Proof of Theorem 2.1

If D has normal crossings at p, then D is free at p, that is, it is either smooth or depth(Jh,O^S,p) = 2 and OSing D,p is Cohen–Macaulay (Aleksandrov’s theorem). The normalization of a normal crossing divisor D =�m

i=1Di is smooth since it is the disjoint union of the smooth components Di (cf. Example A.27). So it remains to show that for a point p ∈ Sing D the ideal J^h is radical at p. This is done by direct computation: since D has normal crossings at p ∈ Sing D, we can assume that D =�m

i=1(Di, p)is given by the equation h = x1· · · x^m, 1 < m ≤ n where each xⁱ corresponds to an irreducible component Di passing through p. Then

Jh=

�m i=1

(x1· · · ˆxⁱ· · · x^m).

Using facts about primary decomposition of monomial ideals, see e.g. [54], it follows that

This irredundant primary decomposition shows that Jh is the intersection of prime ideals of height 2. Thus Jh is clearly radical.

Conversely, suppose that Jh =√

Jh and O^{Sing D,p} is Cohen–Macaulay of dimension (n − 2) and moreover that the normalization π∗OD,p� is Gorenstein (here π : �D→ D denotes the normaliza-tion morphism). Prop. 2.48 implies that each Di is free at p and has a radical Jacobian ideal.

So we may assume that D is irreducible. By our hypothesis, Piene’s theorem A.42 and Remark A.43 yield the equality of ideals

CDIπOD,p� = JhOD,p� .

Since by Lemma 1.69 one has Jh = CD in O^D,p, this implies CD = CDIπ in π∗OD,p� . By Nakayama’s lemma, it follows that Iπ = OD,p� . Hence Ω¹_D/D� = 0. If �D is smooth at π⁻¹(p) then a similar argument as in the proof of Thm. 1.63 yields that D is already smooth at p: then OD� ∼=C{z1, . . . , z_n−1} for some independent variables z¹, . . . , z_n−1. Hence one has an inclusion

By Theorem A.24, D is normal at p. By Aleksandrov’s theorem, D is then already smooth at p. For (D, p) = �m

i=1(Di, p)this means that we are in the situation of the second corollary of Proposition 2.48 and the assertion follows.

Remark 2.49. We can also give a different proof of (2) ⇒ (1) of Thm. 2.1 using the characteri-zation of normal crossings by the logarithmic residue of Thm. 1.63: let (D, p) =�m

i=1(Di, p)be the decomposition into irreducible components and suppose that Jh=√

Jh. Then the singular locus of the singular locus Sing(Sing D) is of dimension less than or equal to (n − 3). By Lemma 2.17, D has normal crossings at smooth points of Sing D. Hence D has normal crossings in codimension 1. From Lemma 1.80 it follows that the logarithmic residue is holomorphic on the

normalization, that is, ρ(Ω¹_S(log D)) = π_∗OD�. Then Theorem 1.63 shows that D is a normal crossing divisor.

Remark 2.50. We do not know whether the condition on the normalization of D in Theorem 2.1 is necessary. If (D, p) is free and has a radical Jacobian ideal, then by Lemma 1.65 the normalization ( �D, π⁻¹(p)) is Cohen–Macaulay. One can use Piene’s theorem only if �D is Gorenstein because then the canonical sheaf ωD� is invertible. More precisely, one can prove the following: ωD� = CDOD� if and only if �D is Gorenstein (see Prop. 3.5 of [66]). Moreover, �D is Gorenstein if and only if it is isomorphic to the blowup of D in the conductor CD (by Thm. 2.7 of [101]).

Question 2.51. Let D ⊆ S be a divisor in a complex manifold S that is locally at a point p given by h = 0 and denote by π : �D→ D its normalization. Suppose that D is free at p and that Jh=√Jh. Is then the normalization �D of D already Gorenstein at π⁻¹(p)?

Jacobian ideals of hypersurfaces

In this chapter we have two different aims: the first one is to classify divisors with radical Jacobian ideals. The second one is to study two possible generalizations of normal crossing divisors, namely splayed divisors and mikado divisors. We consider some of their properties and also try to characterize them in terms of their singular loci given by their Jacobian ideals.

First we ask for an analogue of Theorem 2.1 for radical Jacobian ideals of higher codimension. In low ambient dimension, that is, dim S ≤ 3 divisors with radical Jacobian ideal can be described with the help of Thm. 2.1 (see Prop. 3.2). However, it is not clear how to classify divisors with radical Jacobian ideal in higher dimensional ambient spaces, since then also embedded components of the Jacobian ideal have to be taken into account. Here we have results in special cases and conjectures for more general situations. The second topic of this chapter is splayed divisors (also see Chapter 2), which are a natural generalization of normal crossing divisors. The difference between the two classes of divisors is that irreducible components of splayed divisors may have singularities. Here we present a characterization of splayed divisors in terms of their Jacobian ideals (corresponding to the geometry) and compute their Hilbert–Samuel polynomials, which satisfy a certain additivity property. Finally we consider another generalization of normal crossing divisors, so-called mikado divisors. The irreducible components of a mikado divisor are smooth and all possible intersections between them are also smooth. The difference to normal crossing divisors is that probably more than n components can meet at a point. We give a characterization of a mikado divisor D ⊆ S in terms of its Jacobian ideal for dim S = 2. Finally we ask for a generalization to higher dimensions.

