The weighted homogenous case

The aim of this section is to describe more in detail the case of germs of free divisors defined by a weighted homogeneous equation. We show that in this case, the space of infinitesimal admissible deformations is finite dimensional and that, in the case of curves, its dimension depends only on the weights and the degree of the defining equation.

Proposition 3.7.1. Let (D,0) ⊂ (Cn,0) be a germ of a free divisor defined by a

weighted homogeneous polynomial of degreek. Then an element of F T1(D) can be represented by f0 ∈C[x1, . . . , xn]k, where C[x1, . . . , xn]k is the space of polynomials

of weighted degreek.

Proof. Letf be a defining equation for (D,0). Becausef is weighted homogeneous, then there existsχ∈Der(−logD) such thatχ(f) =f.

Consider (X, x) an infinitesimal admissible deformation of (D,0). By Remark 3.2.4, we can supposed it is defined by the equationf+·f0, wheref0 ∈ O_Cn_,0. Let us

suppose thatf0 is weighted homogeneous of degreeβ. Because (X, x) is admissible, thenχlifts. This means that there existsχ0 ∈Der_Cnsuch that (χ+·χ0)(f+·f0) =

(1 +·α)(f +·f0) and so χ0(f) +χ(f0) = αf+f0, where α ∈ O_Cn_,0. Because

f0 is weighted homogeneous of degree β, then χ(f0) = βf0. Hence, the previous expression becomes (χ0 −α)f = (1−β)f0. Because f is weighted homogenous, (χ0−α)f is a combination of the partial derivative off and so (1−β)f0 is in the

Jacobian ideal ofD. Iff0 is in the Jacobian ideal, then the admissible deformation is trivial, by [30], Chapter II, 1.4. Otherwise β = 1 and so f0 is of weighted degree

kasf.

If f0 is not weighted homogeneous, we can apply the previous argument to each of its weighted homogeneous parts.

Notice that the previous result highlights another difference between the theory of admissible deformations and the classical deformation theory of singularities. In the latter, the weighted homogeneity is not preserved under deformations like in the case of admissible deformations.

Corollary 3.7.2. Let (D,0) ⊂ (Cn,0) be a germ of a free divisor defined by a

weighted homogeneous polynomial of degree k. Then

dim_CF T1(D)≤dim_CC[x1, . . . , xn]k/J(D)∩C[x1, . . . , xn]k,

where J(D) is the Jacobian ideal ofD.

Proof. Because (D,0) is defined by a weighted homogeneous polynomial, then a basis ofF T1(D) can be chosen to be made of monomials. Then it is a consequence of Proposition 3.7.1 and thatJ(D) defines only trivial deformations.

Corollary 3.7.3. Let (D,0) ⊂ (Cn,0) be a germ of a free divisor defined by a

weighted homogeneous polynomial. ThenFDD has a hull.

Proof. By Corollary 3.7.2, condition (H3) from Definition A.1.13 is satisfied. Then the result follows from Theorem 3.1.11 and Theorem A.1.17.

In Corollary 3.5.3, we have seen that for a germ of a linear free divisor LFDD

has a hull. Because each germ of a linear free divisor (D,0)⊂(Cn,0) is defined by a homogeneous equation of degree n, then by the previous Corollary, we also have that

Corollary 3.7.4. Let (D,0) ⊂ (Cn,0) be a germ of a linear free divisor. Then

FDD has a hull.

In the case of plane curves, we can be more precise. In fact we have the following:

Theorem 3.7.5. Let (D,0) ⊂ (C2,0) be a germ of a free divisor defined by a weighted homogeneous polynomial of degree k. Then F T1_(D) ∼_{= C[x, y]}

k/J(D)∩

Proof. Letf be a defining equation for (D,0). Becausef is weighted homogeneous, then there exists χ ∈ Der_Cn such that χ(f) = f. Consider δ = ∂f /∂x∂/∂y −

∂f /∂y∂/∂x. Because (D,0) has an isolated singularity, then δ, χ form a basis of Der(−logD).

By Proposition 3.7.1, we know that if (X, x) is an infinitesimal admissible defor- mation of (D,0) defined byf +·f0, thenf0∈_C[x, y]k.

On the other hand, if f0 ∈ _C[x, y]k and we consider (X,0) defined by f +·

f0 = F, then it is an infinitesimal admissible deformation because both δ and χ

lift. In fact, we can consider δ0 = ∂F/∂x∂/∂y−∂F/∂y∂/∂x and χ as elements of Der(−logX/T).

We have to go moduloJ(D)∩C[x, y]kto avoid trivial admissible deformations. Remark 3.7.6. The previous Theorem is false in higher dimensions.

