For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes... more For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main result in this note is a combinatorial formula for the coefficients of these matroid Kazhdan-Lusztig polynomials in terms of flag Whitney numbers. This formula gives insight into some vanishing behavior of the matroid Kazhdan-Lusztig polynomials.
In this note we study the freeness of the module of derivations on all moduli of the X 3 arrangem... more In this note we study the freeness of the module of derivations on all moduli of the X 3 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.
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the d... more Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.
This paper has been withdrawn by the authors due to crucial error in the main proof (located in S... more This paper has been withdrawn by the authors due to crucial error in the main proof (located in Section 2.4). The authors apologize for any inconveniences.
The fundamental combinatorial structure of a net in CP 2 is its associated set of mutually orthog... more The fundamental combinatorial structure of a net in CP 2 is its associated set of mutually orthogonal latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in CP 2 . Then we count these equivalence classes for small cases. Finally, we prove that the realization spaces of these classes in CP 2 are empty to show some non-existence results for 4-nets in CP 2 .
A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be en... more A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-trivial Massey products of the subspace arrangements complex complement. This is accomplished by studying a spectral sequence associated to the Lie coalgebras of Sinha and Walter. arXiv:0903.4221v1 [math.AT]
... a subspace arrangement A contained in the vector space V is derivation radical if and only if... more ... a subspace arrangement A contained in the vector space V is derivation radical if and only if A â K (or equivalently A Ã {â }) is derivation radical in the larger vector space V â K. In this vein, it is natural to wonder whetherAâ {0 ... [17] Yohannes Tadesse, Derivations ...
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], ... more 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 Euler multiplicity, of a restricted multiarrangement. We compute the Euler 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.
Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian s... more 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 be a free divisor.
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the d... more Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.
Transactions of the American Mathematical Society, 2007
In this paper we consider an effective divisor on the complex projective line and associate with ... more 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 θ such that θ(I i ) ⊂ I m i 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 vanishing of certain Schur functions. As application and motivation, we show that our results imply Terao's conjecture (about the combinatorial nature of the freeness of hyperplane arrangements) for certain new classes of arrangements of lines in the complex projective plane.
For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes... more For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main result in this note is a combinatorial formula for the coefficients of these matroid Kazhdan-Lusztig polynomials in terms of flag Whitney numbers. This formula gives insight into some vanishing behavior of the matroid Kazhdan-Lusztig polynomials.
In this note we study the freeness of the module of derivations on all moduli of the X 3 arrangem... more In this note we study the freeness of the module of derivations on all moduli of the X 3 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.
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the d... more Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.
This paper has been withdrawn by the authors due to crucial error in the main proof (located in S... more This paper has been withdrawn by the authors due to crucial error in the main proof (located in Section 2.4). The authors apologize for any inconveniences.
The fundamental combinatorial structure of a net in CP 2 is its associated set of mutually orthog... more The fundamental combinatorial structure of a net in CP 2 is its associated set of mutually orthogonal latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in CP 2 . Then we count these equivalence classes for small cases. Finally, we prove that the realization spaces of these classes in CP 2 are empty to show some non-existence results for 4-nets in CP 2 .
A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be en... more A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-trivial Massey products of the subspace arrangements complex complement. This is accomplished by studying a spectral sequence associated to the Lie coalgebras of Sinha and Walter. arXiv:0903.4221v1 [math.AT]
... a subspace arrangement A contained in the vector space V is derivation radical if and only if... more ... a subspace arrangement A contained in the vector space V is derivation radical if and only if A â K (or equivalently A Ã {â }) is derivation radical in the larger vector space V â K. In this vein, it is natural to wonder whetherAâ {0 ... [17] Yohannes Tadesse, Derivations ...
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], ... more 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 Euler multiplicity, of a restricted multiarrangement. We compute the Euler 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.
Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian s... more 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 be a free divisor.
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the d... more Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.
Transactions of the American Mathematical Society, 2007
In this paper we consider an effective divisor on the complex projective line and associate with ... more 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 θ such that θ(I i ) ⊂ I m i 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 vanishing of certain Schur functions. As application and motivation, we show that our results imply Terao's conjecture (about the combinatorial nature of the freeness of hyperplane arrangements) for certain new classes of arrangements of lines in the complex projective plane.
Uploads
Papers by Max Wakefield