, a_{j,j}, j ∈ Λ_{j}, j = 1, ..., n, as functions of

c_{k}, k ∈ Λ, c_{j,j}, j ∈ Λ_{j}, j = 1, ..., n.  (14)

Hence the family F̃_{t} can be reparameterized by the free parameters (14):

F̃_{t}(x) = ∑_{k∈Λ} c_{k}x^{k} + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} c_{j,j} [x^{j} \frac{∂F}{∂x_{j}} ] + ∑_{l(k)≤0}(A_{k} + ψ_{k}^{(2)})x^{k}

with ψ_{k}^{(2)} being smooth functions of (14) vanishing at zero.

Now we impose the following condition on the parameters (14):

∑_{k∈Λ} c_{k}x^{k} + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} c_{j,j} [x^{j} \frac{∂F}{∂x_{j}} ] = ∑_{l(k)>0} d_{k}t^{l(k)}x^{k},

where d_{k} is close to A_{k} for any k ∈ ℤ_{+}^{n} with l(k) > 0, and t is a small parameter. The latter relation has a solution with respect to (14) as functions of {d_{k}t^{l(k)} : l(k) > 0}. Let γ_{1} = −\frac{∂l}{∂i_{1}} , ..., γ_{n} = −\frac{∂l}{∂i_{n}}. Passing to the parameters \{d_{k}\} and t and rescaling as follows, we obtain the family

F̂_{t}(x) = F̃_{t}(x_{1}t^{γ_{1}}, ..., x_{n}t^{γ_{n}})t^{−l(0)} = ∑_{l(k)>0} d_{k}x^{k} + ∑_{l(k)≤0} t^{−l(k)}(A_{k} + tψ_{k}^{(3)})x^{k} ,

where ϕ_{k}^{(3)} are smooth functions of {d_{k}} and t.

1.6. The family F̂_{t}(x) is included in the space F of functions in x ∈ ℂ^{n} analytic in the ball B^{2n}. The subfamily F̂_{0}(x) (depending on parameters d_{k}, l(k) > 0) is a germ at Φ_{z} of the space of polynomials P(σ) introduced in subsection 2.5. Due to the strong S-transversality, F̂_{0}(x) transversally intersects the germ M ⊂ F at Φ_{z} of functions which are S-equisingular to Φ_{z} in B^{2n}. One can choose Λ′ ⊂ {i ∈ ℤ_{+}^{n} : l(i) > 0} so that the family F̂′_{0}(x) with fixed parameters d_{k} = A_{k}, k ∉ Λ′, and free parameters d_{k}, k ∈ Λ′, intersects M transversally and only at Φ_{z}. Assume that for any small fixed t, fixed parameters d_{k} = A_{k}, k ∉ Λ′, and free parameters d_{k}, k ∈ Λ′, the family F̂′_{t}(x) meets M at one point as well, and this point corresponds to the parameters d_{k}(t), k ∈ Λ′. These values of d_{k} define the initial parameters a_{i}, i ∈ Λ, as functions of t so that the deformation (8) is S-equisingular outside a neighborhood of z and realizes the given model Φ_{z} in a neighborhood of z.

1.7. It remains to justify the latter argument, since the implicit function theorem in general is not valid in infinite-dimensional spaces. Let μ be the maximal Milnor number of the singular points x_{1}, ..., x_{q} of Φ_{z}. Due to Tougeron’s theorem [28], the condition for an element Ψ ∈ F, close to Φ_{z}, to be S-equisingular to Φ_{z} at x_{i} is expressed by a system of analytic equations

ψ_{i,1}^{(4)}(jet_{x_{i},μ+1}(Ψ)) = ... = ψ_{i,s_{i}}^{(4)}(jet_{x_{i},μ+1}(Ψ)) = 0 ,

whose variables are the coefficients of the (μ + 1)-jet jet_{x_{i},μ+1}(Ψ) of Ψ at x_{i}, and whose Jacobian with respect to these variables has rank s_{i} at Ψ = Φ_{z}. Introducing new qn variables being the coordinates of points \overline{x}_{1}, ..., \overline{x}_{q} ∈ ℂ^{n}, close to x_{1}, ..., x_{q}, respectively, one can express the relation F̂′_{t}(x) ∈ M as

ψ_{i,j}^{(4)}(jet_{\overline{x}_{i},μ+1}(F̂′_{t})) = 0, j = 1, ..., s_{i}, i = 1, ..., q.  (15)

Since the coefficients of the jets jet_{\overline{x}_{i},μ+1}(F̂′_{t}) are analytic functions in

\overline{x}_{1}, ..., \overline{x}_{q}, d_{k}, k ∈ Λ′, t,  (16)

(15) turns into a system of s_{1} +...+s_{q} equations in the s_{1} +...+s_{q} +1 variables (16). The transversality of the intersection of the family F̂′_{0} with M means that, for t = 0, the determinant of the Jacobian of functions (15) with respect to variables (16) is not zero. Hence the system (15) has a solution d_{k} = d_{k}(t), k ∈ Λ′ (and \overline{x}_{i} = \overline{x}_{i}(t), i = 1, ..., q), which completes the proof of Theorem 1 in case S_{2} = {z}, S_{3} = ∅.

1.8. At last, note that the transversality of the intersection of the family F̂′_{t} with M, established in 1.6, 1.7, means that the germ at W_{t} (t ≠ 0) of the S-equisingular stratum in |W |, corresponding to the singular points of W_{t} in a neighborhood of z, is smooth, has an expected dimension and intersects transversally with Σ′, hence so does with Σ ⊃ Σ′, the germ at W of the S-equisingular stratum corresponding to points in S. In the language of zero-dimensional schemes, this is equivalent to (4).

Step 2. Assume that S_{1} = S_{2} = S_{4} = S_{5} = ∅, S_{3} = {z}.

In this case the proof goes word-for-word as the proof in Step 1 with the following changes:

• substitute Ĩ_{z}^{sqh} for I_{z}^{sqh}, Ĩ(F) for I(F),

• substitute in 1.2-1.6 (in summations etc.) the set {i ∈ ℤ_{+}^{n} : l(i) ≥ 0} for {i ∈ ℤ_{+}^{n} : l(i) > 0}, the set {i ∈ ℤ_{+}^{n} : l(i) < 0} for {i ∈ ℤ_{+}^{n} : l(i) ≤ 0}.

Step 3. Assume that S_{1} = S_{2} = S_{3} = S_{4} = ∅, S_{5} = {z}, and the Newton diagram of a representative of the germ (W, z) has at least two facets (otherwise we get the case in Step 2). Here we combine the argument from Step 1 and from the proof of Theorems 3.1, 3.2 [26].

