On some generalizations of Newton non degeneracy for hypersurface singularities

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by  Dmitry Kerner

On some generalizations of Newton non-degeneracy for hypersurface singularities
DmitryKerner
Ben Gurion University,Israel
0
1 Abstract. WeintroducetwogeneralizationsofNewtonnon-degeneratesingularitiesofhypersurfaces. Roughlyspeak-
0 ing,anisolatedhypersurfacesingularityiscalledtopologicallyNewtonnon-degenerate ifthelocalembeddedtopological
2 singularitytypecanberestoredfromacollectionofNewtondiagrams(forsomecoordinatechoices). Asingularitythat
isnottopologicallyNewtonnon-degenerateiscalledessentiallyNewton-degenerate. Forplanecurveswegiveanexplicit
n characterizationoftopologicallyNewtonnon-degenerate singularities;forhypersurfacesweprovideseveralexamples.
a Next,wetreatthequestion: whetherNewtonnon-degenerateortopologicallyNewtonnon-degenerateisaproperty
J
ofsingularitytypesorofparticularrepresentatives. Namely,isthenon-degeneracypreservedinanequi-singularfamily?

8 Thisfact isproved forcurves. For hypersurfaces wegivean exampleof aNewton non-degenerate hypersurfacewhose
1 equi-singulardeformationconsistsofessentiallyNewton-degenerate hypersurfaces.
Finally,wedefinethedirectionallyNewtonnon-degenerategerms,asubclassoftopologicallyNewtonnon-degenerateones.

] ForsuchsingularitiestheclassicalformulasfortheMilnornumberandthezetafunctionoftheNewtonnon-degeneratehy-
G
persurfacearegeneralized.
A
. Contents
h
t
a 1. Introduction 1
m 2. What is determined by the collection of Newton diagrams? 4
[ 3. Topologically Newton non-degenerate hypersurfaces 5

4. Directionally Newton non-degenerate singularities 10
3
5. Some numerical singularity invariants 12
v
5 References 17
3
1
5
.
7 1. Introduction
0
Weworkwithgermsofcomplexalgebraic(orlocallyanalytic)hypersurfacesinCn,mostlywithisolatedsingularities.
8
0 Bythesingularitytypewealwaysmeanthelocalembeddedtopologicaltypeofahypersurfacegerm. Forthestandard
: notions of singularity theory see [AGLV-book] and [GLS-book].
v
i
X 1.1. To every germ of singular hypersurface (with fixed local analytic coordinates) the Newton diagram Γf is as-
sociated. A germ V = f = 0 (Cn,0) is called Newton non-degenerate (or non-degenerate with respect to its
r f
{ } ⊂
a Newton diagram Γ ) if for each face σ Γ , the truncation f of f to σ is non-degenerate (i.e. the corresponding
f f σ
hypersurface has no singular points in t∈he torus (C∗)n). A germ is called generalized Newton non-degenerate if it is
Newton non-degenerate for some choice of coordinates.
The Newton diagramof a Newton non-degenerategerm is a complete invariantof the singularity type of the germ.
Namely, if (V ,0) and (V ,0) are two Newton non-degenerate germs, such that Γ = Γ then they have the same
f g f g
embedded topological type [Oka79]. This distinguishes the generalized Newton non-degenerate germs as especially
simple to deal with. For them many topological invariants of the singularity can be expressed explicitly (or at least
estimated) via the geometry of the Newton diagram in a relatively simple manner. For example:
the Milnor number [Kouchnirenko76] (cf. also [GLS-book, I.2.1, pg.122])
•
themodality(withrespecttorightequivalence)forfunctionsoftwovariables(conjecturedin[Arnol’d74,9.9],proved
•
in [Kouchnirenko76, Proposition 7.2])
Date:January18,2010.
2000 Mathematics Subject Classification. Primary14B05;32S25; Secondary14J17; 14J70;32S05; 32S10;
Key words and phrases. Newtonnon-degenerate singularities,Newtondiagram,equisingulardeformation.
The research was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics
(Mathematics DepartmentofBenGurionUniversity,Israel).
Part of the work was done inMathematische Forschungsinstitute Oberwolfach, duringthe author’s stay as an OWL-fellow. Some of the
resultswerepublishedinthepreprint[Kerner-OWP].
1
2 On some generalizations of Newton non-degeneracy for hypersurface singularities
the zeta function of monodromy [Varchenko76] (cf. also [AGLV-book, II.3.12])
•
the spectrum [Steenbrink76, Khovanski-Varchenko85](cf. also [Kulikov98, II.8.5])
•
the Hodge numbers hp,q [Danilov-Khovanski87]
•
the Bernstein-Saito polynomial [BGMM89]
•
a bound on the L ojasiewitc invariant [Abderrahmane05] and the L ojasiewitc-type inequalities on the sufficiency
•
[Fukui91].
Unfortunately, the condition to be generalized Newton non-degenerate is very restrictive, even in the case of plane
curves.
k
Example 1.1. For the germ (C,0) (C2,0) consider the tangential decomposition: C = C .
α
⊂ i=∪1
HereeachC hasunique tangentline l (but maycontainseveralbranches),andthe linesl
ttaoisharrnedtehngianeelpanlmirtdyu>tislomtt1iiapunnαfllcoitytcir.piotaltTyethhoepmefror.o(ieCnsTfttoαhrt,pewe01,po)t=,ohihi’en.set..ne.iqcs=eutaPhptakiotp=niiif=o1(fC.pt,Nh0ieos)tttiesahneathgmgeaentuntlpetciripoa=lnliicez1ietTdyiCffNoCfies(wαC{tli,o1ps0n1)a·n.·soF·mnlokp-orkdoee=txghae0mnb}erp,ralawnet,chehfeog,rreenraopmntii ..............C.................................................................1..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................C.........................................................................................................................................................................................................................................................2.....................................................................................................................................................................................................................................................................................................................................C.................3....
Indeed, first notice that (in some local coordinates) if (C,0) is non-degenerate, then each (C ,0) is non-degenerate.
α
Moreover, for some fixed i, if p > 1 and C is non-degenerate with respect to its diagram, then a coordinate axis
i α
must be tangentto C (to reflectthe fact that some monomials areabsent). Hence, in general,there are“notenough
α
coordinateaxes”toencodethesingularity. Inparticular,manysingularitieswithsmallMilnornumbersandquitesim-
ple defining equations are not generalizedNewton non-degenerate. For example, the union of three cuspidal branches
(A ) with pairwise distinct tangents are so.