Proof. Considerf = 4x3y2−16x4z+ 27y4−144xy2z+ 128x2z2−256z3 ∈_C[x, y, z]. It is weighted homogeneous of degree 12 with weights (2,3,4) and it defines a germ of a free divisor (D,0) ⊂ (C3,0). A Macaulay 2 computation shows that dim_CC[x, y, z]12/J(D)∩C[x, y, z]12= 3 but F T1(D) = 0.

Corollary 3.7.7. Let (D,0) ⊂ (C2,0) be a germ of a free divisor defined by a

homogeneous polynomial of degree k. Then dimCF T1(D) =k−3 if k≥3, and is

zero otherwise.

Proof. If k = 1, then J(D) = C[x, y] and if k = 2, then J(D) = (x, y) and so, by Theorem 3.7.5, in both casesF T1_{(D) = 0.}

Let us suppose now that k ≥ 3. We have that dim_CC[x, y]k = k+ 1 and that

J(D)∩_C[x, y]k gives us 4 relations: x∂f /∂x, x∂f /∂y, y∂f /∂x, y∂f /∂y. Because

(D,0) is an isolated singularity, then∂f /∂x, ∂f /∂y form a regular sequence and so the Koszul relation generates the relations between the partial derivative off. Be- cause the Koszul relation is of degreek−1>1, thenx∂f /∂x, x∂f /∂y, y∂f /∂x, y∂f /∂y

are linearly independent. Hence, dim_CC[x, y]k/J(D)∩C[x, y]k=k+ 1−4 =k−3.

We conclude by Theorem 3.7.5.

Example 3.7.8. 1. Considerf =xy(x−y)(x+y)∈C[x, y]as defining equation of the germ of a free divisor(D,0)⊂(C2,0). Then F T1(D) is1-dimensional

and it is generated byx2y2. See Appendix C.2 for more details.

2. Consider f = x5 + y4 ∈ _C[x, y] and the germ of a free divisor (D,0) = (V(f),0) ⊂ (C2,0). A direct computation shows that F T1(D) = 0 and so

Notice that the last Example is a particular case of the following:

Proposition 3.7.9. Let (D,0) = (V(f),0) ⊂ (C2,0) be a germ of an irreducible

binomial curve, i.e. f =xa+yb for some coprime positive integers a, b∈_N. Then

F T1(D) = 0 and so(D,0)is formally rigid.

Proof. Becausea, bare coprime,fis weighted homogeneous of degreeabwith respect to weights (b, a). Now C[x, y]ab has basis {xa, yb}. However, both elements belong

toJ(D)∩_C[x, y]ab. We conclude by Theorem 3.7.5.

Remark 3.7.10. Let (D,0) ⊂ (Cn,0) be a germ of a free divisor defined by a

weighted homogeneous equation. Then we can compute the cohomology ofC• degree by degree, because each module and map involved is degree preserving.

Theorem 3.7.11. Let (D,0) ⊂ (Cn,0) be a germ of a free divisor defined by a

weighted homogeneous polynomial. ThenF T1_(D) ∼_{= (H}1_(C•₎

0)0, where (H1(C•)0)0

is the weight zero part of H1_(C•₎

0.

Proof. Letf be a defining equation for (D,0) weighted homogeneous of degreekand let (X, x) be an infinitesimal admissible deformation of (D,0). By Proposition 3.7.1, we can suppose (X, x) has defining equationf+·f0, withf0 weighted homogeneous of degreek.

By Proposition 1.1.24, we can takeδ1, . . . , δn∈Der(−logD) a weighted homoge-

neous basis. By Proposition 3.2.7, Der(logX/T) is generated by δ1+·δ˜1, . . . , δn+

·˜δn such that the determinant of their coefficients is f +·f0. Because f and f0

are both weighted homogenous of the same degree, then each ˜δi is weighted homo-

geneous of the same degree asδi, for all i= 1, . . . , n.

As seen in the proof of Theorem 3.4.1, there exists ψ∈ C1 _{such that ψ(δ}

i) = ˜δi.

So by the previous argumentψ is a weight preserving map and so it represents an element of (H1_(C•₎

0)0.

Corollary 3.7.12. Let (D,0) ⊂ (Cn,0) be a germ of a linear free divisor. Then

F T1(D)∼=LF T1(D).

Proof. It is clear that (H1_(C•₎

0)0 =H1(C0•)0. Then we can conclude by Theorems

3.5.1 and 3.7.11.

Corollary 3.7.13. Let (D,0)⊂(Cn,0)be a germ of a reductive linear free divisor. ThenF T1_{(D) = 0 and hence, it is formally rigid also as free divisor.}