3.1. As in Step 1, from the exact sequence (5) and (3) we obtain

H^{0}(V, O_{V} (W)) = H^{0}(V, J_{X/V} ⊗ O_{V} (W)) ⊕ O_{V,z}/Ĩ_{z}^{nnd} .  (17)

For the point z we have an isomorphism φ : O_{V,z} → O_{ℂ^{n},0} and a Newton nondegenerate representative F_{z} = F : (ℂ^{n}, 0) → (ℂ, 0) such that F (x) = 0 is an equation of the germ (W, z) in the coordinates defined in a neighborhood of z in V by φ. We can write

F (x) = ∑_{i∈Δ(F)} A_{i}x^{i} .  (18)

On the other hand the model Φ_{z} of deformation for z is given by the data Δ_{1}, ..., Δ_{N}, ν : ℝ_{+}^{n} → ℝ, {A_{i} : i ∈ K(Γ(F))\Γ(F)}, as described in section 2.6.

3.2. We choose (finite) sets Λ, Λ_{1}, ..., Λ_{n} ⊂ ℤ_{+}^{n} as in 1.2 and take a basis of H^{0}(V, O_{V} (W)) consisting of elements represented in O_{V,z} = O_{ℂ^{n},0} by

F_{i}(x) = x^{i} + ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj} [x^{j} \frac{∂F}{∂x_{j}} ] + A_{i,0}[F] + ∑_{m∈Δ(F)} A_{im}x^{m}, i ∈ Λ,

F_{i}(x) = ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj} [x^{j} \frac{∂F}{∂x_{j}} ] + A_{i,0}[F] ∑_{m∈Δ(F)} A_{im}x^{m}, i = 0, ..., s,

(here F is nonzero in the ideal Ĩ(F)+ < \frac{∂F}{∂x_{1}} , ..., \frac{∂F}{∂x_{n}} > if Γ(F) contains at least two facets). Denote by B_{i}, i ∈ Λ, B_{i}, i = 0, ..., s, the coordinates in H^{0}(V, O_{V} (W)) with respect to this basis. Without loss of generality, assume that F is encoded by B_{i} = 0, i ∈ Λ, B_{0} = 1, B_{i} = 0, i = 1, ..., s.

3.3. Consider a deformation of F in φ_{z}(H^{0}(V, O_{V} (W))) ⊂ O_{ℂ^{n},0} defined by (8) and depending on small parameters a_{i}, i ∈ Λ, a_{i}, i = 1, ..., s.

As in 1.4, the elements of (8) belong to a smooth germ Σ′ ⊂ Σ if the parameters a_{i}, i ∈ Λ, a_{i}, i = 1, ..., s, satisfy (9).

3.4. We introduce the local coordinate change (10), but transform the family F_{t} in a slightly different way:

F̃_{t}(x) = (1 + a_{0})F_{t}(T (x)) = F_{t}(x) + a_{0}F (x) ∑_{j=1}^{n} ∑_{j∈Λ_{j}} a_{j,j}x^{j} \frac{∂F}{∂x_{j}} + O(a_{0}^{2}, \{a_{i}^{2}\}, \{a_{j,j}^{2}\}) .

(19)

As in 1.5, one can introduce new small parameters (14) and c_{0}, and reparameterize F_{t} in the form

F̃_{t}(x) = ∑_{k∈Λ} c_{k}x^{k} + c_{0}[F] + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} c_{j,j} [x^{j} \frac{∂F}{∂x_{j}} ] + ∑_{k∈Δ(F)\Γ(F)}(A_{k} + ψ_{k}^{(2)})x^{k}

with ψ_{k}^{(2)} being smooth functions of (14) and c_{0} vanishing at zero. Then we impose the following condition on the parameters (14) and c_{0}:

∑_{k∈Λ} c_{k}x^{k} + c_{0}[F] + ∑_{j=1}^{n} ∑_{j∈Λ_{j}} c_{j,j} [x^{j} \frac{∂F}{∂x_{j}} ] = ∑_{k∈K(Γ(F))} d_{k}t^{ν(k)}x^{k},

where d_{k} is close to A_{k} for any k ∈ K(Γ(F)), and t is a small parameter. The latter relation has a solution with respect to (14) as functions of {d_{k}t^{ν(k)} : k ∈ K(Γ(F))\Γ(F)}, and leads us to the family

F̂_{t}(x) = ∑_{k∈K(Γ(F))} d_{k}t^{ν(k)}x^{k} + ∑_{k∈Δ(F)\Γ(F)}(A_{k} + tψ_{k}^{(3)})x^{k} ,  (20)

where ϕ_{k}^{(3)} are smooth functions of {d_{k}} and t.

To complete the proof in the case considered, we should show that the parameters d_{k}, k ∈ K(Γ(F)), can be defined as functions of t equal to A_{k} as t = 0 and such that the singularities of F̂_{t} = 0 in a neighborhood of the origin are Sing(F_{1}, ..., F_{N}) (up to S-equivalence). Then by construction and Proposition 1 one would obtain that the deformation (8) satisfies (9) and realizes the given model (ℂK(Γ(F)), Φ_{z}) in a neighborhood of z.

3.5. The proof of the latter claim essentially coincides with the proof of Theorems 3.1, 3.2, 3.6 in [26], section 3.3: it is indeed the same if F is a polynomial, and it should be completed with the argument from 1.7 if F is a power series.

Remark 5 If V = ℙ^{n} then the situation considered in Step 3 is a particular case of Theorems 3.1, 3.6 [26] with the following difference. Taking the Viro-type polynomial in Theorems 3.1, 3.6 [26] as a deformation of a polynomial F with a Newton nondegenerate singular point, one would have all the coefficients at i ∈ K(Γ(F)) as independent free parameters, whereas in our situation, in the beginning, one has the free coefficients a_{i}, i ∈ Λ, only. The rest of necessary free parameters (as in formula (20)) one obtains from the transformation (19).

Step 4. Assume that S_{1} = S_{2} = S_{3} = S_{5} = ∅, S_{4} = {z}. Then the proof goes as in Step 3 with the following changes (cf. Step 2):

• one should get rid of [F], a_{0}, c_{0} in all the formulae,

• the ideals Ĩ_{z}^{nnd}, Ĩ(F) should be replaced by the ideals I_{z}^{nnd}, I(F), respectively,

• the sets Δ(F)\Γ(F), K(Γ(F)) (in summations etc.) should be replaced by the sets Δ(F), K(Γ(F))\Γ(F), respectively.

Step 5. Assume that S_{1} = ∅, but S_{2}, S_{3}, S_{4}, S_{5} may be nonempty.

The exact sequence (5) implies that H^{0}(V, O_{V} (W)) splits into the direct sum

H^{0}(V, J_{X/V} ⊗ O_{V} (W)) ⊕ ⨁_{z∈S_{2}} \frac{O_{V,z}}{I_{x}^{sqh}} ⊕ ⨁_{z∈S_{3}} \frac{O_{V,z}}{Ĩ_{x}^{sqh}} ⊕ ⨁_{z∈S_{4}} \frac{O_{V,z}}{I_{x}^{nnd}} ⊕ ⨁_{z∈S_{5}} \frac{O_{V,z}}{Ĩ_{x}^{nnd}} .