2
On the other hand, a locally irreducible plane curve singularity (branch) is generalized Newton non-degenerate if
andonlyifithasonlyonePuiseuxpair. AnexampleofnotgeneralizedNewtonnon-degeneratebranchis(x2+y3)2+
x2y3+m, m>0 with µ=15+2m (named as W♯ in [AGLV-book, I.2.3]).
1,2m §
1.2. Motivation. The present work has originated from the observation that many germs of curves are “almost”
generalizedNewtonnon-degenerate. Namely,formanyofthemthesingularitytype(andthusmanyoftheirproperties)
is reflected on Newton diagram, one just has to take several choices of coordinates.
k
Example 1.2. Continue the previous example, and consider again tangential decomposition C = C . For each
α
α∪=1
1 α k, let C(α) be a germ of curve with the tangential decomposition: C(α) = p−pαL C . Here L are
j α j
≤ ≤ j∪=1 ∪ { }
some lines, such that any two are distinct and none is tangent to C (but arbitrary o(cid:16)therwise)(cid:17). Call such a germ: the
α
directional approximation of (C,0). (The germ is non-unique, but its topological singularity type is unique and any
two such approximations are connected by a µ constant family.)
−
If C is generalized Newton non-degenerate then so is each C(α), and the singularity type 6
tsohefgeCmαdαeinactgarnhaambsesorlfeospttheoer−eod1r,ifgrtoihnmealrseogmmeraeminNi(neCgw,t0po)anrttdoi‘oai,sg’rtathhmeesodinfiagCgu(rlαaa)mri(tcyoff.tyCtphαee).poifScti(nuCcr,ee0,)twhiihssecrdeoiamtghprealemfiterlsiyst ppα•............................................................................•.....................................................................•.........Γ....................C....................(........α...............)..............................................................-
determined from the collection of Newton diagrams (corresponding to all the directional p−pα •
approximations). The precise statement is in 3.3.
§
NotethatifatleastonebranchofthecurveisnotgeneralizedNewtonnon-degeneratethennochoiceofcoordinates
can help to recognize the singularity type in such a way.
1.3. Results. In this work we address some natural questions arising from these examples.
Let (V ,0) (Cn,0) be an isolated hypersurface-germ. Suppose all of its Newton diagrams are known, for any
f
• ⊂
choice of coordinates. Which properties of the germ are determined by this information?
It turns out, e.g. that the collection of all possible diagrams (each labeled with the local coordinate system which
determines it) fix the projectivized tangent cone completely. Namely, if f = f +f +... is the Taylor expansion
p p+1
(insomefixedcoordinatesystem),thenf isdetermineduptoscalingfromthecollectionofalldiagrams(Proposition
p
2.1). In fact, even more information can be retrieved (cf. 2), e.g., if f is reduced and n>2, then f is also fixed
p p+1
§
(up to multiplication of f by an invertible germ).
What are the germs whose embedded topological type is completely determined by all the possible Newton diagrams?
•
We call such germs: topologically Newton non-degenerate singularities (the name was suggested by E. Shustin). The
On some generalizations of Newton non-degeneracy for hypersurface singularities 3
precise definition is in 3. It is not clear currently, how broad this class is (in particular all the generalized Newton
§
non-degenerate germs are such), or how to classify such germs.
Forplanecurves(n=2)wegivethecompleteclassificationin 3.3: agermistopologicallyNewtonnon-degenerateiff
§
eachbranchofitis generalizedNewtonnon-degenerateandthe unionofanytwobranchesisgeneralizedNewtonnon-
degenerate.
Forhypersurfaces(n>2)the situationismuchmorecomplicated. We givesomeexamplesoftopologicallyNewton
non-degenerate germs in 3. For example all the singularities of Yomdin type (f +f with f reduced and f
p p+k p p+k
§
generic and the ideal (f ,f ) is radical) are such. Germs that are not topologically Newton non-degenerate are
p p+k
called essentially Newton-degenerate.
Is topologically Newton non-degenerate (or generalized Newton non-degenerate) a property of the singularity type or
•
of the germ? Namely, suppose an embedded topological type has a topologically Newton non-degenerate (or Newton
non-degenerate)realization. IsthegenericrealizationofthetypetopologicallyNewtonnon-degenerate(orgeneralized
Newton non-degenerate)? Or, is this notion preserved in a µ constant deformation?
−
This is true for quasi-homogeneous singularities (when the Newton diagram is constant along the µ constant
−
stratum)by[Varchenko82],cf. also[AGV-book,III.14.3Theorem8]. ForNewtonnon-degeneratecasetheNewtondi-
agramcanchangeessentiallyalongtheµ constantstratum[Brianc¸on-Speder75]. In[Altmann91]somecohomological
−
conditions on the constancy of Newton diagram are given.
Our question can be considered as a weakening of the properties above. The answer is yes for the case of
curves (Corollary 3.7) and no for higher dimensions. We give examples in 3.4 of µ∗ constant surface families
(S ,0) (C3,0) C1 with the central fibre (S ,0) Newton non-degenerate, wh§ile the ge−neric fibre not generalized
t 0
⊂ ×
Newton non-degenerate (or even not topologically Newton non-degenerate). In fact this situation is typical.
AnimmediateconsequenceisthecomparisonoftheequisingularstrataversustheND-topologicalstrata. Recallthat
for a fixedNewtonnon-degeneraterepresentative(f−1(0),0) (Cn,0)of a givensingularitytype, the ND-topological
⊂
stratum is defined as the collection of all the hypersurfaces which in some coordinates have the diagram Γ and are
f
non-degenerate with respect to Γ . For quasi-homogeneous singularities the ND-topological and µ constant strata
f
−
coincide [Varchenko82]. As our examples show, in many cases the ND-topological strata have positive codimension
inside the equisingular strata.
Once a germ is proven to be topologically Newton non-degenerate, its singularity type is determined by the asso-
•
ciated collection of Newton diagrams. Therefore, every topological singularity invariant can be expressed (at least
theoretically)viathecombinatoricsofthediagrams. However,astheclassoftopologicallyNewtonnon-degeneratesin-
gularities is so broad, it seems difficult to do this generally. Rather, we restrict to a subclass of directionally Newton
non-degenerate singularities (introduced in 4). These are germs with Newton non-degenerate directional approxi-
§
mation (the natural generalization of the case of curves, example 1.2). The number of Newton diagrams needed to
determine the singularity type is bounded in this case. For example, if the projectivized tangent cone PT of the
(Vf,0)
hypersurfacehas only isolatedsingularities,then the singularity type of(V ,0) canbe determined by Sing(PT )
f | (Vf,0) |
coordinate choices.
For directionally Newton non-degeneratesingularities we generalizesome classicalformulas. Inparticular in 5 the
§
formulasforthe Milnornumberandforthe zetafunctionofmonodromyaregeneralized. Forcurveswegeneralizealso
the formula for modality.
