1. CMB Online first
 MirandaNeto, Cleto Brasileiro

A moduletheoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, 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)
4. CMB 2014 (vol 57 pp. 697)
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
nonlocal irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2dimensional 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 