This allows us to introduce the following base in H^{0}(V, O_{V} (W)):

F_{i}^{(z)}(x) = x^{i} + ∑_{j∈Λ_{j}^{(z)}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{l^{(z)}(m)≤0} A_{im}^{(z)}x^{m}, i ∈ Λ^{(z)}, z ∈ S_{2},

F_{i}^{(z)}(x) = x^{i} + ∑_{j∈Λ_{j}^{(z)}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{l^{(z)}(m)<0} A_{im}^{(z)}x^{m}, i ∈ Λ^{(z)}, z ∈ S_{3},

F_{i}^{(z)}(x) = x^{i} + ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{m∈Δ^{(z)}(F)} A_{im}^{(z)}x^{m}, i ∈ Λ^{(z)}, z ∈ S_{4},

F_{i}^{(z)}(x) = x^{i} + ∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + A_{i,0}^{(z)}[F]^{(z)} + ∑_{m∈Δ^{(z)}(F)\K(Γ^{(z)}(F))} A_{im}^{(z)}x^{m},

i ∈ Λ^{(z)}, z ∈ S_{5},

and F_{i}, i = 0, ..., s, where (i) each F_{i}^{(z)} is represented by its image in O_{V,z} = O_{ℂ^{n},0}, (ii) the sets Λ^{(z)}, Λ_{1}^{(z)}, ..., Λ_{n}^{(z)} and the symbols [∗]^{(z)} are defined for any point z ∈ S_{2} ∪ ... ∪ S_{5} as described above, (iii) for any i = 1, ..., s and any z ∈ S_{2} ∪ ... ∪ S_{5}, the element F_{i} ∈ H^{0}(V, O_{V} (W)) has an image

∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{l(m)≤0} A_{im}^{(z)}x^{m}, or

∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{l(m)<0} A_{im}^{(z)}x^{m}, or

∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + ∑_{m∈Δ^{(z)}(F)} A_{im}^{(z)}x^{m}, or

∑_{j∈Λ_{j}^{1≤j≤n}} A_{i,jj}^{(z)} [x^{j} \frac{∂F}{∂x_{j}} ]^{(z)} + A_{i,0}^{(z)}[F]^{(z)} + ∑_{m∈Δ^{(z)}(F)\K(Γ^{(z)}(f))} A_{im}^{(z)}x^{m},

according as z ∈ S_{2}, or S_{3}, or S_{4}, or S_{5}. Finally, we assume that F_{0} = F.

The existence of such a basis in H^{0}(V, O_{V} (W)) allows us to perform the above construction of the required deformation for all singular points z ∈ S_{2} ∪ ... ∪ S_{5} independently, and leads to the desired result.

Step 6. Assume that S_{1} ≠ ∅.

Considering the set S_{1} as a part of S and taking X_{z}^{S}X_{z}^{ea}, z ∈ S_{1}, we construct the required deformation of all the points z ∈ S_{2} ∪ ... ∪ S_{5}, while the points z ∈ S ∪ S_{1} keep their S-type (may be, slightly moving in V). For an element W_{t}, t ≠ 0, of the family constructed, we have (4). In this equality X_{z}^{S} = X_{z}^{ea} for all z ∈ S′_{1} (the singular points coming from S_{1} ⊂ Sing(W)); hence the germ Σ′′ at W_{t} of the Sequisingular stratum in |W |, corresponding to the singular points in Sing(W_{t})\S′_{1}, is a joint versal deformation of all the points z ∈ S′_{1}, which are analytically equivalent to the corresponding singular points of W and have the same deformations. Thus, one can deform W_{t} inside Σ′′ realizing prescribed deformations of the singular points z ∈ S′_{1} simultaneously.

It remains to prove (4) for the hypersurfaces in the latter deformation. This follows immediately from the fact that all equisingular (in the analytic sense) strata in a versal deformation base of an isolated hypersurface singularity are smooth and have an expected dimension. □

4 Numerical criteria for the h^{1}-vanishing in Theorem 1

Here we restrict ourselves to the case V = ℂP^{n} or ℝP^{n} and W ⊂ V is a hypersurface of degree d.

Lemma 2 In the notation of Theorem 1, (3) holds if

deg X < 4(d − 1) .

Proof. Since X ⊂ ⋃_{z∈Sing(W)} X_{z}^{ea}, the required statement can be derived in the same way as Theorems 1, 2 [27] (see also [9]). □

In the case n = 2 there is a number of h^{1}-vanishing criteria for zero-dimensional schemes associated with singular plane algebraic curves, some of them we present below.

First, we recall some singular point invariants.

Definition 1 (see [26], section 4.1, and [24]) Let z be an isolated singular point of a plane curve C, T be its embedded resolution tree, C_{(q)} be the strict transform of C corresponding to an infinitely near point q ∈ T, and E_{(q)} be the reduced exceptional divisor. We define topological invariants associated with z:

• Let z be of type A_{2k−1}, k ≥ 1 (equivalent to y^{2} + x^{2k}). Then C has at z two non-singular branches P_{1}, P_{2}, and we put b(C, P_{1}) = b(C, P_{2}) = k − 1.

• Let z be of type A_{2k}, k ≥ 1 (equivalent to y^{2} + x^{2k+1}). Then C has at z one branch P, and we put b(P, C) = 2k − 1.

• Let z be not of type A_{k}. A point q ∈ T is called terminal if it is non-singular both for C_{(q)} and E_{(q)}. We remove from T all the points following terminal points. For a local branch Q of C centered at z, define b′(C, Q) to be the sum of the multiplicities of the strict transforms Q_{(q)} of Q at non-terminal points q ∈ T, and define b′′(C, Q) to be (Q_{(q)} ·E_{(q)})−1 at the (unique) terminal point q on Q, where (∗ · ∗) is the intersection number. Put b(C, Q) = b′(C, Q) + b′′(C, Q).

Let z ∈ C be a GSQH point with a representative F_{z} and the main edge σ of the Newton diagram Γ(F_{z}). We assume that if the straight line through σ meets a coordinate axis at an integral point than this point belongs to σ (clearly, such a representative F_{z} exists). Put b̃(C, Q) = b(C, Q) for branches Q which are not determined by σ, and b̃(C, Q) = p + q for branches Q which are determined by σ, where F_{z}^{σ}(x, y) = x^{α}y^{β} ∏_{k=1}^{m}(y^{q} + λ_{k}x^{p}), (p, q) = 1.

Let z ∈ C be a GNND point with a representative F_{z} and the Newton diagram Γ(F_{z}). Put b̃(C, Q) = p + q for a branch Q determined by an edge σ ⊂ Γ(F), where F_{z}^{σ}(x, y) = x^{α}y^{β} ∏_{k=1}^{m}(y^{q} + λ_{k}x^{p}), (p, q) = 1.