1.4. Conventions and notations. In general, we work in the space of the locally analytic hypersurface germs in
Cn. Sometimes we pass to the space of germs of (high) bounded degrees (to have a finite dimensional space, to use
algebraicity and Zariski topology). As the singularities are isolated this is always possible by finite determinacy.
Note that for u a locally invertible function: Γ =Γ . Hence the Newton diagram is well defined by the zero set
f uf
V . The Newton diagrams are assumed to be commode or convenient (Γ intersects all the coordinate axes), unless
f f
explicitly stated. Denote by f the restriction of the function f to the face σ Γ . Denote by Γ− the set of real
σ ∈ f f
points on or below Γ .
f
Throughout the paper we use equisingular deformations and µ constant deformations. In most cases the two
−
notions coincide: for n= 3 this follows by [Lˆe-Ramanujam76], for n =3 and deformations f linear in t it follows by
t
6
[Parusin’ski99]. In fact our µ constant deformations are often even µ∗ constant.
− −
1.5. Acknowledgements. This work would be impossible without numerous important discussions with G.M.
Greuel, A. N´emethi and E. Shustin. A. N´emethi pointed an uncountable number of errors in the first N 0
≫
versions of the text. Many thanks are also to V. Goryunov for important advices.
4 On some generalizations of Newton non-degeneracy for hypersurface singularities
PartofthisworkwasdoneduringmystayattheMathematischeForschungsinsituteOberwolfach(Germany). Many
thanks to the staff for the excellent working atmosphere.
2. What is determined by the collection of Newton diagrams?
Inthissection(V ,0)and(V ,0) (Cn,0)aregermsofisolatedhypersurfacesingularities,suchthatforany choice
f g
⊂
of coordinates Γ = Γ . Let f = f +f +... and g = g +g +... be the Taylor expansions in some fixed
f g p p+1 p p+1
local coordinate system. For any k 0, let Ik(f) := fp,fp+1,..,fp+k := (Cn,0) be the ideal generated by the
≥ h i ⊂ O O
corresponding homogeneous forms. Finally, let Rad(I) denote the radical of the ideal I.
Proposition 2.1. Under the above assumptions one has:
(1) Foranyk 0theradicalsofidealscoincide: Rad(I (f))=Rad(I (g)). Inparticular,fork >0, /Rad(I (f))
k k k−1
≥ O ⊃
Rad( [f ] )=Rad( [g ] ) /Rad(I (g)).
p+k p+k k−1
h i h i ⊂O
(2) Moreover, regarding the tangent cone, one has f =g , up to multiplication by a constant.
p p
k
(3) For a fixed k > 0, suppose that I (f) is radical for any 0 i k. Then I (f) = I (g) and f =
i k k p+i
≤ ≤
i=0
k P
(a +b (x)g ), where a C∗ and b (x):(Cn,0) (C,0) for all i.
i i p+i i i
∈ →
i=0
P
Proof. (1) Considerthe projectivehypersurfaces PV Pn−1 . Suppose x k PV . By GL(n)action,
{ fp+i ⊂ }0≤i≤k ∈i=∩0 fp+i
onecanassumethatx=[0,...,0,1]. Thus,noneoftheequationsf containsthe monomialxp+i. Inparticular,the
p+i n
diagram Γ intersects the x -axis at a point higher than (p+k). By the assumption on f and g, the same fact holds
f n
forΓ . Thereforex k PV . Hence, (settheoretically) k PV = k PV ,andby Nullstellensatz
g ∈i=∩0 gp+i i=∩0 fp+i red i=∩0 gp+i red
(1) follows. (cid:16) (cid:17) (cid:16) (cid:17)
l l
(2) Write f = fni and g = gmi for the prime decompositions of f and g, where f = g up to scaling,
p p,i p p,i p,i p,i
i=1 i=1
cf. part (1). Let xQ∈ Vfp,i be a geneQric point, so that x is a smooth point of the reduced cone VQfp,i. Apply linear
transformation φ to (Cn,0) in order to get x = [1,0,...,0]. Then the monomial xdeg(fp,i) does not appear in f ,
1 p,i
while for any j = i the monomial xdeg(fp,j) does appear in f . Thus, the number p deg(f )n can be restored
6 1 p,j − p,i p,i
from the Newton diagram Γ by checking the monomial containing the highest power of x . Hence, by equality of
φ(f) 1
the Newton diagrams, one gets n = m . This shows that the scheme structure of the projectivized tangent cone is
i i
also restored from the collection of the Newton diagrams.
(3)Thecoincidenceoftheidealsisprovedbyinduction. First,notethat f = g . Supposethatforall0 i k
p p
h i h i ≤ ≤
the ideals I (f) are radical and I (f)=I (g). Hence, by (1) and induction, I (f)=Rad(I (f))=Rad(I (g))=
i k−1 k−1 k k k
Rad( f ,...,f ,g ). Henceg =αf + β f . Usinggradingonegetsthatαisa(zeroornon-zero)
h p p+k−1 p+ki p+k p+k i<k i p+i
constant. Analyzing both cases, I (f)=I (g) follows. (cid:4)
k k
P
Remark 2.2. The possible attempts to strengthen/generalize the above proposition are obstructed:
(1) It is important to take radicals (for the first statement), as the following example shows. Suppose f = xp
p 1
with p 3 and f contains the monomial xp+1. Then all the relevant coordinate transformations have the form
≥ p+1 2
x x +Q(x ,...,x )withQquadratic,andtheothercoordinates(x ,...,x )aremovedbyalineartransformation
1 1 1 n 1 n
7→
of GL(Cn−1 ) (which preserves the hyperplane x = 0). Write f as f˜ (x ,...,x )+x f˜. Then the possible
x2,...,xn 1 p+1 p+1 2 n 1 p
Newton diagrams fix the homogeneous form f˜ completely, but impose no restrictions on f˜. E.g., f = xp+xp+1
p+1 p 1 2
and g =xp+x xp+xp+1 have equal Newton diagrams in any coordinates, but of course I (f)=I (g).
1 1 2 2 1 6 1
(1’) Similarly, the property /Rad(I (f)) Rad( [f ] ) = Rad( [g ] ) /Rad(I (g)) does not hold
k−1 p+k p+k k−1
O ⊃ h i h i ⊂ O
without the radicals, as it is exemplified by the plane curves defined by the polynomials f = x2y2 +x5 +y5 and
g = x2y2 + x5 y5. Note that the corresponding Newton diagrams coincide in any coordinate system. (By the
−
direct check, a change of coordinates whose linear part is identity has no influence on the diagrams.) But, clearly,
/Rad( f ) [f ] = [g ] /Rad( g ).
