Milnor number and Unfolding
µ-constant stratum and modality
Milnor number, Unfolding and Modality of
Isolated Hypersurface Singularities in
Positive Characteristic
Nguyen Hong Duc
June 26, 2015
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Setting
Hypersurface singularity: f ∈ K [[x]] := K [[x1 , . . . , xn ]],
K = K̄ , char(K ) = p ≥ 0.
isolated ⇐⇒ µ(f ) < ∞.
Right equivalence:
f ∼r g ⇐⇒ ∃φ ∈ Aut(K [[x]]), f = φ(g ).
Unfolding (or, deform. w. section) of f over variety T , t0 :
ht := H(x, t) ∈ O(T )[[x]] s.t.
H(x, t) − f ∈ mT ,t0 OT ,t0 and H(x, t) ∈ hxi.
Complete unfolding: H(x, t) over T , t0 is complete if any
unfolding G (x, s) over S, s0 is “isomorphic” to a pullback
of H after passing to some étale neighbourhood.
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Setting
Semiuniversal unfolding of f over AN , 0:
X
fλ := F (x, λ) = f (x) +
λi ϕi ,
i
where {ϕi }Ni=1 is a basis of m/m · j(f ). Here m ⊂ K [[x]]
maximal ideal and j(f ) jacobian ideal.
([Greuel-N 2014]): Semiuniversal unfolding is complete.
For each unfolding ht := H(x, t), by theorems on
determinacy (cf. [Boubakri, Greuel, Markwig, 2012]) we
may assume, for fixed t, that ht polynomial and defines
ht : An → A, x 7→ H(x, t)
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Semicontinuity of Milnor number
ht : An → A, x 7→ H(x, t)
Theorem
∃U = U(0) ⊂ An , V = V (t0 ) ⊂ T , W = W (0) ⊂ A, s.t.
(1) 0 ∈ U is the only singular point of f = h0 : U → W , and
ht has only isolated singular points in U for all t ∈ V .
(2) One has
X
µ(f ) =
µ(ht , x)
x∈Sing(ht )
In particular, for each t ∈ V , µ(f ) = µ(h0 ) ≥ µ(ht ).
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Singularities in positive characteristic
Remark
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Singularities in positive characteristic
Remark
Deformation of formal power series over variety.
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Singularities in positive characteristic
Remark
Deformation of formal power series over variety.
Zariski topology
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Singularities in positive characteristic
Remark
Deformation of formal power series over variety.
Zariski topology
étale topology
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Singularities in positive characteristic
Remark
Deformation of formal power series over variety.
Zariski topology
étale topology
Example: ϕ : X , x0 → Y , y0 quasi-finite.
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Parity of Milnor number
Corollary
Let f ∈ K [[x1 , . . . , xn ]] with char(K ) = 2 and n odd, be an
isolated singularity. Then µ(f ) is even.
Proof.
Consider the semiuniversal deformation fλ and apply the
theorem to get open neighbourhoods U, V , W .
Show that, ∃λ ∈ V s.t. fλ has only ordinary quadratic
singularities.
P
µ(f ) = x∈Sing(fλ ) µ(fλ , x).
Suffices to show: µ(fλ , x) is even for all x ∈ Sing(fλ ).
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Openness of completeness of unfoldings
Theorem
{t1 ∈ V : H(x, t) is a complete unfolding of f over T , t1 }
is open in V .
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
µ-constant stratum
Theorem
Let fλ (x) be the semi-universal unfolding of f over AN , 0,
∆µ := {λ ∈ AN : µ(fλ ) = µ(f )} the µ-constant stratum of f .
Then the restriction of F on ∆µ is a modular unfolding.
That is, for any λ in some étale neighbourhood U ⊂ ∆µ of 0
there exist only finitely many λ0 ∈ U such that F (x, λ) is right
equivalent to F (x, λ0 ).
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
µ-constant stratum and modality
Corollary
µ-constant stratum is a coarse moduli space for isolated
singularities with constant Milnor number (modulo a finite
group action) w.r.t. right equivalence.
Corollary
rmod(f ) ≥ dim ∆µ .
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
µ-constant stratum and modality
Conjecture
rmod(f ) = dim ∆µ .
Partial results:
Univariate power series [N, 2014]:
rmod(f ) = [µ/p] = dim ∆µ .
Simple, unimodal and bimodal singularities [N, 2015]
using classification of: simple sing. by Greuel-N 2014,
unimodal and bimodal sing. by N. 2015.
Complex analytic case, by Gabrielov, 1974 using
Picarrd-Leftschetz theory.
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Happy Birthday! Gert-Martin
Wishing you hapiness, health and all the best
in your 70s years!
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic
Milnor number and Unfolding
µ-constant stratum and modality
Thanks!
Nguyen Hong Duc
Isolated Hypersurface Singularities in Positive Characteristic