At last, put b(C, z) (resp. b̃(C, z)) be the sum of b(C, Q) (resp. b̃(C, Q)) over all branches centered at z.

Note that b(C, z) is equal to μ(C, z) − mod (C, z) − isod_{z}(N_{C/ℂP^{2}}^{es}, O_{C}) with μ being the Milnor number, mod the modality, isod_{z}(N_{C/ℂP^{2}}^{es}, O_{C}) the isomorphism defect defined in [11], sections 3, 4.

Definition 2 (see [14], section 2.1) Let z be an isolated singular point of a plane curve C, and Y ⊂ X_{z}^{ea} be a zero-dimensional scheme. Then we define

γ(C, Y) = max_{D} \frac{(deg(D ∩ Y) + min\{(D · C)_{z} − deg(D ∩ Y), deg(D ∩ Y)\})^{2}}{min\{(D · C)_{z} − deg(D ∩ Y), deg(D ∩ Y)\}} ,

where D runs over all curves through z which have no component in common with C.

According to [14], Lemma 2.2, γ(C, Y) ≤ (deg Y + 1)^{2}.

Lemma 3 In the notation of Theorem 1, let W be a reduced plane curve of degree d. Then (3) holds if one of the following conditions is satisfied:

(1) ∑_{z∈Sing(W)}(deg X_{z} − isod_{z}(J_{X/ℂP^{2}}, O_{W})) < 4(d − 1) , where isod_{z}(J_{X/ℂP^{2}}, O_{W′}), defined as in [11], section 3, always is a nonnegative integer, or

(2) W is irreducible and

∑_{z∈Sing(W)} γ(W, X_{z}) ≤ (d + 3)^{2} ,

where X_{z} is a component of the scheme X concentrated at z, or

(3) W is irreducible and

∑_{z∈S^{a}∪S_{1}}(τ(W, z) − isod_{z}(N_{W/ℂP^{2}}^{ea}, O_{W})) + ∑_{z∈S^{t}∪S_{2}∪S_{4}} b(W, z) + ∑_{z∈S_{3}∪S_{5}} b̃(W, z) < 3d ,

where isod_{z}(N_{W/ℂP^{2}}^{ea}, O_{W}), defined as in [11], section 3, always is a positive integer, and S^{a}, S^{t} are the parts of S, where the singular points are considered up to analytical or topological equivalence, respectively, or

(4) W splits into irreducible components W_{1}, ..., W_{r}, the sets S^{a}, S_{1} are empty, and

∑_{z∈S∪S_{2}∪S_{4}} ∑_{Q_{z}^{(i)}} b(W, Q_{z}^{(i)}) + ∑_{z∈S_{3}∪S_{5}} ∑_{Q_{z}^{(i)}} b̃(W, Q_{z}^{(i)}) < 3 · deg W_{i}, i = 1, ..., r . (21)

This follows from [14], Proposition 2.1, [11], Corollary 3.9, [26], Theorem 4.1.

Corollary 1 Let W ⊂ ℂP^{2} be an irreducible curve of degree d, and Sing(W) = S_{2} consist of ordinary singular points of multiplicities m_{1}, ..., m_{s}. Then (3) holds if

4 · ∑_{m_{i}=2} 1 + 18 · ∑_{m_{i}=3} 1 + \frac{16}{7} · ∑_{m_{i}>3} m_{i}^{2} ≤ (d + 3)^{2} ,

or

∑_{i=1}^{s} m_{i} < 3d .

This follows from Lemma 3 and the upper bound for γ(W, X_{z}^{sqh}) in [14], Corollary 2.5.

5 Criteria for the S-transversality of deformation models for GSQH and GNND singular points

Let, in the notation of section 2.5, (B^{2n}, Φ ∩ B^{2n}) be a model for a deformation of a GSQH singular point z ∈ W ⊂ V, defined by a polynomial P ∈ P(σ, F). Denote d = deg P. The homogeneous ideals generated by the monomials x_{0}^{d−i_{1}−...−i_{n}}x_{1}^{i_{1}}...x_{n}^{i_{n}}, l(i_{1}, ..., i_{n}) ≥ 0, or by the monomials x_{0}^{d−i_{1}−...−i_{n}}x_{1}^{i_{1}}...x_{n}^{i_{n}}, l(i_{1}, ..., i_{n}) > 0, define in ℂP^{n} subschemes Y and Ỹ, respectively. Put

Z = Y ∪ X(P), Z̃ = Ỹ ∪ X(P), X(P) = ⋃_{w∈Sing(P)\Y} X_{w}^{S} .

Proposition 4 The model (B^{2n}, Φ ∩ B^{2n}) is S-transversal if and only if h^{1}(ℂP^{n}, J_{Z/ℂP^{n}}(d)) = 0, and it is strongly S-transversal if and only if h^{1}(ℂP^{n}, J_{Z̃/ℂP^{n}}(d)) = 0.

Proof. Note that

H^{0}(ℂP^{n}, J_{Y/ℂP^{n}}(d)) = { ∑_{l(i_{1},...,i_{n})≥0} a_{i_{1}...i_{n}}x_{0}^{d−i_{1}−...−i_{n}}x_{1}^{i_{1}}...x_{n}^{i_{n}} } ,

H^{0}(ℂP^{n}, J_{Ỹ /ℂP^{n}}(d)) = { ∑_{l(i_{1},...,i_{n})>0} a_{i_{1}...i_{n}}x_{0}^{d−i_{1}−...−i_{n}}x_{1}^{i_{1}}...x_{n}^{i_{n}} } .

Hence the exact sequences

0 → H^{0}(ℂP^{n}, J_{Z/ℂP^{n}}(d))  → H^{0}(ℂP^{n}, J_{Y/ℂP^{n}}(d)) → H^{0}(X(P), O_{X(P)})

→ H^{1}(ℂP^{n}, J_{Z/ℂP^{n}}(d)) → H^{1}(ℂP^{n}, J_{Y/ℂP^{n}}(d)) = 0,

0 → H^{0}(ℂP^{n}, J_{Z̃/ℂP^{n}}(d))  → H^{0}(ℂP^{n}, J_{Ỹ /ℂP^{n}}(d)) → H^{0}(X(P), O_{X(P)})

→ H^{1}(ℂP^{n}, J_{Z̃/ℂP^{n}}(d)) → H^{1}(ℂP^{n}, J_{Ỹ /ℂP^{n}}(d)) = 0,

prove the required statement. The h^{1}-vanishing for the ideal sheaves of Y and Ỹ pointed out in the latter expressions, can easily be proven by induction on n and deg Y, deg Ỹ: using the fact that Y and Ỹ are contained in a d-multiple hyperplane {x_{0} = 0}, one subsequently divides the schemes Y, Ỹ by {x_{0} = 0} and applies the “Horace’s method” (see detail in [15, 16], or in section 6.2 below). □