4 5 5 4
O h i ⊃h i6 h i⊂O h i
(2) It is not possible to consider the filtration determined by the Newton diagram instead of Taylor expansion.
Indeed, even in the case of a quasi-homogeneous filtration, the lowest order parts do not necessarily coincide. As an
example,considerf(x,y,z)=(z3+x4y5)r+x8r+y10r+f˜andg(x,y,z)=z3r+x4ry5r+x8r+y10r+f˜. Herer 2and
≥
f˜consists of monomials above the hyperplane z + x + y = 1. Note that f,g are Newton non-degenerate and for
3r 8r 10r
any coordinatesystem Γ =Γ . This last statementcan be verifiedas follows. Take a locally analytic transformation
f g
φ of (C3,0). If its linear part mixes the coordinates (i.e. if it is not diagonal or a permutation) then obviously
On some generalizations of Newton non-degeneracy for hypersurface singularities 5
Γφ∗(f) = Γφ∗(g). Hence, one can assume that the linear part is the identity. Analyzing the non-linear parts, all the
relevant cases are z z+ϕ, where ϕ x2,xy,xy2,y2,y3 . By a direct verification, one gets again Γφ∗(f) = Γφ∗(g).
7→ ∈ h i
But the lowest order parts of f,g differ significantly.
(3) (Regarding the last statement of the proposition.) If I is radical then I is such i.e. f is reduced. However,
k 0 p
in general,the fact that f is reduced seems to be not enough to prove that the other intermediate ideals are reduced
p
as well (at least not by general theory of ideals). Below is an example, communicated to me by D. Eisenbud and B.
Ulrich to whom I am very grateful, which shows the subtlety of the problem.
a2 d3 f4
Consider the ideal in C[a,b,c,d,e,f,g] generated by the 3 minors of the matrix . The 3 equations
bc e3 g4
(cid:18) (cid:19)
f ,f ,f have degrees 5,6,7. The total ideal is reduced (this can be checkedusing e.g. [GPS-Singular]). But the ideal
5 6 7
generated by the first two equations is non-reduced, e.g. it contains a2f but not af .
7 7
Note that this example is general, it does not use all the assumptions of the proposition above. In particular the
singularity of this example is non-isolated. It would be interesting to give a counterexample which fit exactly to our
situation.
Corollary 2.3. With the assumptions of proposition 2.1 one has:
(1) Let f = f +f + be the Taylor expansion and suppose that the ideal f ,f is radical. Then g is
p p+k p p+k
··· h i
contact equivalent to f +f +some higher order terms (which might be different from those of f).
p p+k
(2) Letf =f +f +f + with q >k be the Taylorexpansionandsuppose that the ideal f ,f ,f
p p+k p+q p p+k p+q
is radical. Then g is contact··e·quivalent to f +f +λf +some higher order terms, for somh e λ C. i
p p+k p+q
∈
Proof. We prove the second claim, the first is proved similarly. The last part of proposition 2.1 gives: g =
af +bf +cf + fora,b,clocallyanalyticandlocallyinvertible. Hencethereexistsalocallyanalyticcoordinate
p p+k p+q
···
scaling: (x , ,x ) δ(x , ,x )suchthat a = b . Applyitanddividegby a toget: g f +f +λf +...
1 ··· n → 1 ··· n δp δp+k δp ∼ p p+k p+q
(cid:4)
Remark 2.4. Regarding the possible converse of proposition 2.1 we note the following:
p+k p+k
(a)Supposethatthe componentsofthe Taylorexpansion(forsomechoiceofcoordinates)satisfy: f = g (a +
i i i
i=p i=p
α (x)) with a C∗ and α :(Cn,0) (C,0). Then the Newton diagrams (in that fixed coordinatPe systemP) coincide
i i i
∈ →
up to the order p+k (i.e. the parts of Γ ,Γ lying below the hyperplane x =p+k coincide).
f g i i
This follows immediately from the fact that Γ =Γ for a C∗ and Γ =Γ .
Pifi Piaifi i ∈ P Pifi Pifi(1+αi(x))
p+k p+k
On the other hand, the equality f = g (a +α (x)) in some coordinate system does not imply that the
i i i i
i=p i=p
(truncated) Newton diagrams coincidPe for anyPchoice of coordinates. An elementary example is: f = xy +y4 and
g =xy y4.
−
(b)The equalityofradicalidealsdoesnotimply anyrelationofNewtondiagramsorsingularitytypes. Asanexample
consider f =xp+xyp+yq and g =xp+yq (for q >p+1).
3. Topologically Newton non-degenerate hypersurfaces
3.1. Preparations for the definition. Start from the following observation. Let (V ,0) = f = 0 (Cn,0) be
f
a generalized Newton non-degenerate isolated singularity. Fix some coordinates (x ,...,x ). {Let φ (cid:8)} ⊂(Cn,0) be a
1 n
locally analytic coordinate change,such that φ∗(f) is non-degeneratewith respect to its diagramΓφ∗(f). In the space
of all the locally analytic series at the origin (or, in some of its truncations if it is necessary) consider the stratum:
(1) Σ(φ,Γφ∗(f)) :={g ∈C{x} | Γφ∗(g) =Γφ∗(f)}.
Here the closure is taken in the classical topology (for the coefficients of the defining series). This stratum is irre-
ducible. Then for the generic point g Σ the local embedded topological types of (V ,0) and (V ,0) coincide
∈ (φ,Γφ∗(f)) f g
[Kouchnirenko76].
This can be rephrased as follows: any small deformation of f inside Σ is µ constant (cf. 1.4).
(φ,Γφ∗(f)) − §
Recall the notion of Newton weight function [AGLV-book, I.3.8] associated to every commode Newton diagram.
Namely, λ :Rn R is defined uniquely by the conditions: λ (α~x)=αλ (~x) (for any α R ) and λ (Γ)=1.
Γ + → + Γ Γ ∈ + Γ
Given two diagrams we say Γ Γ if λ (Γ ) 1 (or λ (Γ ) 1).
1 ≥ 2 Γ1 2 ≤ Γ2 1 ≥
Suppose a collectionofpairs (φ ,Γ ) is given(with φ (cid:8)(Cn,0)localcoordinatechangesand Γ Newtondiagrams).
i i i i
{ }
6 On some generalizations of Newton non-degeneracy for hypersurface singularities
Definition 3.1. The stratum of hypersurfaces germs, associated with the collection (φ ,Γ ) is the closure of the set
i i
{ }
of all the germs giving the prescribed diagrams in the prescribed coordinates, i.e.
