On the Monodromy at Infinity of a Polynomial Map, II

R. García López^{1} and A. Némethi^{2}

Universidad de Barcelona, Dept. de Algebra y Geometría. Gran Via, 585, 08007 Barcelona, Spain.

The Ohio State University, 231 West 18th Avenue, Columbus OH 43210, USA.

§1. Introduction

In the last years a lot of work has been concentrated on the study of the behaviour at infinity of polynomial maps (see for example [8], [11], [9], [10], [5], among others). This behaviour can be very complicated, therefore the main idea was to find special classes of polynomial maps which have, in some sense, nice properties at infinity. In this paper, we completely determine the complex algebraic monodromy at infinity for a special class of polynomial maps (which is complicated enough to show the nature of the general problem).

Next, we give the precise definitions: Let f : C^{n+1} → C be a map given by a polynomial with complex coefficients (which will be also denoted by f). Then there exists a finite set Γ ⊂ C such that the map

f |_{C^{n+1}−f^{−1}(Γ)}: C^{n+1} − f^{−1}(Γ) → C − Γ

is a locally trivial C^{∞}-fibration ([12]). We denote by Γ_{f} the smallest subset of the complex plane with this property. Γ_{f} contains the set Σ_{f} of critical values of f, but in general it is bigger. Fix t_{0} ∈ C such that |t_{0}| > max{|t| : t ∈ Γ_{f}}. The complex algebraic monodromy associated with the path s ↦ t_{0}e^{2πis}, s ∈ [0, 1], is denoted by

(T_{f}^{∞})^{∗} : H^{∗}(f^{−1}(t_{0}), C) → H^{∗}(f^{−1}(t_{0}, C).

This isomorphism is called the monodromy at infinity of f. As we will see later, (T_{f}^{∞})^{∗} is a very delicate invariant of f.

1Supported by the DGICYT

2Partially supported by OSU Seed Grant

On studying topological properties of polynomial maps, one usually imposes some condition which insures the absence of vanishing cycles “at infinity” for a suitable compactification of the map f (tameness, Malgrange condition,... cf. [11]). From this point of view, a class of polynomial maps which looks natural to study is the following:

Definition: A polynomial f ∈ C[X_{1}, …, X_{n+1}] will be called a (∗)-polynomial if it verifies the following condition:

(∗) \{ \begin{matrix}For t ∈ C − Σ_{f}, the closure in P^{n+1} of the affine \\ hypersurface \{f = t\} is non-singular. \end{matrix}

The goal of this article is the computation of (T_{f}^{∞})^{∗} for (∗)-polynomials. We will assume that n ≥ 2. The case n = 1 is completely clarified in [5].

If d =deg(f) and f = f_{d} + f_{d−1} + ... is the decomposition of f into homogeneous components, condition (∗) is equivalent to

{x ∈ C^{n+1} | gradf_{d}(x) = 0, f_{d−1}(x) = 0} = {0},

where grad denotes the gradient vector. In the first part [5] of this sequence of papers, the following results are given (besides others):

a) A (∗)-polynomial f satisfies Γ_{f} = Σ_{f} and any fiber of f has the homotopy type of a bouquet of n-dimensional spheres (cf. [4]). In particular, the only interesting monodromy transformation is (T_{f}^{∞})^{n}, which in the sequel will be denoted simply by T_{f}^{∞}.

b) The hypersurface X^{∞} ⊂ P^{n} given by f_{d} = 0 has only isolated singularities, and the monodromy at infinity (actually, the whole topology at infinity) depends only on the hypersurface X^{∞}.

c) The characteristic polynomial of T_{f}^{∞} is computable in terms of the characteristic polynomials of the local monodromies of the isolated singularities of X^{∞} (cf. Corollary 2).

d) On the other hand, the nilpotent part of X^{∞} cannot be determined only from local data attached to the isolated singularities of X^{∞}, it depends essentially on the position of these singular points.

Part of the global information about the position of the singular points of X^{∞} is already encoded in its Betti numbers. More subtle invariants are hidden in the complement P^{n} − X^{∞} of X^{∞}, or in the cyclic coverings of P^{n} branched along X^{∞}. For algebraic surfaces, O. Zariski related this kind of invariants with the defect (or superabundance) of some linear systems, respectively with some Betti numbers of cyclic coverings. In the sequel we give the numerical invariants of X^{∞} which will provide our description of T_{f}^{∞}.

For X a quasi-projective variety, denote by b_{q}(X) (respectively, p_{q}(X)) the dimension of H^{q}(X, C) (respectively, the dimension of the q-th primitive cohomology of X). The numbers p_{n}(X^{∞}) = b_{n−1}(P^{n}−X^{∞}) and p_{n−1}(X^{∞}) = b_{n}(P^{n} −X^{∞}) are in general global invariants of X^{∞} (Here, if n = 2, we define p_{2}(X^{∞}) = b_{2}(X^{∞}) − 1)). We define a map h : π_{1}(P^{n} − X^{∞}) → Z/dZ as follows: If n > 2 then h is just the Hurewicz map (in fact, isomorphism):

π_{1}(P^{n} − X^{∞}) → H_{1}(P^{n} − X^{∞}, Z) = Z/dZ.

If n = 2, let r denote the number of irreducible components of X^{∞}, of degrees d_{1}, …, d_{r}. Then h is defined as the composition

π_{1}(P^{n} − X^{∞}) → H_{1}(P^{n} − X^{∞}, Z) = \frac{Z^{r}}{(d_{1}, …, d_{r})} →^{α} Z/dZ,

where α is defined by α[(a_{1}, …, a_{r})] = \overline{∑ a_{i}}. The composition of h with the characters ρ_{s} : Z/dZ → C^{∗} defined by ρ_{s}(1) = e^{2πis/d} (for 1 ≤ s ≤ d − 1) provide one dimensional flat bundles V_{s} over P^{n+1} − X^{∞} with monodromy representation ρ_{s} ∘ h.

Let j : P^{n} −X^{∞} ↪ P^{n} denote the inclusion map. It is not difficult to see that the direct image sheaf j_{∗}V_{s} (= R^{0}j_{∗}V_{s}) coincides with the extension by zero j_{!}V_{s}, s = 1, … d − 1. We define the “equivariant defect” by:

β_{s} = \{ \begin{matrix}p_{n}(X^{∞}) = b_{n−1}(P^{n} − X^{∞}) if s = 0; \\ b_{n+1}(P^{n}, j_{∗}V_{s}) if s = 1, …, d − 1. \end{matrix}

Here, b_{n+1}(P^{n}, j_{∗}V_{s}) is the dimension of the sheaf cohomology H^{n+1}(P^{n}, j_{∗}V_{s}). This vector space is the e^{2πis/d}- eigenspace of H^{n+1}(X′_{0}), where X′_{0} is the d-th cyclic covering of P^{n+1} branched along X^{∞} and the action is induced by the natural Galois action (cf. (2.12), see also §2, VIII).

