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Search: MSC category 32S22 ( Relations with arrangements of hyperplanes [See also 52C35] )

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1. CMB Online first

Miranda-Neto, Cleto Brasileiro
A module-theoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor.

Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor
Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

2. CMB 2016 (vol 59 pp. 449)

Abdallah, Nancy
On Hodge Theory of Singular Plane Curves
The dimensions of the graded quotients of the cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$ with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on $H^2(U,\mathbb C)$.

Keywords:plane curves, Hodge and pole order filtrations
Categories:32S35, 32S22, 14H50

3. CMB 2016 (vol 59 pp. 279)

Dimca, Alexandru
The Poincaré-Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined
Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

Keywords:line arrangement, Milnor fiber, monodromy, mixed Hodge structures
Categories:32S22, 32S35, 32S25, 32S55

4. CMB 2014 (vol 57 pp. 697)

Bailet, Pauline
On the Monodromy of Milnor Fibers of Hyperplane Arrangements
We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

Keywords:hyperplane arrangements, Milnor fiber, monodromy, local systems
Categories:32S22, 32S55, 32S25, 32S40

5. CMB 2014 (vol 57 pp. 658)

Thang, Nguyen Tat
Admissibility of Local Systems for some Classes of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.

Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance variety
Categories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50

6. CMB 2010 (vol 54 pp. 56)

Dinh, Thi Anh Thu
Characteristic Varieties for a Class of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\mathcal{P}$ in $\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for which $H_0$ is a diagonal.

Keywords:local system, line arrangement, characteristic variety, resonance variety
Categories:14C21, 14F99, 32S22, 14E05, 14H50

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