(2) Σ{(φi,Γi)} :={g ∈C{x} | Γφ∗i(g) ≥Γi for all i}.
Lemma3.2. Foranycollection (φ ,Γ ) ,asabove, theassociatedstratumΣ isa(non-trivial)linearsubspace
{ i i i} {(φi,Γi)}
of the space of all locally analytic functions at the origin. In particular it is closed, irreducible and the notion of the
generic point is well defined.
Proof. The condition Γφ∗i(g) ≥ Γi means the absence of some monomials in the Taylor expansion of φ∗i(g). This
says that some directional derivatives of φ∗i(g) vanish: aj1..jn∂yj11..∂yjnn(g◦φi) = 0. Here {yj = φi(xj)} are the new
coordinates. And these conditions are linear in g in any coordinate system. (cid:4)
P
3.2. The main definition.
Definition 3.3. The function f C x is called topologically Newton non-degenerate if there exist a finite number
of coordinate choices (i.e. locally∈ana{lyt}ic φi (cid:8) (Cn,0)) and the Newton diagrams Γi such that Γφ∗(f) = Γi and
{ } { i }
any small deformation of f inside the stratum Σ is µ constant.
{(φi,Γi)} −
Recall that for a locally invertible u C x one has Γ =Γ and (V ,0)=(V ,0). Moreover, f is topologically
uf f uf f
Newton non-degenerate iff uf is such.∈Th{us}the hypersurface germ (V,0) (Cn,0) is defined to be topologically
⊂
Newtonnon-degenerateifone(andhenceany)ofitslocallydefiningfunctionsistopologicallyNewtonnon-degenerate.
The definition3.3is equivalenttothe following: the generalpointofΣ correspondsto ahypersurfacegerm,
{(φi,Γi)}
whose singularity type is that of (V ,0). General here means: lying in the complement of a proper analytic subset.
f
Some comments are in order. Note that the µ constant deformation is equisingular (for n = 3 use the linearity of
−
the space Σ andsee the remarkin 1.4). Thus by semi-continuity of µ we deduce that the subsetof the germs
{(φi,Γi)} §
f Σ with the given topologicaltype is open; and for germs in the complement the Milnor number is strictly
∈ {(φi,Γi)}
larger. So f is topologically Newton non-degenerate iff it belongs to this (Zariski) open set.
Example 3.4. (1) Every generalized Newton non-degenerate germ is topologically Newton non-degenerate. In
this case, by definition, just one pair (φ,Γ) suffices.
k
(2) Let C = C be the tangential decomposition of a plane curve singularity (cf. example 1.1). If each of
α
i=∪1
C is generalized Newton non-degenerate then C is topologically Newton non-degenerate. Indeed, make k
α
choices of coordinates (x(i),x(i)) with xˆ(i) axis generic and xˆ(i) axis chosen such that the germ C is Newton
1 2 2 1 α
non-degenerate. Then get the collection of Newton diagrams similar to those of example 1.2. Obviously, this
collectionof diagrams specifies the singularity type uniquely (the diagramΓ specifies the type of C and the
i α
fact that the tangent line of C is not tangent to any other C .
α j6=i
(3) The curve germ (x2 y3)(x2 y3+x3)=0 is the union of two branches,each being Newton non-degenerate,
− −
but the union is not topologically Newton non-degenerate. It is easy to see that the family C = (x2
t=6 0
{ −
y3)(x2 ty3+x3)=0 has constant Newton diagram for any choice of coordinates. But µ(C )<µ(C ).
t=6 1 1
(4) (Yomdi−n series.) Let }f = f +f + with f generic with respect to f . Namely, PV and PV
p p+k p+k p f p+k
intersect transversally in Pn−1, in partic·u··lar Sing(PV ) PV = ∅. Assume that the ideal f ,f is
f p+k p p+k
radical. Then (V ,0) (Cn,0) is topologically Newton no∩n-degenerate. h i
f
Indeed, if for some⊂g C x the Newton diagrams Γ ,Γ coincide in any coordinates, then by corollary
g f
∈ { }
2.3 g is contact equivalent with a germ of the form f +f +higher order terms. But all these germs are
p p+k
topologically contact equivalent to f +f , hence g and f are topologically equivalent. Finally note that
p p+k
the set of all the coordinate systems can be replaced by a finite set (by the argument as in the proposition
below).
Webelievethatthecondition” f ,f isradical”isunnecessaryhere,butdonothaveanyrigorousproof.
p p+k
h i
Proposition 3.5. (Consistency of the definition.)
(1) Let (V ,0) (Cn,0) be a topologically Newton non-degenerate germ and (φ ,Γ )k a collection of pairs
f ⊂ { i i i=1}
fulfilling the condition of definition 3.3 (i.e. specifying the singularity type of V uniquely). Then for any
f
additional pair (φ ,Γ ) the collection (φ ,Γ )k+1 also fulfills the condition of the definition.
k+1 k+1 { i i i=1}
(2) Let (φ ,Γ ) beaninfinitecollection suchthat theassociated stratumΣ (definedsimilarly tothe
{ i i i}i∈I {(φi,Γi)}i∈I
case of finite collection) is of positive dimension. Suppose that any small deformation of f inside this stratum
is µ constant. Then there exists a finite sub-collection J I such that the conditions of the definition 3.3 are
− ⊂
satisfied for (φ ,Γ ) and hence (V ,0) is topologically Newton non-degenerate.
i i i∈J f
{ }
On some generalizations of Newton non-degeneracy for hypersurface singularities 7
Proof.
(1) By assumption any small deformation of (V ,0) inside the stratum Σ is µ constant, hence this is
f {(φi,Γi)ki=1} −
true for the substratum Σ as well.
{(φi,Γi)ki=+11}
(2) By finite determinacy one can pass to finite jets J of some high order. Let j (g) be the N-jet of g.
N N
Then µ(V ,0)=µ(V ,0) and any small deformation of (V ,0) inside Σ J is µ constant.
f jN(f) jN(f) {(φi,Γi)}i∈I ∩ N −
PresentthestratumΣ J assuccessiveintersections: Σ J ,whereJ I isanincreasing
{(φi,Γi)}i∈I∩ N (φi,Γi)∩ N ⊂
i∈J
filtration of I. T
At each step we get a linear subspace of the finite dimensional space of N-jets. Each intersection either
decreasesthe dimension or has no influence. Therefore the process stabilizes after a finite number of intersec-
tions. Hence, there exists a finite subset J I satisfying the conditions of the definition. (cid:4)
⊂
3.3. The case of curves. ForcurvesitispossibletogiveaveryexplicitequivalentdefinitionoftopologicalNewton-
non-degeneracy.