3.1 Radical Jacobian ideals

In Chapter 2 it was shown that if a free divisor with a Gorenstein normalization in a complex manifold has a radical Jacobian ideal, then it is already a normal crossing divisor. Now we con-sider a more general problem. Suppose that D is a divisor in a smooth complex n-dimensional manifold S that is locally at a point p given by a reduced equation h ∈ O^S,p = C{x1, . . . , xn}.

Denote by Jh= (∂x1h, . . . , ∂xnh)its Jacobian ideal and suppose that Jhis radical. Which ideals I⊆ O^S,pcan be such radical Jacobian ideals Jh? More precisely: given a radical ideal I ⊆ O^S,p,

when does there exist a divisor (D, p) = {h = 0} such that I = Jh?

The case of dim S = 2 was treated in Chapter 2: if D is a reduced curve in S, then its singular locus consists of isolated points. Thus locally at such a singularity, the ideal Jh is an m-primary ideal and if Jh = √

Jh is radical then it has to be the maximal ideal. With the theorem of Mather–Yau (or with the Corollary of Theorem 3.49) it follows that (D, p) is a normal crossing singularity. For dim S = 3 we need a little preparation.

Lemma 3.1. Let (R, m) be an n-dimensional regular local ring, I ⊆ R an ideal of height (n − 1) and suppose that R/I is reduced. Then R/I is a one-dimensional Cohen–Macaulay ring.

Proof. Since I has height (n − 1) in R and R is Cohen–Macaulay, it follows from the height-equality that R/I is of dimension 1. Since R/I is reduced, I is radical and can be written as as a finite intersection of minimal prime ideals p1∩· · ·∩p^k, where height(pi) ≥ n−1. If height(pi) = n holds for some i, then pi= m (any prime ideal is contained in a maximal ideal). But m cannot be a minimal element of the primary decomposition of I since at least one pi is of height (n −1) and hence strictly contained in m. Thus all pi have height (n − 1) and I is equidimensional. Now it remains to show that the depth of R/I is 1. The maximal ideal of R/I is m, where m is the image of m under the canonical projection. We show that m is not contained in Ass(R/I) = {p ∈ R prime: p = ann(¯a), for an ¯a ∈ R/I}. Suppose therefore that m were contained in Ass(R/I). This means that there exists an ¯a �= 0 such that m · ¯a = ¯0. If ¯a �= ¯0 were also contained in m, then

¯a² = ¯0 would hold, which is a contradiction to R/I reduced. Hence ¯a ∈ (R/I)^∗, that is, there exists some ¯b ∈ R/I such that a · b = ¯1. Then ab · m = m and by Nakayama’s lemma ab = 0.

Contradiction. Thus there exists a ¯c ∈ m such that for all ¯a ∈ R/I we have ac �= ¯0, that is, R/I contains a nonzerodivisor. Hence depth(R/I) ≥ 1 and since its dimension is already one, it follows that R/I is Cohen–Macaulay.

Proposition 3.2. Let S be a 3-dimensional manifold and let D ⊆ S be a divisor such that at a point p, D is defined by h ∈ O^S,p and has radical Jacobian ideal Jh�= (1). Suppose moreover that the normalization �D of D is Gorenstein. Then one of the two cases occurs:

(i) depth(Jh,O^S,p) = 3 and (D, p) is an A1-singularity.

(ii) depth(Jh,O^S,p) = 2 and D has normal crossings at p.

Proof. (i) Since Jhis of depth 3 in a three-dimensional local ring, it defines an isolated singularity.

From the radicality of Jhfollows that Jhhas to be the maximal ideal m ⊆ O^S,p. The rest is the content of Prop. 3.9.

(ii) If depth(Jh,O^S,p) = 2, then Jh defines a reduced curve C in S. Since O^S,p/Jh is a reduced one-dimensional local ring, it is Cohen–Macaulay by Lemma 3.1. Hence it follows by Theorem 2.1 that D has normal crossings at p.

Question 3.3. Does there exist a surface (D, p) ⊆ (C³, p) such that (D, p) is free and Jh =

√Jh�= (1) but ( �D, ˜p)is not Gorenstein?

For dim S ≥ 4 the situation is more complicated. We split this part into two subsections.