Freeness and multirestriction of hyperplane arrangements

Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.Comment: 8 page

Irregularity of hypergeometric systems via slopes along coordinate subspaces

We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(\beta)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $\beta\in\mathbb{C}^d$. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector $L$ on torus equivariant generators. To this end we introduce the $(A,L)$-umbrella, a simplicial complex determined by $A$ and $L$, and identify its facets with the components of the associated graded ring. We then establish a correspondence between the full $(A,L)$-umbrella and the components of the $L$-characteristic variety of $M_A(\beta)$. We compute in combinatorial terms the multiplicities of these components in the $L$-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities. We deduce from this that slopes of $M_A(\beta)$ are combinatorial, independent of $\beta$, and in one-to-one correspondence with jumps of the $(A,L)$-umbrella. This confirms a conjecture of Sturmfels and gives a converse of a theorem of Hotta: $M_A(\beta)$ is regular if and only if $A$ defines a projective variety.Comment: 44 pages, 3 figures, choose PS or PDF to see figures, new Lemma 2.8 fills gap in previous version of Lemma 2.12, error in previous version of Theorem 3.2 repaired by considering L-holonomic modules in Sections 3.2 and 4.

Cohen-Macaulayness and computation of Newton graded toric rings

Let $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let $K[H]$ be the associated semigroup ring over a field $K$. We investigate heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every $H$ there exist generating sets $A$ for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on $A$-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.Comment: 20 pages, 4 figure

Quasihomogeneity of curves and the Jacobian endomorphism ring

We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos' normalization algorithm. In the process, we give a simplified proof of the Kunz-Ruppert criterion.Comment: 9 page