Proposition 3.6. Let (C,0) = (C ,0) be the tangential decomposition. Then (C,0) is topologically Newton non-
i α
∪
degenerate iff each (C ,0) is topologically Newton non-degenerate. Moreover (C ,0) is topologically Newton non-
α α
degenerate iff the following two conditions are satisfied:
Each branch of C is generalized Newton non-degenerate, i.e. locally it is of the type xp+yq with (p,q)=1.
i
•
In addition, the union of any two singular branches is a generalized Newton non-degenerate singularity. More
•
precisely, there does not exist a pair of singular branches in C with local equation (in some coordinates): (xp+yq+
α
)(xp+yq+ ). Here the dots mean higher order terms (i.e. monomials lying over the Newton diagram of xp+yq).
··· ···
The last condition can be rephrased as follows: for any pair of singular branches C ,C with the Puiseux pairs
i j
(p ,q )and(p ,q )either (p ,q )=(p ,q )or(p ,q )=(p ,q )but the intersectionmultiplicity mult (C C ) p q .
i i j j i i j j i i j j 0 i j i i
6 ∩ ≤
Proof. For the first part note that the collection of φ ,Γ that specifies the type of (C,0) does the same for each
i i
{ }
(C ,0) independently. Conversely, let φ ,Γ be a collection that specifies the type of (C ,0). Then the total
α i α,i α,i α
∪ { }
collection φ ,Γ specifies the type of (C,0).
i,α α,i α,i
∪ { }
Thus, in the sequel we assume that all the branches of (C,0) have a common tangent.
⇛ Suppose (C,0) contains a branch (C ,0) which is not generalized Newton non-degenerate. Choose coordinates in
i
which the defining equation of C can be written as: (xar + +ybr)+h = 0. Here 1 < a < b, (a,b) = 1, the part
i
···
(xar+ +ybr) is quasi-homogeneousand degenerate andh is of higher order with respect to the weights above. Let
···
(C′,0) be a generic germ with the Newton diagram of C , in particular C′ is Newton non-degenerate. Note, that the
i i i
Newton diagrams of C ,C′ coincide in any coordinates.
α i
ConsiderthegermC′ = C C′. BytheconstructionC,C′ havedifferentsingularitytypes,buttheirdiagrams
j6=i j∪ i
coincide in all coordinates.
S
The same applies to the case of (xp+yq + )(xp+yq + ): no choice of coordinates can distinguish this from
··· ···
the pair of branches (xp+yq+ )(xp yq+ ) (which is certainly of different type).
··· − ···
⇚ We want to determine the singularity type of C by choosing different coordinates. For each (generalized Newton
non-degenerate) branch C let (p ,q ) be its Puiseux pair (for smooth branches take (1, )). Let φ be a choice of
i i i i
coordinatesforwhichC isdefinedbyxp+yq+ =0. (Thesmoothbranchesarerectified∞,i.e. the equationbecomes
i i i ···
x=0).
LetΓ be the Newtondiagramof(C,0)in the coordinatesdefined byφ . ThenΓ containsasegmentofslopep /q
i i i i i
(for smooth branches the Newton diagram is non-commode). Then the collection φ ,Γ r , for r the number of
{ i i}i=1 −
branches, suffices to determine the singularity type of (C,0).
Indeed, let C(t) Σ be a small deformation. By construction, after the change of coordinates φ , the
⊂ {φi,Γi}i=1..r i
family contains (as a subvariety) the family defined by (xpimα1(t)+ +yqimαm+1(t))+ = 0. Here inside the
··· ···
terms brackets correspond to a quasi-homogeneousform, the dots outside correspond to the higher order monomials.
Byconstruction,the quasi-homogeneousformxpimα1(0)+ +yqimαm+1(0)isnon-degeneratewithα1(0)=0,hence
··· 6
the same holds for t small enough.
Thus, the family C(t) can be decomposed as C (t) and the type of each branch is preserved. (Moreover, all the
i
∪
smooth branches of C(0) stay constant.) Finally the intersection degrees of the branches are constant (fixed by the
set φ ,Γ ). (cid:4)
i i
{ }
The proof of the proposition gives an upper bound for the number of the needed choices of coordinate system to
restore the singularity type: the number of branches. This bound can be improved (e.g. in each coordinate system
one takes both axes tangent to some branches). But, e.g. for r singular pairwise non-tangent branches one certainly
needs at least r/2 coordinate choices.
8 On some generalizations of Newton non-degeneracy for hypersurface singularities
The proposition also allows to answer positively the question from the introduction: for curves being generalized
Newton non-degenerate or topologically Newton non-degenerate are properties of singularity types and not only of
their representatives.
Corollary3.7. Let(C,0) (C2,0)beageneralizedNewtonnon-degenerate(ortopologicallyNewtonnon-degenerate)
germ of curve. Let (C′,0) ⊂(C2,0) be a germ of the same singularity type as (C,0). Then (C′,0) is also generalized
⊂
Newton non-degenerate (or topologically Newton non-degenerate).
Proof. For the topologicallyNewton non-degeneratecase the statement follows immediately from proposition 3.6
•
(as the conditions are formulated in terms of the singularity types of the branches). Next we consider the other case.
The topological characterization of generalized Newton non-degenerate curve-germs is well known in the folklore,
•
but we could not find any reference, except for the preprint [GaLePl 07].
Another way to prove the statement is as follows. By [GLS-book, Proposition 2.17(2), pg. 287] the miniversal
equisingulardeformationofany(isolated)Newtonnon-degenerateplanecurvesingularitycanberealizedbymonomials
not below the Newton diagram, hence consists of Newton non-degenerate germs. Let S be the singularity type of the
Newton non-degenerate germ (C,0), recall that the equisingular stratum ΣS of germs of the singularity type S is
globally irreducible. Thus we get: there exists a (Zariski) open dense subset of ΣS, whose points correspond to
generalized Newton non-degenerate curve-germs.
On the other hand, for (C,0) = f−1(0), let Γ be the Newton diagram in the fixed coordinates (so that f is Γ
f f
non-degenerate). Consider the stratum of curves that (in some coordinates) can be brought to Γ or to a bigger
f
diagram:
(3) ΣΓf :={g ∈C{x}| for some change of coordinates φ(cid:8)(C2,0) Γφ∗(g) ≥Γf}
By definition the stratum is closed, its points correspond either to generalized Newton non-degenerate curve-germs
of type S or to higher types adjacent to S. The natural morphism ΣΓf → ΣS is defined by g → (g−1(0),0). By the
remark above this morphism is dominant, hence in fact is surjective. Which means: every point of ΣS corresponds to
a generalized Newton non-degenerate curve germ. (cid:4)
3.4. Germs vs types. In this subsection we discuss the questions of 1.3 in higher dimensional cases.
§
For n 3, being topologicallyNewton non-degenerateor generalizedNewton non-degenerateis a property of germ
≥
representatives (or of analytic singularity types) but not of the topological types. We construct equisingular families
of surfaces (V ,0) (C3,0) such that V is Newton non-degenerate in the classical sense but V is degenerate (in
t 0 t=6 0
⊂
various senses).