Set Sing(X^{∞}) = {p_{1}, ..., p_{k}}, and let F_{i}, μ_{i}, T_{i} be respectively the local Milnor fiber, the Milnor number and the local algebraic monodromy H^{n−1}(F_{i}, C) → H^{n−1}(F_{i}, C) of the isolated hypersurface singularity (X^{∞}, p_{i}). We will call an invariant local if it can be expressed in terms of the local operators {T_{i}}_{i=1}^{k} and the numbers n and d. We define the following local numerical invariants:

χ_{0} = −Σ_{i=1}^{k}μ_{i} + \frac{(−1)^{n} + (d − 1)^{n+1}}{d} + (−1)^{n+1}
χ_{s} = χ_{0} + (−1)^{n} for s = 1, …, d − 1.

Now we are ready to formulate our main result. If T is an operator, let T_{α} denote its restriction to its generalized α-eigenspace and let #_{l}T_{α} be the number of Jordan blocks of T_{α} of size l. Set #T_{α} = Σ_{l≥1}#_{l}T_{α}. With this notations one has:

Main Theorem:

I. If α = e^{2πis/d}, s = 0, …, d − 1, then:

a) #_{1}(T_{f}^{∞})_{α} = χ_{s} + 2β_{s} − Σ_{i=1}^{k}#(T_{i})_{α}.

b) #_{2}(T_{f}^{∞})_{α} = −β_{s} + Σ_{i=1}^{k}#_{1}(T_{i})_{α}.

c) #_{l+1}(T_{f}^{∞})_{α} = Σ_{i=1}^{k}#_{l}(T_{i})_{α} for l ≥ 2.

II. If α^{d} ≠ 1, then (T_{f}^{∞})_{α} = ⊕_{i=1}^{k}α^{d} · (T_{i})_{α^{1−d}}, i.e. #_{l}(T_{f}^{∞})_{α} = Σ_{i=1}^{k}#_{l}(T_{i})_{α^{1−d}} for all l ≥ 1.

Corollary 1:

a) #_{l}(T_{f}^{∞})_{α} = 0 for l ≥ n + 2.

b) #_{n+1}(T_{f}^{∞})_{1} = 0 if α^{d} ≠ 1 or α = 1.

Corollary 2: [5, (3.3)]The characteristic polynomial of T_{f}^{∞} is given by the following local formula:

det(λ·Id−T_{f}^{∞}) = (λ−1)^{(−1)^{n+1}} ·(λ^{d} −1)^{\frac{(d−1)^{n+1}+(−1)^{n}}{d}} · ∏_{i=1}^{k} \frac{det(λ^{d−1} · Id − T_{i})}{(λ^{d} − 1)^{μ_{i}}}.

Another byproduct of the main theorem is the following:

Corollary 3: If α = e^{2πis/d}, s = 0, …, d − 1, then:

\frac{∑_{i=1}^{k} #(T_{i})_{α} − χ_{s}}{2} ≤ β_{s} ≤ ∑_{i=1}^{k} #_{1}(T_{i})_{α}.

If ∑_{i=1}^{k} #(T_{i})_{α} ≤ χ_{s}, then the lower bound given by Corollary 3 is useless, but in some cases it gives even the right value of β_{0}. For example, if n = 2 and l_{1}, …, l_{d} ∈ C[X, Y, Z] are linear forms defining an arrangement of lines in P^{2} such that no two of them meet at a point, then β_{0} = d − 1, and this is exactly the bound given by corollary 3 above. From the discussion in [3, p.161] follows that if (X^{∞}, p_{i}) are all non-degenerate singularities (i.e., Σ_{i=1}^{k}#(T_{i})_{1} = 0), then the defect β_{0} = 0. Also, in Zariski’s book [19] we can find similar criteria for n = 2. Notice that our bound gives a sharper criterion: If ∑_{i=1}^{k} #_{1}(T_{i})_{1} = 0 then β_{0} = 0 (cf. also Remark 2.30 below).

For another corollary of the main theorem, see (2.16).

§2. Proof of the main theorem

I. The main construction and two exact sequences

Let f : C^{n+1} → C be a polynomial map which satisfies the condition (∗). By [5, (2.6)], we can assume that f is of the form f_{d} + x_{n+1}^{d−1}, where f_{d} is homogeneous of degree d (and no singularity of X^{∞} is on the hyperplane x_{n+1} = 0). Set (cf. [5, §5]):

X = {([x], t) ∈ P^{n+1} × D : t(f_{d}(x_{1}, …, x_{n+1}) + x_{0}x_{n+1}^{d−1}) = x_{0}^{d}},

where D denotes a disk of sufficiently small radius in the complex plane with center at the origin. Then the map π : X → D given by π([x], t) = t induces a locally trivial C^{∞}-fibration over D − {0} with projective fibers, these are exactly the projective closures of the fibers of f. Moreover, if we denote by T the algebraic monodromy H^{n}(\overline{f^{−1}(t_{0})}) → H^{n}(\overline{f^{−1}(t_{0})}) of the projective closure \overline{f^{−1}(t_{0})} associated with the path s ↦ t_{0}e^{2πis} (s ∈ [0, 1], |t_{0}| sufficiently large), then the monodromy of π over ∂D (with its natural orientation) is exactly T^{−1}. Theorem (4.6) in [5] says basically that the knowledge of T is equivalent to that of T_{f}^{∞}.

(2.1) Theorem [5, (4.6)]

a) For any α ≠ 1, (T_{f}^{∞})_{α} = T_{α}.

b) For α = 1 one has

i) #_{1}(T_{f}^{∞})_{1} = b_{n}(X^{∞}) + p_{n−1}(X^{∞}) − #T_{1}.

ii) #_{2}(T_{f}^{∞})_{1} = #_{1}T_{1} − b_{n}(X^{∞}).

iii) #_{l+1}(T_{f}^{∞})_{1} = #_{l}T_{1} for l ≥ 2.

