Projective dimensions of hyperplane arrangements

Takuro Abe

Home/Archive/arXiv

arXiv · 2009.04101

Projective dimensions of hyperplane arrangements

Takuro Abe

View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.

Explore related subjects

math.AG math.AC math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Takuro Abe. 2021-07-01. Projective dimensions of hyperplane arrangements. https://arxiv.org/abs/2009.04101

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Harnack Rigidity

The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under strong smoothness and nondegeneracy assumptions. For every two-dimensional lattice polygon, except unimodular triangles, we construct a smooth Newton-nondegenerate curve with smooth logarithmic critical locus and smooth embedded contour satisfying $\mathcal C(\mathscr A_f)=\partial\mathscr A_f$, although the curve is not Harnack. We also provide explicit primitive and nonprimitive families that are not torus-equivalent to simple Harnack curves. A concrete primitive example is certified by exact elimination and Sturm root counting. These results disprove contour--boundary rigidity and show that the contour as a set does not detect the real structure or the multiplicity of coincident critical sheets, thereby refining the compensation problem proposed by Lang, Shapiro, and Shustin.

math.AG↗

arXiv2026

A generalized Saito criterion, from line arrangements to curves

We formulate a generalized Saito criterion for complex projective plane curves, building on a recent generalized Saito criterion for plus-one generated arrangements of hyperplanes due to Junyan Chu.

math.AG↗

arXiv2026

Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen--Macaulay $R$-modules, introduced by Iyama--Wemyss and Iyama--Kalck--Wemyss--Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen--Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama--Kalck--Wemyss--Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.

math.AG↗