Remark 3.8. Two observations are useful. Let (V ,0) (C3,0) with fixed coordinates, such that f is non-degenerate
f
for Γ . So, the coordinates in P2 =Proj(C3) are fixed⊂too.
f
Suppose the projectivizedtangentconePT P2 isirreducible,reduced(hence withisolatedsingularities). Then
• (Vf,0)⊂
itssingularlocusliesin[1,0,0],[0,1,0],[0,0,1] P2. InparticularPT P2 canhaveatmostthreesingularpoints.
Let pt Sing(PT ) and suppose the (plan∈e curve)singularity (P(TVf,0)⊂,pt) is not an ordinarymultiple point. Let
(V,0) (V,0)
T• = lp1∈ lpk = 0 be the tangent cone of (PT ,pt), assume p > 1. Then (as f is Newton non-degenerate) l
{ 1 ··· k } (V,0) 1 1
coincides with one of the coordinate axes.
Theexamplesbelowarebasedontwoideas: movingseveralmildsingularitiesofthe tangentcone(forthe caseofgen-
eralizedNewtonnon-degenerate)ordeformingone strongsingularityofthe tangentcone (for the caseof topologically
Newton non-degenerate).
Example 3.9. Consider a super-isolatedsingularity [Luengo87] V = f +f =0 (C3,0). Here f is generic
0 p p+1 p+1
and the projective curve f = 0 P2 has three cusps (assume p {is big enough,}e.⊂g. p 6). According to the
p
remark above, arrange the{coordin}at⊂es such that the cusps are at [1,0,0], [0,1,0], [0,0,1] P≥2 (corresponding to the
coordinate axes xˆ, yˆand zˆof C3). ∈
Note thata GL(3) actionwhichfixes this points is atmosta permutation. To make V Newtonnon-degenerateas-
0
sume that the tangents to the cusps are oriented along the coordinate axes, e.g. f = zp−3(zx2+y3)+xp−3(xy2+
p
z3)+yp−3(yz2+x3).
Let V be the equi-singular family, with the cusps staying at their points xˆ,yˆ,zˆ, but their tangents changing freely.
t
For example, f(t)=f (t)+f with f (t)=zp−3(z(x ty)2+y3)+xp−3(x(y+tz)2+z3)+yp−3(y(z tx)2+x3).
p p+1 p
− −
Then V is topologically Newton non-degenerate but not generalized Newton non-degenerate. Indeed, to bring
t6=0
V to a Newton non-degenerate form one should keep the cusps at the points xˆ,yˆ,zˆ and at the same time keep
t6=0
On some generalizations of Newton non-degeneracy for hypersurface singularities 9
their tangents along the axes. So, only GL(3) transformations are relevant. But, as was noted above, the only GL(3)
transformations which keep the cusps at the points xˆ,yˆ,zˆare permutations.
Example 3.10. (cf. also [Altmann91, example 5.3]) Consider the family of surfaces f = f (t)+f = x5+z(zx+
t 5 6
ty2)2+y5+z6. This is a super-isolated singularity because Sing(f =0) (f =0)=∅. The projectivized tangent
5 6
coneofthesesurfaceistheplanequintic f (t)=0 P2withoneA point∩at[0,0,1] P2. Thusµ=68=(5 1)3+4
5 4
{ }⊂ ∈ −
(see 5.1 for the general formula). The family is equisingular in t, e.g. because each surface V is resolved by one
t
§
blowup of the origin and the singularity type of exceptional divisor in the surface is independent of t.
The singularityV isNewtonnon-degenerate(bydirectcheck). Onthe otherhandfort=0the singularityisnot
t=0
6
generalizedNewtonnon-degenerate. Toshowthis,weprovethattherestrictionoff tothefaceΓ Span(x5,y5,z5)is
degenerateforanychoiceofcoordinates. Letφ(cid:8)(C3,0)bealocalcoordinatechantge. Aswearefitn∩terestedintheface
whose monomials correspond to the tangent cone, the non-linear part of φ is irrelevant. So, assume φ GL(C3) and
actsonPT = f =0 P2. ThusthegoalistobringthesingularityofthisquintictoaNewtonnon-de∈generateform.
S 5
{ }⊂
But this is impossible for t =0, because in local coordinates the curve is defined by (x+ty2)2+x5+y5 = 0. So, to
bring it to a Newtonnon-de6generateformwe must do a non-linear transformationx x ty2 onC2, which does not
arise from GL(3)(cid:8)(C3,0). → −
Example 3.11. Equisingular deformation to essentially Newton-degenerate singularities. In the last example all the
fibres are topologically Newton non-degenerate (by corollary 4). The next example, in which the generic fibre is
essentially Newton-degenerate (i.e. not topologically Newton non-degenerate), is a simple alteration. The goal is to
construct a Newton diagram with the properties: any linear change of coordinates erases the essential information of
the singularity, while any coordinate change whose linear part is the identity preserves the diagram.
For this, one changes the inclinations of the face on which the degeneration occurs (Span(x5,z3x2,zy4,y5) in the
last example) and adds some other faces.
v•NCC((yb54eoeox,)+xwnn5aits,vtkzyia6o(d(ny,n1ce1dbor−,)dnzxitdaco2h4bignzet)lrikiee+oa+hnmah2yxs,bap:(ycsoe2bo4vrzntse−hkusti)ers1hfat−aekpsnc4brpdeoe)lfvaC=fnitoohe1un=e,Ssvts(pehoxxaxra2nteazh(e+mkxe+fp2ayc2azlocb,e(keny.+2+)sd42zi(,tzkcyiScf,o14.uznz+p−cskt)ph,.za4obyerkTsb)e(ep):hz>.ixic(sTta1+uc,harabeen,ny)ced2b:,q)ek2uCc.)ae(ont1anis(oruvFner(oxe4orsdafu),(bct>xahhy2,i1bzsnt,.kehpc+xal,a2ttk,nc)yetobhn=i)es-, x......•............................................2....................x..............................................z......................................a........................................k.................................................................+...........................................................................................2...........................................................•..........................................................................................................................................................................................................................................•...................................................................................................................................................................................................................................................................................................................................z.z..................................................................................................c..c....................................................................................................................................................................................•.................................................................................................................................................y.......................................................................4..........................................................z.....................................................................k................................................•................y................b.....
b − k k − b
zc and y4zk lie above the plane Span(xa,x2zk+2,yb). The equation of the plane is x + y + 1−a2z = 1, so the
• a b k+2
conditions are
k+2 k k+2
(5) c> and > .