The big disadvantage of the map π is that its central fiber π^{−1}(0) is nonreduced. For this reason we consider the following construction: Let D′ be again a disc of small radius and consider δ : D′ → D given by δ(t) = t^{d}. Then the normalization X′ of X ×_{δ} D′ can be identified with

X′ = {([x], t) ∈ P^{n+1} × D′ | f_{d}(x_{1}, …, x_{n+1}) + tx_{0}x_{n+1}^{d−1} = x_{0}^{d}}

Now π′ : X′ → D′ (π([x], t) = t) induces a locally trivial C^{∞}-fibration over D′ − {0} with algebraic monodromy T^{−d} : H^{n}((π′)^{−1}(t_{0})) → H^{n}((π′)^{−1}(t_{0})).

Notice that now both X′ and the central fiber X′_{0} = (π′)^{−1}(0) have only isolated singularities: Sing(X′) = Sing(X^{∞}) × {0}. In fact, the central fiber is the d-fold cyclic covering of P^{n} branched along X^{∞}, in particular if we set Sing(X′_{0}) = {p′_{1}, …, p′_{k}}, then the isolated singularities (X′_{0}, p′_{i}) are the d-th suspensions of the singularities (X^{∞}, p_{i})_{i=1}^{k} and the map π′ provides their smoothings. Let F′_{i} (respectively, T′_{i}) be the Milnor fiber of (X′_{0}, p′_{i}) (respectively, the monodromy H^{n}(F′_{i}) → H^{n}(F′_{i}) corresponding to the smoothing given by π′), 1 ≤ i ≤ k. Then the exact sequence of vanishing cycles is:

(2.2) 0 → H^{n}(X′_{0}) → H^{n}(X′_{t}) → ⊕_{i=1}^{k}H^{n}(F′_{i}) → P^{n+1}(X′_{0}) → 0

where X′_{t} = (π′)^{−1}(t) (for some fixed t ≠ 0) and P^{n+1}(X′_{0}) is the primitive cohomology Ker[H^{n+1}(X′_{0}) → H^{n+1}(X′_{t})] (for details see [5, (10), (5.3)]).

Our second exact sequence is given by the generalized invariant cycle theorem proved in the Appendix of [5]:

(2.3) 0 → H^{n}(X′_{0}) → Ker((T^{−d})_{1} − Id) → ⊕_{i=1}^{k}H_{{p′_{i}}}^{n+1}(X′) → 0.

Both sequences are exact sequences of mixed Hodge structures and there is a natural monodromy action on them, which at the level of H^{n}(X′_{t}) is T^{−d}.

The main point of the paper is the construction of an action on these exact sequences which at the level of H^{n}(X′_{t}) is T^{−1}. More precisely: we would like to understand the monodromy of π, but this map has a non-reduced central fiber, which makes the study difficult. Then we go to the normalization of the d-fold covering π, which is π′, and we lift the monodromy of π to the level of π′.

First, notice that π′ : X′ → D′ has a natural Galois action of the cyclic group Z/dZ over π : X → D, that is, we have a commutative diagram:

[figure: D′; X′; D; D′; X′; X; G; π′; π′; π; (2.4)]

where if we set ξ = e^{2πi/d}, the horizontal map D′ → D′ is given by t ↦ tξ^{−1} and G is given by G([x_{0} : …: x_{n+1}], t) = ([ξx_{0} : …: x_{n+1}], tξ^{−1}).

Now we lift the geometric monodromy of π (over D − {0}) to the level of π′ : X′ → D′. Fix a point t_{0} ∈ D′ − {0}, consider the circle S_{t_{0}}^{1} = {z ∈ D′ : |z| = t_{0}}, and take E′ = (π′)^{−1}(S_{t_{0}}^{1}). The fibration E′ → S_{t_{0}}^{1} is still denoted by π′, its monodromy transformation is T^{−d}. Take a local trivialization over the positive arc [t_{0}, t_{0}ξ] i.e., a diffeomorphism h such that

[figure: [0, \frac{1}{d}] × X′_{t_{0}}; h; ≃; (s, x) ↦ t_{0}e^{2πis}; π′; S_{t_{0}}^{1}; (π′)^{−1}(arc[t_{0}, t_{0}ξ]); (2.5)]

Then the geometric monodromy of π can be identified at the level of π′ with the composition

(2.6) X′_{t_{0}} →^{h( \frac{1}{d} ,·)} X′_{t_{0}ξ} →^{G} X′_{t_{0}}.

This lifting construction can be extended over D′ as follows: Since X′_{0} = (π′)^{−1}(0) has only isolated singularities, it is possible to construct a flow

[figure: [0, 1] × D′; [0, 1] × X′; D′; X′; id × π′; π′; φ; ϕ]

such that the above diagram is commutative, φ(s, t) = te^{2πis}, and ϕ(s, x) = x for any x ∈ X′_{0} (see, for example, [2]). Now consider the composition G ∘ ϕ(\frac{1}{d}, ·) over D′

[figure: X′; X′; X′; ϕ(\frac{1}{d}, ·); G; π′; π′; D′; (2.7)]

This will be called the “lifted geometric monodromy”. In the next subsections we will determine the isomorphisms induced by it on the vector spaces which appear in the exact sequences (2.2) and (2.3). Obviously, on H^{n}(X′_{t}) the induced “lifted geometric monodromy” is exactly T^{−1}.

The action on the spaces H^{q}(X′_{0}) can be determined as follows. Since ϕ(s, x) = x for any x ∈ X′_{0}, the isomorphism ϕ(\frac{1}{d}, ·) restricted to X′_{0} is the identity. Therefore, the action on H^{q}(X′_{0}) is induced by the Galois action G : X′_{0} → X′_{0}, G([x_{0} : …: x_{n+1}]) = [ξx_{0}, …, x_{n+1}]. This action will be denoted by G^{q}.

II. The action on ⊕_{i=1}^{k}H^{n}(F′_{i}).

If φ : H → H is a linear map, we will denote by c_{l}(φ) : H^{⊕l} → H^{⊕l} the linear map defined by c_{l}(φ)(x_{1}, …, x_{l}) = (φ(x_{l}), x_{1}, …, x_{l−1}). Then we have:

(2.8) Theorem: a) If S(F_{i}) denotes the suspension of F_{i} then we have a homotopy equiva-

lence F′_{i} ∼ ⋁_{d−1} S(F_{i}), therefore an isomorphism H^{n}(F′_{i}) ≃ H^{n−1}(F_{i})^{⊕(d−1)}

b) Under the isomorphism above, the “lifted monodromy action” on H^{n}(F′_{i}) is c_{d−1}(T_{i}).

