Characterization of hypersurface singularities in positive characteristic

The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized by M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 61(109) (2018), no. 3, 333–343] for the case when the corank of hypersurface singularities is ≤ 2. The aim of this article is to characterize the right unimodal and bimodal hypersurface singularities of corank 3 in an algorithmic way by means of easily computable invariants such as the multiplicity, the Milnor number of the given equation, and its blowing-up. On the basis of this characterization we implement an algorithm to compute the type of the right unimodal and bimodal hypersurface singularities without computing the normal form in the computer algebra system Singular.


Introduction
Let K[[x 1 , . . . , x n ]] be the local ring of formal power series in n variables, m its maximal ideal, K an algebraically closed field of characteristic p > 0, and R = Aut K (K[[x 1 , . . . , x n ]]), the set of all K-automorphisms of K[[x 1 , . . . , x n ]]. Let f and g ∈ m. f is said to be right equivalent to g, f ∼ r g, if there exists an automorphism φ ∈ R such that φ(f ) = g. In case of two (resp. three) variables, we will later use K [[x, y]] (resp. K [[x, y, z] Arnold introduced in the seventies [2,3,4] the notion of modality in singularity theory for real and complex singularities. He classified simple, unimodal, and bimodal hypersurface singularities with respect to right equivalence. These are also classifications with respect to contact equivalence. Simple hypersurface singularities in charateristic p > 0 were classified by Greuel and Kröning [7] with respect to contact equivalence. Greuel and Nguyen [8] classified the simple hypersurface singularities in characteristic p > 0 with respect to right equivalence. These classifications are characterized in [1].
Nguyen gave the classification of right unimodal and bimodal hypersurface singularities in positive characteristic [10]. In this article we use the results of [10] in order to characterize this classification for unimodal and bimodal hypersurface singularities of corank 3 in terms of certain invariants. We use the names of the singularities from [10], where normal forms are given. In some cases we use blowing-up as a tool to differentiate certain types. We use the right-modality as defined in [8] and used in [10]: , λ in some Zariski neighbourhood of 0 ∈ K N , falls into finitely many families of right equivalence classes depending on r parameters, then f is called right r-modal. If r = 1 (resp. r = 2), then f is called unimodal (resp. bimodal).

Type
Normal form modality x r + y s + z t + axyz 1 U 12 x 3 + xz 2 + y 4 + axyz 2 1 In this section we characterize all right unimodal and bimodal hypersurface singularities of corank 3 in terms of multiplicity, Milnor number, and blowing-ups. We have only to consider the cases m(f ) = 2 and m(f ) = 3, since f is not uni or bimodal if m(f ) ≥ 4 or corank(f ) > 3 (see Theorems 120 and 121 in [10]). Moreover, we only need to consider p ≥ 5, because for p = 2 and p = 3 there is no unimodal or bimodal hypersurface singularity of corank 3 (see Theorem 136 and Theorems 147 to 151 in [10]).  Proof. The result follows from Theorems 74 to 119 in [10].  [10], it follows that if j 3 (f ) ∼ r x 2 y or j 3 (f ) ∼ r x 3 then rmod(f ) ≥ 3. Therefore we only need to discuss the rest of the seven possibilities (I-VII) of j 3 (f ).
Then q, r, and s can be computed by using blowing-ups.
Proof. Consider the blowing-up in the first chart defined by x → x, y → xy, and z → xz; then we have The strict transform is g(x, xy, xz) Then obviously the Milnor number of g(x,xy,xz) Similarly we obtain r − 4 and s − 4 from the other charts.

) is the intersection of three planes and has as singular locus the union of three lines
Moreover, the triple (q, r, s) modulo permutation is an invariant under right equivalence for this type of singularities.
Proof. If the zero-set V (j 3 (f )) is the intersection of three planes * and has as singular locus the union of three lines To prove the statement we use induction on m. We write g = xyz +α 1 (x)+β 1 (y)+ To prove the invariance of (q, r, s) modulo permutation, we assume that f = xyz + terms of order ≥ 4 ∼ r g = xyz + terms of order ≥ 4, i.e. there exists an such that ϕ(f ) = g. This implies that ϕ(xyz) = xyz mod (x, y, z) 4 , i.e. the linear part of ϕ is a permutation of the variables. Since we have to prove the invariance of (q, r, s) modulo permutation, we may assume that ϕ = id mod (x, y, z) 4 . This implies that in the charts the Milnor number of the blowing-up of f and g is the same. Using Lemma 2.4, the result follows.

the intersection of three planes and has as singular locus the intersection of three lines
Proof. If the zero-set V (j 3 (f )) is the intersection of three planes and has as a singular locus the intersection of three lines with q ≤ r ≤ s. Moreover, by using Lemma 2.4 we can compute q, r, s and hence the type of f .
is the intersection of a plane and a node and has as singular locus the union of the lines V (x, y) ∪ V (x, z) and the point V (x, y, z) as embedded point, Proof. Since the zero-set V (j 3 (f )) is the intersection of a plane and a node and has as singular locus the union of the lines V (x, y) ∪ V (x, z) and the point V (x, y, z) as embedded point, we can assume that f Similarly to the previous case, we use induction on m.