1 2 1 4 1 2
− a − b − a
xa and yb lie above the plane Span(zc,x2zk+2,y4zk). The equation of the plane is z +x1−k+c2 +y1−kc =1 giving:
• c 2 4
k+2 b k
(6) c> and (1 )>1.
1 2 4 − c
− a
Assume further that b<a<k+4<c and also: if φ(cid:8)(C2,0) is any locally analytic transformation whose linear
part is identity (i.e. (x,y,z) (x+φx,y+φy,z+φz) with φi m2) then Γf =Γφ∗(f). This can be achieved e.g. if
→ ∈
for each face the angles with all the coordinate planes are bounded 1 <tan(α)<2.
2
For this it is enough to assume: zc−1 lies below the plane Span(xa,x2zk+2,yb).
Summarizing, all the restrictions above are implied by the following inequalities:
k+2 k
(7) b< < <min(c,2b), b<a<min(2b,k+4)<c<k+6.
1 2 1 4
− a − b
This implies c = k+5 and k > 10. We consider (possibly the simplest case): f = x14+y13+z16+z11(zx+ty2)2.
t
By direct check this family is equisingular (e.g. µ =2220=const, can be calculated using [GPS-Singular]).
t
The generic fibre f−1(0) is essentially Newton-degenerate by the following proposition.
t
Lemma 3.12. Let f as above and g =x14+y13+z16+z11(z2x2+y4), Newton non-degenerate. Then f and g
t=6 0 t=6 0
have thesame Newton diagram in any coordinate system. But µ(g)=2219<µ(f )=2220,so f is not topologically
t t=6 0
Newton non-degenerate.
10 On some generalizations of Newton non-degeneracy for hypersurface singularities
Proof. Letφ(cid:8)(C2,0)be a locallyanalytic changeofcoordinateswhose linearpartis identity. By the construction
it preserves the Newton diagram. Therefore it is enough to consider only linear coordinate changes. But then only
the monomials xa,yb are relevant and their coefficients are the same in both cases. (cid:4)
Remark 3.13.
The families in the examples above are not just µ constant but µ∗ constant (by direct computation). In fact, one
• − −
can show that in these families all the polar multiplicities are preserved, i.e. the surfaces are c-cosecant in the sense
of [Teissier77].
In the examples above the singular types admit some Newton non-degenerate representatives, but the generic
•
representatives are not generalized Newton non-degenerate. More precisely, consider the equisingular stratum in
the space of miniversal deformation. The locus of generalized Newton non-degenerate surface-germs is of positive
codimension.
4. Directionally Newton non-degenerate singularities
Aswasshownabove,thetopologicallyNewtonnon-degenerategermsformquiteabroadclass,difficulttoworkwith.
Onecouldconsiderotherintermediateclassesofsingularities(betweengeneralizedNewtonnon-degenerateandtopolog-
icallyNewtonnon-degenerate). Westudyheretheminimalgeneralization: directionallyNewtonnon-degenerategerms.
These are higher dimensional analogs of example 1.2.
For a fixed diagram Γ define the subset ∆ Γ of faces ”far from the xˆ axis” as follows. Let xˆ be the unit
n n
⊂
normal to the hyperplane Span(x , ,x ). For each top-dimensional face σ Γ let Span(σ) be the supporting
1 n−1
··· ⊂
hyperplane. Let vˆ be the unit normal to Span(σ), oriented such that (xˆ ,vˆ )>0.
σ n σ
We define:
σ Γ 1
(8) ∆:= ⊂ (xˆn,vˆσ) .
∪ top-dimensional| ≥ √n
n o
Since the union is over the top dimensional faces, ∆ can be empty. Here the topological closure is needed to add the
relevant non-top-dimensional faces.
Example 4.1. In the case of plane curves, with Γ R2 , one gets that ∆ is the union of all the faces whose angle
⊂ ≥0
with the xˆ is not bigger than π/4.
1
For the Newton diagram of the surface x x x +xp+xq +xr with p,q,r 3 one has: ∆=Conv(x x x ,xp,xq).
For the Newton diagram of the surface xN1 +2 x3p+x1p wi2th N3>p one has:≥∆=∅. 1 2 3 1 2
3 2 3
In the sequel we assume that the projectivizedtangent cone PT has only isolatedsingularities,in particularit
(Vf,0)
is reduced.
Definition 4.2. The isolated hypersurface singularity (V ,0) (Cn,0) is directionally Newton non-degenerate if for
f
each singular point x Sing(PT ) there exists a coordinat⊂e-change φ(cid:8)(Cn,0) such that φ(x) =[0, ,0,1] and
∈ (Vf,0) ···
the restriction φ∗(f) is non-degenerate.
|Γ\∆
generalized
directionally topologically
Lemma 4.3. ( Newton-non-degenerate )⇛( ) ⇛( ).
Newton-non-degenerate Newton-non-degenerate
with Sing(PT ) isolated
(Vf,0)
Proof. The first implication is obtained as follows. Suppose f is Newton non-degenerate in some fixed coordinates
and pt Sing(PT ) Pn−1. Let p = mult(V ,0). By the non-degeneracy on the face Γ Span(xp, ,xp)
∈ (Vf,0) ⊂ f f ∩ 1 ··· n
one gets that pt (C∗)n−1, so by a permutation of coordinates we can assume: pt = [0, , , ] Pn−1. Now
6∈ ∗ ··· ∗ ∈
consider the restriction of f to the boundary ∂ Γ Span(xp, ,xp) . Checking the non-degeneracy on the top-
f ∩ 1 ··· n
dimensional components one has (after a permu(cid:16)tation) pt=[0,0, , ,(cid:17)] Pn−1. After several similar steps we get
∗ ··· ∗ ∈
that pt=[0,0, ,0,1].
···
Forthesecondimplicationsuppose(V ,0)isdirectionallyNewtonnon-degenerate. Considerthecorrespondingstratum
f
(9) Σ = g Γ =Γ in all coordinate systems .
f g f
{ | }
Firstweprovethatanyfamily(V(t),0) Σ fortsmallenoughconsistsofdirectionallyNewtonnon-degeneratehy-
f
• ⊂
persurfaces. Recall that the tangent cone is constant in this family (cf. proposition 2.1). For any point pt
Sing(PT )=Sing(PT ) choose coordinates with pt=[0,..,0,1] and f(0) non-degenerate. ∈
V(0) V(t) |Γ\∆