Max Wakefield

The Euler multiplicity and addition-deletion theorems for multiarrangements

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the e-multiplicity, of a restricted multiarrangement. We compute the e-multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements.

Derivations of an effective divisor on the complex projective line

In this paper we consider an effective divisor on the complex projective line and associate with it the module D consisting of all the derivations 0 such that θ(I i ) C I mi i for every i, where I i is the ideal of p i . The module D is graded and free of rank 2; the degrees of its homogeneous basis, called the exponents, form an important invariant of the divisor. We prove that under certain conditions on (m i ) the exponents do not depend on {p i }. Our main result asserts that if these conditions do not hold for (m i ), then there exists a general position of n points for which the exponents do not change. We give an explicit formula for them. We also exhibit some examples of degeneration of the exponents, in particular, those where the degeneration is defined by the vanishing of certain Schur functions. As an application and motivation, we show that our results imply Teraos conjecture (concerning the combinatorial nature of the freeness of hyperplane arrangements) for certain new classes of arrangements of lines in the complex projective plane.

Local cohomology of logarithmic forms

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

Formality of pascal arrangements

In this paper we construct a family of subspace arrangements whose intersection lattices have the shape of Pascal’s triangle. We prove that even though the intersection lattices are not geometric, the complex complement of the arrangements are rationally formal.

Free Multiplicities on the moduli of X 3

In this note we study the freeness of the module of derivations on all moduli of the

Skeleton simplicial evaluation codes

X_3

The characteristic polynomial of a multiarrangement

arrangement with multiplicities. We use homological techniques stemming from work of Yuzvinsky, Brandt, and Terao which have recently been developed for multi-arrangements by the first author, Francisco, Mermin, and Schweig.

The Kazhdan–Lusztig polynomial of a matroid

For a subspace arrangement over a finite field we study the evaluation code defined on the arrangements set of points. The length of this code is given by the subspace arrangements characteristic polynomial. For coordinate subspace arrangements the dimension is bounded below by the face vector of the corresponding simplicial complex. The minimum distance is determined for coordinate subspace arrangements where the simplicial complex is a skeleton. A few examples are presented with high minimum distance and dimension.