Proof: Part a) was proved already in [5, (5.3)], but it follows also from our discussion here. For b) we use a similar construction as in [loc. cit.]. The map-germ (X′, p′_{i}) →^{π′} (D′, 0) can be identified with

Y_{i} := {(y_{0}, y, t) : g_{i}(y) + ty_{0} = y_{0}^{d}} →^{π′_{i}} (D′, 0)

where y_{0} and y = (y_{1}, …, y_{n}) are local affine coordinates (y_{i} = \frac{x_{i}}{x_{n+1}} , i = 0, …, n), g_{i} is the local equation of (X^{∞}, p_{i}) ⊂ (P^{n}, p_{i}) and π′_{i} is given by π′_{i}(y_{0}, y, t) = t. The Galois action on Y_{i} is G(y_{0}, y, t) = (y_{0}ξ, y, tξ^{−1}). As in the global situation (2.5), consider the (local) locally trivial fibrations

induced by π′, E′_{i} = (π′)^{−1}(S_{t_{0}}^{1}) ∩ Y_{i} →^{π′_{i}} S_{t_{0}}^{1}, with fiber F′_{i}. Consider the local trivializations over the positive arc [t_{0}, t_{0}ξ]

[figure: [0, \frac{1}{d}] × F′_{i}; h_{i}; ≃; (s, x) ↦ t_{0}e^{2πis}; π′_{i}; S_{t_{0}}^{1}; (π′_{i})^{−1}(arc[t_{0}, t_{0}ξ])]

Then the “lifted geometric action” on F′_{i} := (π′_{i})^{−1}(t_{0}) is the composition

(2.9) (π′_{i})^{−1}(t_{0}) →^{h_{i}( \frac{1}{d} ,·)} (π′_{i})^{−1}(t_{0}ξ) →^{G} (π′_{i})^{−1}(t_{0})

which will be denoted h′_{i}. We will prove that this geometric action induces c_{d−1}(T_{i}) at the cohomology level. Remark: It is not difficult to see that (h′_{i})^{d} is the monodromy of π′_{i}. In [5] this is identified with c_{d−1}(T_{i}^{d}) ≈ [c_{d−1}(T_{i})]^{d}.

As in [5, (5.3)], consider the isolated complete intersection singularity given by Y′_{i} →^{φ} D′ × D′, φ(y_{0}, y, t) = (t, y_{0}). The discriminant of φ is Δ = {ty_{0} = y_{0}^{d}} and φ is a locally trivial fibration over D′ × D′ − Δ with fiber F_{i}.

For t_{0}e^{2πiβ} ∈ S_{t_{0}}^{1}, the intersection points of the line {t = t_{0}e^{2πiβ}} with the discriminant {ty_{0} = y_{0}^{d}} of φ are

q_{0}(β) = (t_{0}e^{2πiβ}, 0) and q_{j}(β) = (t_{0}e^{2πiβ}, e^{2πi \frac{j+β}{d−1}} · ^{d−1} \sqrt{t_{0}})

for j = 1, …, d − 1. We will use the following notations:

I_{j}(β) = segment [q_{0}(β), q_{j}(β)] (in {t_{0}e^{2πiβ}} × D′),
I(β) = ⋃_{j=1}^{d−1} I_{j}(β),
B = ⋃_{β∈[0,1]} I(β) ⊂ D′ × D′,
r_{j}(β) = middlepoint of I_{j}(β) = (t_{0}e^{2πiβ}, \frac{1}{2} ^{d−1} \sqrt{t_{0}} e^{2πi \frac{j+β}{d−1}} ).

It is obvious that, for all 1 ≤ j ≤ d − 1, φ^{−1}(q_{j}(β)) is contractible and φ^{−1}(r_{j}(β)) is exactly F_{i}. Therefore φ^{−1}(I_{j}(β)) can be identified with the suspension S(F_{i}) and φ^{−1}(I(β)) ∼ ⋁_{d−1} S(F_{i}).

The inclusion B ⊂ S_{t_{0}}^{1} × D′ admits a strong deformation retract which can be lifted. Consider the torus T = S_{t_{0}}^{1} × S_{\frac{1}{2} ^{d−1} \sqrt{t_{0}}}^{1} which contains the points r_{j}(β). By the identification of F′_{i} = (π′_{i})^{−1}(t_{0}) = φ^{−1}({t_{0}} × D′) with φ^{−1}(I(0)) ∼ ⋁_{d−1} S(F_{i}), the homology of F′_{i} is generated by a wedge of suspensions of cycles which lie above the points r_{j}(0), 1 ≤ j ≤ d − 1. When we move t on the positive arc [t_{0}, t_{0}ξ], then these points move on the path [0, \frac{1}{d}] → T given by β ↦ r_{j}(β). We denote these paths by γ_{j}, with endpoints r_{j}(0) and r_{j}(\frac{1}{d}), i.e.

γ_{j}(s) = (t_{0}e^{2πis}, \frac{1}{2} ^{d−1} \sqrt{t_{0}} e^{2πi \frac{j+s}{d−1}} ) , s ∈ [0, \frac{1}{d}].

The local trivialization over ∪_{j}γ_{j} corresponds to h_{i}(\frac{1}{d}, ·) in (2.9) (we will explain this identification more precisely later). Next, we identify the Galois action with some local trivialization over some paths. Consider the paths τ_{j} : [0, \frac{1}{d}] → T defined by

τ_{j}(s) = (t_{0}e^{2πi( \frac{1}{d} −s)}, \frac{1}{2} ^{d−1} \sqrt{t_{0}} e^{2πi( \frac{j−1}{d−1} + \frac{1}{d(d−1)} +s)}),

which connects r_{j−1}(\frac{1}{d}) and r_{j}(0).

[figure: \frac{1}{d}; 1; \frac{1}{d−1}; \frac{2}{d−1}; \frac{d−2}{d−1}; 1; argument of t_{0}; argument of y_{0}; the points r_{j}(β); γ_{d−1}; γ_{1}; γ_{2}; γ_{d−2}; τ_{1}; τ_{2}; τ_{d−1}]

Notice that the Galois action (y_{0}, y, t) ↦ (y_{0}ξ, y, tξ^{−1}) induces

G : φ^{−1}(r_{j−1}(\frac{1}{d})) →^{∼} φ^{−1}(r_{j−1}(0)).

Consider the isomorphism (up to isotopy) given by the local trivialization of φ above the oriented path τ_{j}:

T r_{j} : φ^{−1}(r_{j−1}(\frac{1}{d})) →^{∼} φ^{−1}(r_{j−1}(0)).