In the notation of section 2.6, let (ℂK(Γ(F)), Φ) be a model of deformation for a GNND singular point z ∈ W ⊂ V, defined by a subdivision K(Γ(F)) = Δ_{1}∪...∪Δ_{N} and polynomials F_{1}, ..., F_{N} with Newton polyhedra Δ_{1}, ..., Δ_{N}, respectively. Given an oriented graph G ∈ G and a triple (Δ_{k}, Δ_{k,+}(G), F_{k}), 1 ≤ k ≤ N, assume that F_{k} is divided by maximal possible powers of x_{1}, ..., x_{n} and Δ_{k} is shifted in ℝ_{+}^{n} correspondingly, and introduce the scheme Y_{k} ⊂ ℂP^{n} defined by the homogeneous ideal, whose generators are • for n ≥ 3, the monomials x_{0}^{d_{k}−i_{1}−...−i_{n}}x_{1}^{i_{1}}...x_{n}^{i_{n}}, (i_{1}, ..., i_{n}) ∈ Δ_{k}\Δ_{k,+}(G), where

d_{k} = deg F_{k};

• for n = 2, the monomials x_{0}^{d_{k}−i_{1}−i_{2}}x_{1}^{i_{1}}x_{2}^{i_{2}}, (i_{1}, i_{2}) ∈ Δ_{k}\Δ_{k,+}(G), and the truncations F_{k}^{σ}, where σ runs over all connected components of Δ_{k,+}(G).

Put

Z_{k} = Y_{k} ∪ ⋃_{w∈Sing(F_{k})∩(ℂ^{∗})^{n}} X_{w}^{S} .

Proposition 5 The triple (Δ_{k}, Δ_{k,+}(G), F_{k}) is S-transversal if and only if h^{1}(ℂP^{n}, J_{Z_{k}/ℂP^{n}}(d_{k})) = 0.

Proof basically coincides with that of Proposition 4 with the following remark: the vanishing of H^{1}(ℂP^{n}, J_{Y_{k}/ℂP^{n}}(d_{k})) is derived again by means of Horace’s method from the fact that Y_{k} is contained in the union of multiple coordinate hyperplanes with the total degree ≤ d_{k}. □

Below we present few numerical sufficient criteria for the S-transversality of deformation models.

Lemma 6 A nonsingular deformation model of a GSQH or GNND singular point is strongly S-transversal.

Proof. Straightforward from Propositions 4, 5 and their proofs. □

Lemma 7 In the above notation, let z ∈ W be an ordinary singular point of multiplicity d. Then a deformation model given by a polynomial P (of degree d) is S-transversal if deg X(P) < 4(d − 1), and is strongly S-transversal if deg X(P) < 3(d − 1).

Proof. Both the statements can be proven as Theorems 1, 2 [27] (see also [9]). □

In the case n = 2 (planar curve singularities) we can say more.

If F is a polynomial in two variables with Newton polygon Δ such that the truncation F^{σ}, for any edge σ ⊂ ∂Δ, has no singular points in (ℂ^{∗})^{2}, then each edge σ determines (up to topological equivalence) certain local branches of the projective closure of the curve F (x, y) = 0, centered on the lines x_{0} = 0, x_{1} = 0, x_{2} = 0. The number of these branches is equal to the number of the minimal segments on σ with integral endpoints. If G is a component of F then ρ(σ, G) is the number of the local branches of G determined by the edge σ.

Lemma 8 In the notation of section 2.6, let n = 2. The triple (Δ_{k}, Δ_{k,+}(G), F_{k} is transversal with respect to the topological equivalence of singular points if • the curve F_{k} = 0 is irreducible in (ℂ^{∗})^{2} and

∑_{z∈Sing(F_{k})∩(ℂ^{∗})^{2}} b(F_{k}, z) < ∑_{σ⊄Δ_{k,+}(G)^{σ⊂∂Δ}} ρ(σ, F_{k}) ,

• the curve {F_{k} = 0} splits into components C_{1}, ...,C_{s}, and

∑_{Q_{i}} b(F_{k}, Q_{i}) < ∑_{σ⊄Δ_{k,+}(G)^{σ⊂∂Δ}} ρ(σ, C_{i}), i = 1, ...,s,

where Q_{i} runs over all local branches of C_{i} centered at Sing(F_{k}) ∩ (ℂ^{∗})^{2}.

The triple (Δ_{k}, Δ_{k,+}(G), F_{k} is transversal with respect to the analytic equivalence of singular points if

• the curve {F_{k} = 0} ∩ (ℂ^{∗})^{2} is irreducible and

∑_{w}(τ(F_{k}, w) − 1) < ∑_{σ⊄Δ_{k,+}(G)^{σ⊂∂Δ}} ρ(σ, F_{k}) ,

where w runs through all singular points of F_{k} = 0 in (ℂ^{∗})^{2},

• the curve {F_{k} = 0} ∩ (ℂ^{∗})^{2} splits into components C_{1}, ...,C_{s} and

∑_{w}(τ(C_{i}, w) − 1 + (C_{i} · Ĉ_{i})_{w}) < ∑_{σ⊄Δ_{k,+}(G)^{σ⊂∂Δ}} ρ(σ, C_{i}), i = 1, ...,s,

where w runs through all singular points of F_{k} = 0 in (ℂ^{∗})^{2}, and (C_{i} · Ĉ_{i})_{w} is the intersection number at w of C_{i} with Ĉ_{i}, the union of all components of {F_{k} = 0} ∩ (ℂ^{∗})^{2} but C_{i}.

This is a reformulation of Theorem 4.1 [26].

Corollary 2 If n = 2 and a deformation model of a GSQH or GNND singular point has nodes as its only singularities, then it is strongly S-transversal.

This is a reformulation of Theorem 4.2 [26].

6 Existence of hypersurfaces with prescribed isolated singularities

6.1 Main construction

Given a linear system |W | on a smooth algebraic variety V of dimension n ≥ 2 and a classification S of isolated hypersurface singular points, we give a sufficient condition for the existence of an element W′ ∈ |W | with a given (up to S-equivalence) collection of isolated singularities.

Definition 3 For a type s ∈ S define e(s) to be the minimal degree of a polynomial F_{s}(x) such that

• the leading e(s)-form of F has no critical points in ℂ^{n}\{0},

• Sing(F_{s}) consists of one point z, and this singular point is of type s,

• the germ at F_{s} of the family of polynomials of degree e(s) with a singular point of type s in a neighborhood of z is smooth and has an expected dimension (i.e. h^{1}(ℂP^{n}, J_{X_{z}^{S}/ℂP^{n}}(e(s))) = 0).

