Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula

Terao, Hiroaki

doi:10.1007/bf01389197

Cited by 173 publications

(139 citation statements)

References 8 publications

Supporting

Mentioning

135

Contrasting

Unclassified

Order By: Relevance

“…a i /; the a i are called the exponents of A. Terao's famous theorem [11] states that if D.A/ '˚S. a i /, then P .X; t/ D Q .1 C a i t/.…”

Section: Introductionmentioning

confidence: 99%

Freeness of conic-line arrangements in $ℙ^2$

Schenck¹,

Tohǎneanu

2009

Comment. Math. Helv.

View full text Add to dashboard Cite

Abstract. Let C D S n i D1 C i Â P 2 be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module 1 .C/ of logarithmic differential forms with pole along C . We also show that the analog of Terao's conjecture (freeness of 1 .C/ is combinatorially determined if C is a union of lines) is false in this setting.

show abstract

“…a i /; the a i are called the exponents of A. Terao's famous theorem [11] states that if D.A/ '˚S. a i /, then P .X; t/ D Q .1 C a i t/.…”

Section: Introductionmentioning

confidence: 99%

Freeness of conic-line arrangements in $ℙ^2$

Schenck¹,

Tohǎneanu

2009

Comment. Math. Helv.

View full text Add to dashboard Cite

show abstract

“…The splitting of these sheaves has some applications to combinatorial problems of hyperplane arrangements. See [6] or [7] for details.…”

Section: Then E Splits Into a Direct Sum Of Line Bundles If And Only mentioning

confidence: 99%

Splitting criterion for reflexive sheaves

Abe

Yoshinaga

2008

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We give a criterion for a reflexive sheaf to split into a direct sum of line bundles. Main theoremVector bundles over the projective space P n K are one of the main subjects in both (algebraic) geometry and commutative algebra. The most fundamental result in this area is the theorem due to Grothendieck which asserts that any holomorphic vector bundle over P 1 K splits into a direct sum of line bundles. When n ≥ 2, vector bundles over P n K do not necessarily split. Indeed, the tangent bundle is indecomposable. In these cases, some sufficient conditions for vector bundles to split have been established. The following is one of such criterions, which we call "Restriction criterion". Theorem 0.1 (Horrocks). Let K be an algebraically closed field, n be an integer greater than or equal to 3, and let E be a locally free sheaf on P n K of rank r (≥ 1). Then E splits into a direct sum of line bundles if and only if there exists a hyperplane H ⊂ P n K such that E| H splits into a direct sum of line bundles. In other words, the splitting of a vector bundle can be characterized by using a hyperplane section. However, vector bundles, or equivalently locally free sheaves, form a small class among all coherent sheaves. There are some important wider classes of coherent sheaves, e.g., reflexive sheaves or torsion free sheaves. The purpose of this article is to generalize the "Restriction criterion" to one for reflexive sheaves, and we also show that it fails in the class of torsion free sheaves. Our main theorem is as follows: Theorem 0.2. Let K be an algebraically closed field, n be an integer greater than or equal to 3, and let E be a reflexive sheaf on P n K of rank r (≥ 1). Then E splits into a direct sum of line bundles if and only if there exists a hyperplane H ⊂ P n K such that E| H splits into a direct sum of line bundles.We give two proofs for Theorem 0.2. The first proof is basically parallel to that of Theorem 0.1, in which we also establish a general principle that the structure of a reflexive sheaf can be recovered from its hyperplane section (Theorem 2.2).

show abstract

“…Thus χ(A, t) is not factored. It follows from Terao's factorization theorem ( [14]) that any extension of A is not free.…”

Section: Introductionmentioning

confidence: 99%

On the Extendability of Free Multiarrangements

Yoshinaga

2009

Arrangements, Local Systems and Singularities

View full text Add to dashboard Cite

Abstract. A free multiarrangement of rank k is defined to be extendable if it is obtained from a simple rank (k + 1) free arrangement by the natural restriction to a hyperplane (in the sense of Ziegler). Not all free multiarrangements are extendable. We will discuss extendability of free multiarrangements for a special class. We also give two applications. The first is to produce totally non-free arrangements. The second is to give interpolating free arrangements between extended Shi and Catalan arrangements.

show abstract

Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula

Cited by 173 publications

References 8 publications

Freeness of conic-line arrangements in $ℙ^2$

Freeness of conic-line arrangements in $ℙ^2$

Splitting criterion for reflexive sheaves

On the Extendability of Free Multiarrangements

Contact Info

Product

Resources

About