Fact: The composition

(2.10) φ^{−1}(r_{j−1}(0)) →^{G^{−1}} φ^{−1}(r_{j−1}(\frac{1}{d})) →^{T r_{j}} φ^{−1}(r_{j−1}(0))

is isotopic to the identity. Proof of the fact: Consider the map δ : D′ ×D′ → D′ given by δ(t, y_{0}) = y_{0}^{d} − ty_{0} (one has δ^{−1}(0) =the discriminant of φ). First notice that δ(r_{j−1}(\frac{1}{d})) = δ(r_{j}(0)). Therefore, the composition (2.10) can be identified with

g_{i}^{−1}(δ(r_{j}(0))) →^{Id} g_{i}^{−1}(δ(r_{j−1}(\frac{1}{d}))) \underrightarrow{_{T r_{j}}} g_{i}^{−1}(δ(r_{j}(0)))

where the first map is the identity y ↦ y (the second component of G) and T̃ r_{j} is the trivialization of g_{i} above the loop s ↦ δ(τ_{k}(s)), (s ∈ [0, \frac{1}{d}]). Now, it is easy to verify that δ(τ_{k}(s)) can be written in the form B · e^{2πi(a+ds)} + A, where |A| > |B|. Therefore the loop s ↦ δ(τ_{k}(s)) is isotopic to zero in D′ − {0}. So, T̃ r_{j} (and hence T r_{j} ∘ G^{−1} too) is isotopic to the identity. □

The above fact shows that the Galois action G can be replaced by the local trivialization above the paths {τ_{j}}_{j}.

Since h(\frac{1}{d}, ·) in (2.9) corresponds to the local trivialization above \{γ_{j}\}_{j} and the Galois action to the local trivialization above {τ_{j}}_{j}, then the composed map in (2.9) corresponds to the trivialization above {τ_{j+1} ∘ γ_{j}}_{j}. Now we identify the fibers of φ above the points

r_{d−1}(0), r_{d−1}(\frac{1}{d}), r_{1}(0), r_{1}(\frac{1}{d}), …, r_{d−1}(0), r_{d−2}(\frac{1}{d})

via the paths:

(2.11) γ_{d−1}, τ_{1}, γ_{1}, τ_{2}, γ_{2}, τ_{3}, …, γ_{d−2}.

The fiber F′_{i} is

Sφ^{−1}(r_{d−1}(0)) ∨ Sφ^{−1}(r_{1}(0)) ∨ … ∨ Sφ^{−1}(r_{d−2}(0))

and the “lifted monodromy action” is induced by

S(τ_{1} ∘ γ_{d−1})_{∗} ∨ S(τ_{2} ∘ γ_{1})_{∗} ∨ … ∨ S(τ_{d−1} ∘ γ_{d−2})_{∗}

But this (because of the identification of fibers via the paths in (2.11)) is exactly the isomorphism c_{d−1}(Q), where Q is the monodromy of φ above the loop

l = γ_{d−1} ∘ τ_{1} ∘ γ_{1} ∘ τ_{2} ∘ … ∘ γ_{d−2} ∘ τ_{d−1}.

The loop l in the complement of the discriminant of φ is homotopic to s ↦ (t_{0}, e^{2πis} \frac{\sqrt[d−1]{t_{0}}}{2} ), s ∈ [0, 1]. The linking number of this (second) loop with {y_{0} = 0} is one, and with {y_{0}^{d−1} = t_{0}} is zero. Therefore Q = T_{i}, in particular the map h′_{i} induced by G ∘ h_{i}(\frac{1}{d}, ·) is c_{d−1}(T_{i}). □

III. The action on H_{{p′_{i}}}^{n+1}(X′).

In this subsection we prove that the “lifted action” on H_{{p′_{i}}}^{n+1}(X′) is trivial. Let K′_{i} be the link of (X′, p′_{i}) = {g_{i}(y) + ty_{0} − y_{0}^{d} = 0} (we use the same notations as in II). The map

π′ : {g_{i}(y) + ty_{0} − y_{0}^{d} = 0} → D′, (y_{0}, y, t) ↦ t

gives an open book decomposition of K′_{i}. Let K_{i} = {t = 0} ⊆ K′_{i} be the link of t, then arg = arg(t) : K′_{i} − K_{i} → S^{1} is a C^{∞}-locally trivial fibration. Consider the flow ϕ : [0, 1] × K′_{i} → K′_{i} such that

a) If x ∈ {t = 0} = K_{i}, then ϕ(s, x) = x for any s.

b) If x ∉ K_{i}, then arg(ϕ(s, x)) = e^{2πis}arg(s).

The wanted geometric action is the composed map

K′_{i} →^{ϕ( \frac{1}{d} ,·)} K′_{i} →^{G} K′_{i}

Now ϕ(\frac{1}{d}, ·) is isotopic to the identity via the flow ϕ(s, ·), s ∈ [0, \frac{1}{d}]. The Galois action is isotopic to the identity as well. To see this consider the isotopy:

(s, (y_{0}, y, t)) ↦ (y_{0}(s), y(s), t(s)) =
= (y_{0}e^{2πis/d}, y, (t − y_{0}^{d−1})e^{−2πis/d} + y_{0}^{d−1}e^{2πis(d−1)/d}).

If s = 0, then (y_{0}(0), y(0), t(0)) = (y_{0}, y, t), if s = 1 then (y_{0}(0), y(0), t(0)) = (y_{0}ξ, y, tξ^{−1}) = G(y_{0}, y, t).

IV. The exact sequences revisited.

We summarize the results of the subsections I-III: One has the following two exact sequences, with the “lifted monodromy action”:

[figure: 0; H^{n}(X′_{0}); H^{n}(X′_{t}); ⊕_{i=1}^{k}H^{n}(F′_{i}); P^{n+1}(X′_{0}); 0; 0; H^{n}(X′_{0}); H^{n}(X′_{t}); ⊕_{i=1}^{k}H^{n}(F′_{i}); P^{n+1}(X′_{0}); 0; ⊕ c_{d−1}(T_{i}); G^{n}; T^{−1}; G^{n+1}; (E.1)]

and (E.2):

[figure: 0; H^{n}(X′_{0}); Ker (T^{−d} − Id); ⊕_{i=1}^{k}H_{{p′_{i}}}^{n+1}(X′); 0; 0; H^{n}(X′_{0}); Ker (T^{−d} − Id); ⊕_{i=1}^{k}H_{{p′_{i}}}^{n+1}(X′); 0; G^{n}; T^{−1}; Identity]

The main theorem will follow from these exact sequences and from some mixed Hodge-theoretical arguments.