Now by using the transformation
x → x, y → y − h 1 , z → h 2 we get f ∼ r xyz + x 3 + β 1 (y) + γ 1 (z) mod m m+1 . Using induction we obtain The proof of invariance of (r, s) is similar to the corresponding proof of Lemma 2.5.
where (r, s) is defined as in Lemma 2.7.
Proof. Since the zero-set V (j 3 (f )) is the intersection of a plane and a node and has as singular locus the union of the lines V (x, y) ∪ V (x, z) and the point V (x, y, z) as embedded point, we have by Lemma 2.7 that with 4 ≤ r ≤ s. Moreover, by using Lemma 2.8 we can compute r, s and hence the type of f . Proof. Since the zero-set V (j 3 (f )) is irreducible and has in the singular locus a fat point of multiplicity 6, we have f ∼ r g = xyz + x 3 + y 3 + i+j+k≥4 a i,j,k x i y j z k . We write g = xyz + x 3 + y 3 + α(x) + β(y) + γ(z) + g 0 (x, y, z) mod m m+1 , where m(α), m(β), m(γ) ≥ 4 and g 0 is a homogeneous polynomial of degree m ≥ 4 with g 0 (x, 0, 0) = g 0 (0, 0, z) = g 0 (0, y, 0) = 0.
After using the transformation Lemma 2.11. Let g = xyz + x 3 + y 3 + v≥s c v z v . Then s can be computed by using blowing-ups.
Proof. The proof is similar to that of Lemma 2.4. Proof. Since the zero-set V (j 3 (f )) is irreducible and has in the singular locus a fat point of multiplicity 6, we have by Lemma 2.7

Proposition 2.12. Let f ∈ K[[x, y, z]] be such that m(f ) = 3. If the zero-set V (j 3 (f )) is irreducible and has in the singular locus a fat point of multiplicity
Moreover, by using the Lemma 2.11 we can compute s and hence the type of f .
Proof. The result follows from step 75 in the singularity determinator of [10].  (2) the zero-set V (j 3 (f )) is the intersection of a plane and a node and has as singular locus the line V (x, z) and j x 2 y,yz 2 ,y 5 (f ) = x 2 z + yz 2 + x 2 y 2 and µ(f ) ≥ p + 9; (3) the zero-set V (j 3 (f )) is the intersection of a plane and a node and has as singular locus the line V (x, z) and j x 2 y,yz 2 ,y 5 (f ) = x 2 z + yz 2 + zy 3 and µ(f ) ≥ p + 9; (4) the zero-set V (j 3 (f )) is the intersection of a plane and a node and has as singular locus the line V (x, z) and j x 2 y,yz 2 ,y 6 (f ) = x 2 z + yz 2 ; (5) the zero-set V (j 3 (f )) is the intersection of three planes and has as singular locus the line V (x, z) and j x 3 ,z 3 ,xy 3 (f ) = x 3 + xz 2 + xy 3 and µ(f ) ≥ p + 13; (6) the zero-set V (j 3 (f )) is the intersection of three planes and has as singular locus the line V (x, z) and j x 3 ,z 3 ,xy 3 (f ) = x 3 + xz 2 ; (7) j 3 (f ) has two linear factors, one of multiplicity 1 and one of multiplicity 2; (8) j 3 (f ) has only one linear factor of multiplicity 3.

Proof.
(1) is a consequence of step 90; (2) is a consequence of step 99; (3) is a consequence of step 101; (4) is a consequence of step 105; (5) is a consequence of step 113; (6) is a consequence of step 117; (7) is a consequence of step 118; and (8) is a consequence of step 119 in the singularity determinator of [10].
Notation. Consider the charts of the blowing-up given by 1 : x → x, y → xy, z → xz, 2 : x → xy, y → y, z → yz, By j k (f i,n ) we denote the k-th jet of the strict transformation of the n-th blowingup corresponding to the i-th chart.
Proof. We may assume that f = x 3 + yz 2 + i+j+k≥4 a i,j,k x i y j z k . The analysis of Theorem 81 to Theorem 89 in [10] gives that f is unimodal of type These three cases can be differentiated by using blowing-ups as follows.
If µ(f ) = 16, then if V (j 2 (f 2,2 )) is the intersection of two planes then f is of type Q 16 and if V (j 2 (f 2,2 )) is a double plane then f is of type Q 2,2 . If µ(f ) = 17, then if f 2,3 is a smooth curve then f is of type Q 17 and if V (j 2 (f 2,3 )) is the intersection of two planes then f is of type Q 2,3 . If µ(f ) = 18, then if f 2,3 is a smooth curve then f is of type Q 18 and if V (j 2 (f 2,3 )) is a double plane then f is of type Q 2,4 .