Lemma 9 Given singularity types s_{1}, ..., s_{r} ∈ S and a hypersurface W′ ∈ |W | having r ordinary singular points of multiplicities e(s_{1}) + 1, ..., e(s_{r}) + 1, respectively, as its only singularities and satisfying

h^{1}(V, J_{X(W′)/V} ⊗ O_{V} (W)) = 0 ,  (22)

where X(W′) = ⋃_{z∈Sing(W′)} X_{z}^{sqh}, then there exists W′′ ∈ |W | having r singular points of types s_{1}, ..., s_{r}, respectively, as its only singularities.

Proof. As we saw in the proof of Theorem 1, condition (22) means that, given coordinates in neighborhoods of the points Sing(W′) = {z_{1}, ..., z_{r}}, the jets jet_{e(s_{i})+1}(W′_{i}) (where W′_{i} is the image of W′ in O_{V,z_{i}}), are homogeneous forms of degrees e(s_{i}) + 1, i = 1, ..., r, and, in addition, any variations of the jets jet_{e(s_{i})}(W′_{i}), i = 1, ..., r, are simultaneously realizable, when varying W′ in |W |. So, first, we deform W′ into W̃′ ∈ |W | such that jet_{e(s_{i})}(W̃′_{i}) is the leading form of the polynomial F_{s_{i}}(x) (which can be assumed nondegenerate without loss of generality), i = 1, ..., r. Then we apply Theorem 1, since X in (22) is ⋃_{z} X̃_{z}^{sqh} for W̃′, and the deformation models F_{s_{i}}, i = 1, ..., r, are S-transversal. □

6.2 Multidimensional generalization of a theorem of Hirschowitz and projective hypersurfaces with prescribed isolated singularities

Let X be the zero-dimensional subscheme of ℂP^{n}, n ≥ 1, concentrated at distinct points z_{1}, ..., z_{r} ∈ ℂP^{n} and defined at z_{i} by the power 𝔪_{i}^{m_{i}} of the maximal ideal 𝔪_{i} ⊂ O_{ℂP^{n},z_{i}}, i = 1, ..., r. We call X the scheme of fat points z_{1}, ..., z_{r} with multiplicities m_{1}, ..., m_{r}, respectively. If r = 1, or n = 1 then the condition m_{1} + ..., m_{r} ≤ d + 1 is necessary and sufficient for

h^{1}(ℂP^{n}, J_{X}(d)) = 0 .  (23)

We shall give a sufficient condition for (23) for r ≥ 2, n ≥ 2. Given k ≥ 1, put

M(n, 2k) = 2 · (\begin{matrix}k + n \\ n \end{matrix}) , M(n, 2k − 1) = (\begin{matrix}k + n \\ n \end{matrix}) + (\begin{matrix}k + n − 1 \\ n \end{matrix}) .

Proposition 10 If z_{1}, ..., z_{r} are generic points in ℂP^{n} and

∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) < M(n, d)  (24)

then (23) holds true.

Remark 6 (1) For n = 2 Proposition 10 coincides with Hirschowitz’s theorem [16].

(2) Estimate (24) is sharp for r ≥ 2 in the following sense. Equality in (24) can be realized by the scheme of two points z_{1}, z_{2} of multiplicities m_{1} = m_{2} = k + 1 if d = 2k, or m_{1} = k + 1, m_{2} = k if d = 2k − 1. In both the cases m_{1} + m_{2} = d + 2 which implies h^{1}(ℂP^{n}, J_{X}(d)) > 0, what one gets restricting the sheaf J_{X}(d) on the straight line through z_{1}, z_{2}.

Proposition 11 Under the condition of Proposition 10, there exists a hypersurface of degree d in ℂP^{n} which has ordinary singularities of multiplicities m_{1}, ..., m_{r} at z_{1}, ..., z_{r}, respectively, and no singular points in ℂP^{n}\{z_{1}, ..., z_{r}}.

Corollary 3 Given singularity types s_{1}, ..., s_{r} ∈ S, for any integer d satisfying

∑_{i=1}^{r} (\begin{matrix}e(s_{i}) + n \\ n \end{matrix}) < M(n, d) ,

there exists a hypersurface of degree d in ℂP^{n} which has r singular point of types s_{1}, ..., s_{r}, respectively, as its only singularities. Such a hypersurface can be chosen irreducible if n ≥ 3, or

n = 2, ∑_{i=1}^{r} (\begin{matrix}e(s_{i}) + 2 \\ 2 \end{matrix}) < \frac{d^{2} + 6d − 1}{4} − [\frac{d}{2}] .

Straightforward from Lemma 9, Proposition 11, and [12], Lemma 5.7 (irreducibility condition for n = 2).

6.3 Proof of Proposition 10

Due to the semicontinuity of h^{1}, the statement of Proposition 10 follows from the next claim: given nonnegative r′ ≤ r such that z_{1}, ..., z_{r′} are generic points in the hyperplane ℂP^{n−1} ⊂ ℂP^{n}, z_{r′+1}, ..., z_{r} are generic points in ℂP^{n}\ℂP^{n−1}, and m_{1}, ..., m_{r} satisfy (24) and

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < M(n − 1, d) ,  (25)

then (23) holds true.

We prove the latter claim by the double induction on d and n. For n ≤ 2 it is proven in [16]. For d ≤ 1 it is trivial. So, assume that n ≥ 3, d ≥ 2.

Step 1. First, we consider the case r′ = r, where (25) turns into

∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < M(n − 1, d) .  (26)

Note that this particular claim implies Proposition 10 as well.

We have the following exact sequence of sheaves

0 → J_{Y} (d − 1) → J_{X}(d) → J_{Z}(d) → 0

which implies the exact sequence

H^{1}(ℂP^{n}, J_{Y} (d − 1)) → H^{1}(ℂP^{n}, J_{X}(d)) → H^{1}(ℂP^{n−1}, J_{Z}(d)) ,  (27)

where Y = X : ℂP^{n−1} is the scheme of fat points z_{1}, ..., z_{r} in ℂP^{n} of multiplicities m_{1} − 1, ..., m_{r} − 1, respectively, Z = X ∩ ℂP^{n−1} is the scheme of fat points z_{1}, ..., z_{r} in ℂP^{n−1} of multiplicities m_{1}, ..., m_{r}, respectively. By the induction assumption and (26), h^{1}(ℂP^{n−1}, J_{Z}(d)) = 0. Hence, to show (23), one needs h^{1}(ℂP^{n}, J_{Y} (d−1)) = 0. The latter relation follows from the induction assumption as well, provided

∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 2 \\ n \end{matrix}) < M(n, d − 1) , ∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 3 \\ n − 1 \end{matrix}) < M(n − 1, d − 1) . (28)

Indeed,

(\begin{matrix}m_{i} + n − 2 \\ n \end{matrix}) = (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) · \frac{m_{i} − 1}{m_{i} + n − 1} ≤ (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) · \frac{d − 1}{d + n − 1} .