We end this subsection with some facts about the Galois action G^{∗} : H^{∗}(X′_{0}) → H^{∗}(X′_{0}), where p : X′_{0} → P^{n} is the d-th cyclic covering branched along X^{∞}. One has that H^{q}(X′_{0}, C) = H^{q}(P^{n}, Rp_{∗}C_{X′_{0}}) = H^{q}(P^{n}, p_{∗}C_{X′_{0}}) and the restriction of p_{∗}C_{X′_{0}} to the complement of X^{∞} is a flat bundle. Its corresponding monodromy representation is given by the composed map π_{1}(P^{n}− X^{∞}) →^{h} Z/dZ →^{r} Aut(Z^{d}) (see §1 for the definition of h), where r(1) := σ : Z/dZ → Z/dZ is the permutation σ(x_{1}, …, x_{d}) = (x_{d}, x_{1}, …, x_{d−1}).

But also the Galois action is induced by σ. So, we have a direct sum decomposition p_{∗}C_{X′_{0}} = C_{P^{n}} ⊕ ⊕_{s=1}^{d−1}j_{∗}V_{s} such that G |_{C_{P^{n}}} is the identity and G |_{j_{∗}V_{s}} is the multiplication by ξ^{s} = e^{2πis/d}. Therefore:

(2.12) (H^{q}(X′_{0}); G^{q}) = (H^{q}(P^{n}) ⊕ (⊕_{s=1}^{d−1}H^{q}(P^{n}, j_{∗}V_{s})); ⊕_{s=0}^{d−1}ξ^{s}).

V. The proof of the main Theorem, case α = 1.

Consider the exact sequence (E.2) with its actions. Since the (generalized) 1-eigenspace of T^{−1} on Ker(T^{−d} − Id) is exactly Ker(T^{−1} − Id), the decomposition (2.12) provides the following exact sequence:

(2.13) 0 → H^{n}(P^{n}) → Ker(T^{−1} − Id) → ⊕_{i=1}^{k}H_{{p′_{i}}}^{n+1}(X′) → 0

This is again an exact sequence of mixed Hodge structures. Now, it is on the one hand clear that the weight of H^{n}(P^{n}) is n and, on the other hand,

dim Gr_{n−l+1}^{W}H_{{p′_{i}}}^{n+1}(X′) = #_{l}(T_{i})_{1} (for l ∈ Z)

(see [5, (5.5)]). Since the weight filtration on H^{n}(X′_{t}) is the monodromy weight filtration of T^{−1} centered at n, one has dim Gr_{n−l+1}^{W}Ker(T^{−1} − Id) = #_{l}(T^{−1}). This shows that

(2.14)  (T^{−1})_{1} = ⊕_{i=1}^{k}(T_{i})_{1} ⊕ \{ \begin{matrix}0  if n is odd \\ id_{C} if n is even \end{matrix}

Now notice that

(2.15) p_{n−1}(X^{∞}) − p_{n}(X^{∞}) = χ_{0}

(see, for example (2.29), or [3]). Hence the result follows from (2.1), (2.14) and (2.15). □ (2.16) Remark: Using the exact sequence of vanishing cycles of X^{∞} ⊂ P^{n}, by standard mixed Hodge theoretical arguments, one can prove that the dimensions dim Gr_{n−l}^{W}P^{n−1}(X^{∞}) (l ≥ 1) are equal to the numbers on the right hand side of the equalities in the Main Theorem, case α = 1. Hence , the main theorem gives:

#_{l}(T_{f}^{∞})_{1} = dim Gr_{n−l}^{W}P^{n−1}(X^{∞}) for l ∈ Z,

where W denotes the weight filtration.

VI. The proof of the main Theorem, case α^{d} = 1, α ≠ 1.

First notice that the Galois action G : X′_{0} → X′_{0} is an algebraic map, therefore G^{q} : H^{q}(X′_{0}) → H^{q}(X′_{0}) preserves the weight filtration. Since H^{n}(P^{n}, j_{∗}V_{s}) is the ξ^{s}-eigenspace of G^{q} (cf. 2.12), it has a natural induced weight filtration (actually, it has a natural mixed Hodge structure). Now let α = e^{2πis/d} = ξ^{s} for 1 ≤ s ≤ d − 1. Then by (E.2) one has:

(2.17) H^{n}(P^{n}, j_{∗}V_{s}) = Ker(T^{−1} − α).

In particular, for l ≥ 1 one has

(2.18) #_{l}(T^{−1})_{α} = dim Gr_{n−l+1}^{W}H^{n}(P^{n}, j_{∗}V_{s}).

Actually, (2.18) together with (2.1) already determine (T_{f}^{∞})_{α}, but this is not exactly the assertion of the main theorem, we want some more information about the right hand side of (2.18).

Consider the exact sequence (E.1). Using (2.17) one has:

[figure: 0; Ker(T^{−1} − α); H^{n}(X′_{t})_{α}; [⊕H^{n}(F′_{i})]_{α}; H^{n+1}(P^{n}, j_{∗}V_{s}); 0; 0; Ker(T^{−1} − α); H^{n}(X′_{t})_{α}; [⊕H^{n}(F′_{i})]_{α}; H^{n+1}(P^{n}, j_{∗}V_{s}); 0; ·α; (T^{−1})_{α}; [⊕c_{d−1}(T_{i})]_{α}; ·α]

By [13], the weight filtration of H^{n}(X′_{t})_{α} is the monodromy weight filtration of (T^{−1})_{α} centered at n, thus the quotient H^{n}(X′_{t})/Ker (T^{−1} − α) has a (polarized) mixed Hodge structure with weight filtration equal to the monodromy weight filtration of the class [T_{α}^{−1}] of (T^{−1})_{α} centered at n+1. On the other hand, H^{n+1}(X′_{0}) is pure of weight n + 1 ([16]) and G^{n+1} preserves the weight filtration, hence H^{n+1}(P^{n}, j_{∗}V_{s}) is pure of weight n + 1 (cf. (2.12)). These two facts show that the weight filtration of [⊕H^{n}(F′_{i})]_{α} is the monodromy weight filtration of (⊕c_{d−1}(T_{i}))_{α}. Now comparing the dimensions of the primitive cohomologies of [T_{α}^{−1}] and (c_{d−1}(T_{i}))_{α} one has:

(2.19)  \{ \begin{matrix}#_{2}(T^{−1})_{α} = −dim H^{n+1}(P^{n}, j_{∗}V_{s}) + Σ_{i=1}^{k}#_{1}(c_{d−1}(T_{i}))_{α} \\ #_{l+1}(T^{−1})_{α} = Σ_{i=1}^{k}#_{l}(c_{d−1}(T_{i}))_{α} for l ≥ 2. \end{matrix}

Since α^{d} = 1 and α ≠ 1, (c_{d−1}(T_{i}))_{α} = (T_{i})_{α^{−1}}. Therefore:

(2.20)  \{ \begin{matrix}#_{2}(T^{−1})_{α} = −β_{s} + Σ_{i=1}^{k}#_{1}(T_{i})_{α^{−1}} \\ #_{l+1}(T^{−1})_{α} = Σ_{i=1}^{k}#_{l}(T_{i})_{α^{−1}} for l ≥ 2. \end{matrix}

Since β_{d−s} = β_{s} (s = 1, …, d − 1), parts (Ib,Ic) follow from (2.20) and (2.1.a). In order to prove (Ia) (i.e., to compute #_{1}(T_{f}^{∞})_{α} = #_{1}T_{α}) notice that from (2.18) one has:

(2.21) #(T^{−1})_{α} = dim H^{n}(P^{n}, j_{∗}V_{s}),

and from (2.20):

(2.22) Σ_{l≥2}#_{l}(T^{−1})_{α} = −β_{s} + Σ_{i=1}^{k}#(T_{i})_{α^{−1}}.

Therefore

#_{1}(T^{−1})_{α} = dim H^{n}(P^{n}, j_{∗}V_{s}) + dim H^{n+1}(P^{n}, j_{∗}V_{s}) − Σ_{i=1}^{k}#(T_{i})_{α^{−1}}.

Now (Ia) follows from this identity and the following result:

(2.23) Proposition: For s = 1, …, d − 1 one has: a) dim H^{q}(P^{n}, j_{∗}V_{s}) = 0 for q ≠ n, n + 1.

b) dim H^{n}(P^{n}, j_{∗}V_{s}) − dim H^{n+1}(P^{n}, j_{∗}V_{s}) = χ_{s}.

(For the definition of χ_{s}, see the introduction). In particular, the Euler characteristic of (P^{n}, j_{∗}V_{s}) is local.

Proof: The first part follows from (2.12), because P^{q}(X′_{0}) = 0 if q ≠ n, n + 1. The second part will be proved (together with some other relations) in subsection VIII, (2.29).

VII. The proof of the main theorem, case α^{d} ≠ 1.

Consider the exact sequence (E.1). Since (G^{∗})^{d} = Id, the generalized α-eigenspaces are:

(2.25) (H^{n}(X′_{t})_{α}, (T^{−1})_{α}) ≃ ((⊕_{i=1}^{k}H^{n}(F′_{i}))_{α}, (⊕c_{d−1}(T_{i}))_{α}).

By (2.1), T_{α} = (T_{f}^{∞})_{α}. Also, if Π_{j}(λ − ξ_{j}) is the characteristic polynomial of T_{i}, then Π_{j}(λ^{d−1} −ξ_{j}) is the characteristic polynomial of c_{d−1}(T_{i}). Therefore, if α is an eigenvalue of c_{d−1}(T_{i}), then α^{d−1} = ξ_{j} for some ξ_{j} and the unipotent (or nilpotent) part of c_{d−1}(T_{i})_{α} and (T_{i})_{ξ_{j}} can be identified. This ends the proof of the main theorem.

VIII. The relation with the Milnor fiber of f_{d} : C^{n+1} → C.

Consider the homogeneous singularity f_{d} : C^{n+1} → C with one-dimensional singular locus, let F be its Milnor fiber. It is well-known that its (reduced) homology is concentrated in H_{n}(F) and H_{n−1}(F). Let h_{q} : H_{q}(F) → H_{q}(F) be the algebraic monodromy of f_{d}, where q = n − 1, n. In this subsection we identify our local and global invariants with numerical invariants given by the transformations h_{n−1}, h_{n}.

Recall that we denoted p : X′_{0} → P^{n} the d-th cyclic covering branched along X^{∞}. Then p^{−1}(X^{∞}) can be identified with X^{∞} and X′_{0}−p^{−1}(X^{∞}) with F. This gives a cyclic (unramified) covering F → P^{n} −X^{∞} with fiber Z/dZ. By duality, H_{q}(F) = H^{2n−q}(X′_{0}, p^{−1}(X^{∞})), therefore the exact sequence of the pair (X′_{0}, p^{−1}(X^{∞})) reads as follows:

[figure: 0; P^{n−1}(X^{∞}); H_{n}(F); P^{n}(X′_{0}); P^{n}(X^{∞}); H_{n−1}(F); P^{n+1}(X′_{0}) → 0; 0; P^{n−1}(X^{∞}); H_{n}(F); P^{n}(X′_{0}); P^{n}(X^{∞}); H_{n−1}(F); P^{n+1}(X′_{0}) → 0; Id; h_{n}^{−1}; G^{n}; Id; h_{n−1}^{−1}; G^{n+1}; (2.26)]

In the above diagram, we have also inserted the corresponding Galois actions. The Galois action on X′_{0} = {f_{d}(x_{1}, …, x_{n+1}) = x_{0}^{d}} is [x_{0} : …: x_{n+1}] ↦ [ξx_{0} : …: x_{n+1}]. If on X′_{0} − p^{−1}(X^{∞}) we take affine coordinates y_{i} = x_{i}/x_{0} , 1 ≤ i ≤ n + 1, then the induced action is (y_{1}, …, y_{n+1}) ↦ ξ^{−1}(y_{1}, …, y_{n+1}). This is the inverse of the geometric monodromy of the Milnor fiber F.

Now, if we consider the generalized eigenspaces of the Galois action in (2.26), one has the following identifications:

(2.27)  \{ \begin{matrix}(H_{n}(F), h_{n}^{−1})_{≠1} = (P^{n}(X′_{0}), G^{n})_{≠1} \\ (H_{n−1}(F), h_{n−1}^{−1})_{≠1} = (P^{n+1}(X′_{0}), G^{n+1})_{≠1} \\ dim H_{n}(F)_{1} = p_{n−1}(X^{∞}) and dim H_{n−1}(F)_{1} = p_{n}(X^{∞}). \end{matrix}

We recall (cf. (2.12)) that (P^{q}(X′_{0}), G^{q})_{≠1} = (⊕_{s=1}^{d−1}H^{q}(P^{n}, j_{∗}V_{s}), ⊕_{s=1}^{d−1}ξ^{s}). In particular, all our global invariants are equivalent to the characteristic polynomial of h_{n−1}, i.e. β_{s} = rankH_{n−1}(F)_{α} (0 ≤ s ≤ d − 1). Now let us consider the zeta function of f_{d} : C^{n+1} → C. This is basically given in [15]:

(2.28) \frac{det(λ · Id − h_{n})}{det(λ · Id − h_{n−1})} = (λ − 1)^{(−1)^{n+1}} · (λ^{d} − 1)^{\frac{(d−1)^{n+1}+(−1)^{n}}{d}} · ∏_{i=1}^{k}(λ^{d} − 1)^{−μ_{i}}

= ∏_{s=0}^{d−1}(λ − e^{2πis/d})^{χ_{s}}.

