Hypertetrahedral arrangements

In this paper, we introduce the notion of a complete hypertetrahedral arrangement $${\mathcal {A}}$$ A in $${\mathbb {P}}^{n}$$ P n . We address two basic problems. First, we describe the local freeness of $${\mathcal {A}}$$ A in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustaţă–Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $${\mathcal {A}}$$ A .

Divisors of expected Jacobian type

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.

SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS

Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

Schemes Supported on the Singular Locus of a Hyperplane Arrangement in $\mathbb P^n$

A hyperplane arrangement in $\mathbb P^n$ is free if $R/J$ is Cohen–Macaulay (CM), where $R = k[x_0,\dots ,x_n]$ and $J$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $ J^{un}$, the intersection of height two primary components, and $\sqrt{J}$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $\mathbb P^3$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $r$, there is an arrangement for which $R/J^{un}$ (resp. $R/\sqrt{J}$) fails to be CM in only one degree, and this failure is by $r$. We draw consequences for the even liaison class of $J^{un}$ or $\sqrt{J}$.

Hypersurfaces with linear type singular loci

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

A comparison formula for residue currents

Given two ideals $\mathcal {I}$ and $\mathcal {J}$ of holomorphic functions such that $\mathcal {I} \subseteq \mathcal {J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal {I}$ and $\mathcal {J}$. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes. One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.