We show that M(n, d − 1)/M(n, d) ≥ (d − 1)/(d + n − 1): if d = 2k, k ≥ 1, then

\frac{M(n, d − 1)}{M(n, d)} ≥ \frac{2k − 1}{2k + n − 1} ⟺

(2k + n − 1) · ((\begin{matrix}k + n \\ n \end{matrix}) + (\begin{matrix}k + n − 1 \\ n \end{matrix})) ≥ 2(2k − 1) · (\begin{matrix}k + n \\ n \end{matrix}) ⟺

(2k + n − 1) · (\begin{matrix}k + n − 1 \\ n \end{matrix}) ≥ (2k − n − 1) · (\begin{matrix}k + n \\ n \end{matrix}) ⟺

(2k + n − 1) ≥ (2k − n − 1)\frac{k + n}{k} ⟺ n^{2} + n ≥ 0 ,

which is true, and if d = 2k + 1, k ≥ 1, then

\frac{M(n, d − 1)}{M(n, d)} ≥ \frac{2k − 1}{2k + n − 1} ⟺

(2k + n) · (\begin{matrix}k + n \\ n \end{matrix}) ≥ 2k · (\begin{matrix}k + n + 1 \\ n \end{matrix}) ⟺

(2k + 2n) ≥ 2k · \frac{k + 1 + n}{k + 1} ⟺ n ≥ 0,

which is true. Similarly one derives the second inequality in (28).

Step 2. Assume that r′ < r, and

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) ≥ M(n, d) − M(n, d − 1).  (29)

Once again we have the exact sequence (27) with Y ⊂ ℂP^{n} being the scheme of fat points z_{1}, ..., z_{r} of multiplicities m_{1} − 1, ..., m_{r′} − 1, m_{r′+1}, ..., m_{r}, respectively, and Z ⊂ ℂP^{n−1} being the scheme of fat points z_{1}, ..., z_{r′} of multiplicities m_{1}, ..., m_{r′}, respectively. By (25) and the induction assumption, h^{1}(ℂP^{n−1}, J_{Z}(d)) = 0. Since by (29)

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n \end{matrix}) + ∑_{i=r′+1}^{r} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) = ∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) − ∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix})

< M(n, d) − (M(n, d) − M(n, d − 1)) = M(n, d − 1) ,

and by (25)

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 3 \\ n − 1 \end{matrix}) ≤ M(n − 1, d − 1) ,

which one derives as the second inequality in (28), the induction assumption for the scheme Y and degree d − 1 implies h^{1}(ℂP^{n}, J_{Y} (d − 1)) = 0, that together with (27) completes the proof of (23).

Step 3. Assume that r′ < r, d = 2k, k ≥ 1, and

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < M(n, d) − M(n, d − 1) = (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) .  (30)

If one of m_{r′+1}, ..., m_{r}, say m_{r′+1}, does not exceed k + 1, then one can specialize the point z_{r′+1} into a generic point in ℂP^{n−1}\{z_{1}, ..., z_{r′}}. Condition (25) for z_{1}, ..., z_{r′+1} still holds:

∑_{i=1}^{r′+1} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) + (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) = M(n − 1, d) ,

hence one arrives either to the situations of Step 1 or 2, or in the beginning of Step 3.

If one of m_{r′+1}, ..., m_{r}, say m_{r}, exceeds k + 2, then one has

(\begin{matrix}m_{r} + n − 2 \\ n − 1 \end{matrix}) ≥ (\begin{matrix}k + n \\ n − 1 \end{matrix}) > (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) = M(n, d) − M(n, d − 1) .  (31)

Let L ⊂ ℂP^{n} be a generic hyperplane through z_{r}. The one has the exact sequence

H^{1}(ℂP^{n}, J_{Y} (d − 1)) → H^{1}(ℂP^{n}, J_{X}(d)) → H^{1}(L, J_{Z}(d)) ,  (32)

where Y = X : L ⊂ ℂP^{n} is the scheme of fat points z_{1}, ..., z_{r} of multiplicities m_{1}, ..., m_{r−1}, m_{r} − 1, respectively, Z = X ∩ L ⊂ L is the scheme of the fat point z_{r} of multiplicity m_{r}. Since m_{r} ≤ d, we have h^{1}(ℂP^{n−1}, J_{Z}(d)) = 0. Since by (31)

∑_{i=1}^{r−1} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) + (\begin{matrix}m_{r} + n − 2 \\ n \end{matrix}) = ∑_{i=1}^{r} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) − (\begin{matrix}m_{r} + n − 2 \\ n − 1 \end{matrix})

< M(n, d) − (M(n, d) − M(n, d − 1)) = M(n, d − 1) ,

and by (30)

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) < (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix})+(\begin{matrix}k + n − 2 \\ n − 1 \end{matrix}) = M(n−1, d−1) ,

by the induction assumption h^{1}(ℂP^{n}, J_{Y} (d − 1)) = 0, that together with (32) completes the proof of (23).

Step 4. Assume that r′ < r, d = 2k + 1, k ≥ 1, and

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < M(n, d) − M(n, d − 1) = (\begin{matrix}k + n \\ n − 1 \end{matrix}) .  (33)

If, for some j = r′ + 1, ..., r,

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) + (\begin{matrix}m_{j} + n − 2 \\ n − 1 \end{matrix}) < M(n − 1, d) ,

then one specializes the point z_{j} on ℂP^{n−1} and comes to Steps 1,2, or 3, or in the beginning of Step 4.

Assume that for any j = r′ + 1, ..., r

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) + (\begin{matrix}m_{j} + n − 2 \\ n − 1 \end{matrix}) ≥ M(n − 1, d) .  (34)

By (33) this means that

(\begin{matrix}m_{j} + n − 2 \\ n − 1 \end{matrix}) > M(n − 1, d) − (\begin{matrix}k + n \\ n − 1 \end{matrix}) = (\begin{matrix}k + n + 1 \\ n − 1 \end{matrix}) ,  (35)

hence m_{j} ≥ k + 3. The latter inequality and (25) imply that

(\begin{matrix}k + n + 2 \\ n \end{matrix}) < M(n, d) = (\begin{matrix}k + n + 1 \\ n \end{matrix}) + (\begin{matrix}k + n \\ n \end{matrix})

⟺ n(n + k + 1) < (k + 2)(k + 1) ⟹ n ≤ k .  (36)

Take a generic hyperplane L ⊂ ℂP^{n} through z_{r} and consider the exact sequence (32). As in Step 3, h^{1}(ℂP^{n−1}, J_{Z}(d)) = 0. By (35),

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) + (\begin{matrix}m_{r} + n − 2 \\ n \end{matrix}) = ∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 1 \\ n \end{matrix}) + (\begin{matrix}m_{r} + n − 2 \\ n \end{matrix})

− (\begin{matrix}m_{r} + n − 2 \\ n − 1 \end{matrix}) < M(n, d) − (\begin{matrix}k + n + 1 \\ n − 1 \end{matrix}) < M(n, d) − (\begin{matrix}k + n \\ n − 1 \end{matrix}) = M(n, d − 1) .