Now, (2.28) and (2.27) give:

(2.29) \{ \begin{matrix}p_{n−1}(X^{∞}) − p_{n}(X^{∞}) = χ_{0} \\ dim H^{n}(P^{n}, j_{∗}V_{s}) − dim H^{n+1}(P^{n}, j_{∗}V_{s}) = χ_{s} (s = 1, …, d − 1). \end{matrix}

This proves (2.15) and (2.23.b).

(2.30)Remark: By (2.27) and corollary 3 one has (for s = 0, …, d − 1)

dim H_{n−1}(F)_{e^{2πis/d}} = β_{s} ≤ Σ_{i=1}^{k}#_{1}(T_{i})_{e^{2πis/d}}

Similar restrictions can be found in [14, (5.4) and §9].

(2.31) Remark: (The relation with π_{n−1}(P^{n} − X^{∞}).) Here we present the connection between the present paper and [7], more precisely, between the defects β_{s} (0 ≤ s ≤ d − 1) and π_{n−1}(P^{n} − X^{∞}).

First, assume that n = 2. Denote G = π_{1}(P^{n} − X^{∞}), G′ = [G, G], and G′′ = [G′, G′]. Then 0 → π_{1}(F) → G → Z/dZ → 0 is an exact sequence, actually π_{1}(F) = G′. Therefore, H_{1}(F) = G′/G′′, and it has a natural action of Z/dZ. By (2.27) one has:

β_{s} = rank((G′/G′′) ⊗ Q)_{α}; α = e^{2πis/d}; s = 0, …, d − 1.

Now, assume that n > 2. From the covering F → P^{n} − X^{∞} and Hurewicz theorem one has: π_{1}(P^{n} − X^{∞}) = Z/dZ, π_{q}(P^{n} − X^{∞}) = 0 if 1 < q < n − 1, and π_{n−1}(P^{n} − X^{∞}) = π_{n−1}(F) = H_{n−1}(F). Hence by (2.27):

β_{s} = rank(π_{n−1}(P^{n} − X^{∞}) ⊗ Q)_{α}; α = e^{2πis/d}; s = 0, …, d − 1,

where the action of π_{1}(P^{n} − X^{∞}) on π_{n−1}(P^{n} − X^{∞}) is the natural one.

We thank Professor A. Libgober bringing in our attention the invariant π_{n−1}, and his helpful comments about it.

§3. Examples

I. Zariski’s plane sextics:

Set d = 6 and let f_{6} ∈ C[X; Y; Z] be a form defining a plane sextic in P^{2} with six cusps and no other singularities. Then χ_{0} = 8 and χ_{s} = 9 for s = 1, …, 5. Moreover (since the characteristic polynomial of the local monodromy of a cusp singularity is t^{2} − t + 1), β_{s} = 0 if s = 0, 2, 3, 4. Our main theorem gives:

a) If α^{d} ≠ 1, then (T_{f}^{∞})_{α} has only one-dimensional Jordan blocks, and (T_{f}^{∞})_{α} = Id_{C^{6}} if α = e^{πiφ}, φ ∈ \{\frac{1}{15}, \frac{7}{15}, \frac{11}{15}, \frac{13}{15}, \frac{17}{15}, \frac{19}{15}, \frac{23}{15}, \frac{29}{15}\} (i.e., if α^{5} = e^{2πis/6} for s = 1 or s = 5 and α^{6} ≠ 1). Otherwise (T_{f}^{∞})_{α} = 0. b) (T_{f}^{∞})_{e^{2πis/6}} has only one-dimensional blocks if s = 0, 2, 3, 4. The number

of them is 8 if s = 0 and 9 if s = 2, 3, 4.

c) (T_{f}^{∞})_{e^{2πis/6}} has only one and two dimensional blocks if s = 1 or s = 5. The number of one-dimensional blocks is 3 + 2β_{s}, and the number of two dimensional blocks is 6 − β_{s}.

Now, by the identification (2.27) and [3, Theorem 2.9] one has β_{s} = 1, (s = 1, 5) if the cusps are on a conic and β_{s} = 0 otherwise.

II. Nodal hypersurfaces:

Assume that f_{d} defines a hypersurface in P^{n} with only nodal (i.e. A_{1}) singularities, let k denote the number of nodes. It follows from the main theorem that the maximal size of a Jordan block of T_{f}^{∞} is two. The numbers β_{s} can be computed using [3, VI, Theorem (4.5)] and (2.27).

If nd is even, set S = C[X_{1}, …, X_{n+1}], q = \frac{dn}{2} − n − 1, and let S_{q} denote the homogeneous component of degree q of S. If Σ ⊂ X^{∞} denotes the set of nodes of {f_{d} = 0}, let S_{q}(Σ) = {h ∈ S_{q} | h_{|Σ} = 0}, and defect(S_{q}(Σ)) := k − codim_{S_{q}}(S_{q}(Σ)).

From the main theorem we get the following possibilities for T_{f}^{∞}:

a) n is odd, d is odd.

Here β_{s} = 0 for all s. Thus T_{f}^{∞} has no Jordan blocks of size two, i.e. it is of finite order.

b) n is odd, d is even.

In this case β_{s} = 0 for s ≠ \frac{d}{2} and T_{f}^{∞} can have Jordan blocks of size two only for eigenvalue −1, the number of them is #_{2}(T_{f}^{∞})_{−1} = k − β_{d/2} = k − defect(S_{q}(Σ)) = codim_{S_{q}}(S_{q}(Σ)).

c) n is even.

In this case, β_{s} = 0 for s ≠ 0 and β_{0} = p_{n}(X^{∞}). It follows that T_{f}^{∞} can have Jordan blocks of size two only for eigenvalue 1, and #_{2}(T_{f}^{∞})_{1} = k − p_{n}(X^{∞}) = k − defect(S_{q}(Σ)) = codim_{S_{q}}(S_{q}(Σ)). This number, in general, is not zero. For example, the defect(S_{5}(Σ)) = β_{0} of the recently constructed quintic hypersurface in P^{4} with k = 130 nodes [18] is 29 [loc.cit., p. 864]. The quintic constructed by Hirzebruch [6] has 126 nodes and defect β_{0} = 25. Actually, there are quintics in P^{4} with 118 nodes and defect 18 ≤ β_{0} ≤ 19 [17].

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