In addition, by (33)

∑_{i=1}^{r′} (\begin{matrix}m_{i} + n − 2 \\ n − 1 \end{matrix}) < (\begin{matrix}k + n \\ n − 1 \end{matrix}) ≤ M(n − 1, d − 1) = 2 · (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) .

The latter inequality follows from

(\begin{matrix}k + n \\ n − 1 \end{matrix}) ≤ 2 · (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) ⟺ (\begin{matrix}k + n − 1 \\ n − 2 \end{matrix}) ≤ (\begin{matrix}k + n − 1 \\ n − 1 \end{matrix}) ,

which is true, because n ≤ k in (36). Hence by the induction assumption for the scheme Y and degree d − 1, h^{1}(ℂP^{n}, J_{Y} (d − 1)) = 0, that together with (32) completes the proof of (23).

6.4 Proof of Proposition 11

The required statement follows from the next one: given nonnegative r′ ≤ r such that z_{1}, ..., z_{r′} are generic points in the hyperplane ℂP^{n−1} ⊂ ℂP^{n}, z_{r′+1}, ..., z_{r} are generic points in ℂP^{n}\ℂP^{n−1}, and m_{1}, ..., m_{r} satisfy (24) and (25), then there exists F ∈ H^{0}(ℂP^{n}, J_{X}(d))\{0} such that the hypersurface {F = 0} ⊂ ℂP^{n} has ordinary singularities of multiplicities m_{1}, ..., m_{r} at z_{1}, ..., z_{r}, respectively, and no singular points in ℂP^{n}\{z_{1}, ..., z_{r}}.

Indeed, given a family of r-tuples z_{1}(t), ..., z_{r}(t) ∈ ℂP^{n}, t ∈ (ℂ, 0), such that, for t ≠ 0, z_{1}(t), ..., z_{r}(t) are generic points in ℂP^{n}, and, for t = 0, z_{1}(0), ..., z_{r′}(0) are generic points in ℂP^{n−1}, z_{r′+1}(0), ..., z_{r}(0) are generic points in ℂP^{n}\ℙ^{n−1}, by Proposition 10 h^{0}(ℂP^{n}, J_{X(t)}(d)) = const > 0, t ∈ (ℂ, 0). Hence any element F_{0} ∈ H^{0}(ℂP^{n}, J_{X(0)}(d)) can be included in a family F_{t} ∈ H^{0}(ℂP^{n}, J_{X(t)}(d)), t ∈ (ℂ, 0). Hence F_{t} possesses the required properties for almost all t if F_{0} does.

Then one proves the claim stated above along the same plan as in the proof of Proposition 10. In all the cases we have h^{1}(ℂP^{n}, J_{Y} (d − 1)) = 0, hence the exact sequence

0 → H^{0}(ℂP^{n}, J_{Y} (d − 1)) → H^{0}(ℂP^{n}, J_{X}(d)) → H^{0}(L, J_{Z}(d)) → 0,

where L stands for ℂP^{n−1}, or a generic hyperplane through z_{r}, and Y, Z have the corresponding meaning. So, we have the canonical decomposition

H^{0}(ℂP^{n}, J_{X}(d)) = L · H^{0}(ℂP^{n}, J_{Y} (d − 1)) ⊕ H^{0}(L, J_{Z}(d)) .

The induction step is as follows: if F_{1} ∈ H^{0}(ℂP^{n}, J_{Y} (d − 1)) defines a hypersurface with ordinary singularities at z_{1}, ..., z_{r} of multiplicities m_{1} − 1, ..., m_{r′} − 1, m_{r′+1}, ..., m_{r}, respectively, and without singularities in ℂP^{n}\{z_{1}, ..., z_{r}}, and F_{2} ∈ H^{0}(L, J_{Z}(d)) defines a hypersurface in L with ordinary singularities at z_{1}, ..., z_{r′} of multiplicities m_{1}, ..., m_{r′}, respectively, and without singularities in L\{z_{1}, ..., z_{r′}}, then, for a generic λ, μ ∈ ℂ, F = λF_{1}L + μF_{2} ∈ H^{0}(ℂP^{n}, J_{X}(d)) defines in ℂP^{n} a hypersurface having the required properties.

7 Deformation of plane real algebraic curves

Nonsingular deformations (smoothings) of real plane singular curves are an essential source of isotopy types of nonsingular plane curves [30, 31, 32]. Original Viro’s method allows simultaneous lower nonsingular deformations of at most three Newton nondegenerate singular points, which should be in a regular position with respect to the projective coordinate axes. Few known constructions which cannot be done by the standard Viro method appeared to be particular cases of the generalization of Viro’s method presented here.

Example 1 For the first time, a non-standard Viro method was used in the construction of a curve of degree 8 of type < 4 + 3 < 5 >> [23]. This construction is based on Lemma [23] which a particular case of Lemma 3(3): the restrictions to singular point invariants are the same, whereas the statement concerns only nonsingular deformations.

Example 2 Two other applications of a non-standard Viro method were proposed in [6, 19]. In particular, Chevallier [6] constructed nonsingular curves of degree 8 with 22 ovals of new isotopy types: < 16+1 < 2+1 < 2 >>, 13+1 < 2+1 < 5 >>>, 2 + 1 < 2 + 1 < 16 >>>, < 5 + 1 < 2 + 1 < 13 >>>. The basic step in these construction is a deformation of a singular point of the union of three or four conics which have one common point, where they intersect with multiplicity 4. Note that such a singular point cannot be semiquasihomogeneous for a curve of degree 6 or 8 (but is topologically equivalent to y^{3} + x^{12} or y^{4} + x^{16}, hence is GSQH). We shall show that this is a very special case covered by Lemma 3(4):

Lemma 12 Let real conics C_{1}, ..., C_{r} have one common point z and (C_{i} · C_{j})_{z} = 4, 1 ≤ i < j ≤ r. Then any nonsingular lower deformation of the GSQH singular point z ∈ C_{1}...C_{r} can be realized when varying C_{1}...C_{r} in the space of curves of degree 2r.

Proof. By Lemma 6 a nonsingular lower deformation is transversal. In the notation of Lemma 3, S = S_{1} = S_{3} = S_{4} = S_{5} = ∅, S_{2} = {z}, C_{i} has only one local branch Q_{i} centered at z, and b(W, Q_{i}) = 4. Hence inequality (21) holds for any i = 1, ..., r:

b(W, Q_{i}) = 4 < 6 = 3 · deg W_{i} □

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Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, 69978 Tel Aviv, Israel. E-mail: shustin@math.tau.ac.il

and

Max-Planck-Institut für Mathematik, 7 Vivatsgasse, 53111 Bonn, Germany. Email: shustin@mpim-bonn.mpg.de