Some remarks on plane curves related to freeness

Let $C$ be a reduced complex projective plane curve, and let $d_1$ and $d_2$ be the first two smallest exponents of $C$. For a free curve $C$ of degree $d$, there is a simple formula relating $d,d_1,d_2$ and the total Tjurina number of $C$. Our first result discusses how this result changes when the curve $C$ is no longer free. For a free line arrangement, the Poincaré polynomial coincides with the Betti polynomial $B(t)$ and with the product $P(t)=(1+d_{1}t)(1+d_{2}t)$. Our second result shows that for any curve $C$, the difference $P(t)-B(t)$ is a polynomial $at+b{t}^2$, with $a$ and $b$ non-negative integers. Moreover $a=0$ or $b=0$ if and only if $C$ is a free line arrangement. Finally we give new bounds for the second exponent $d_2$ of a line arrangement $\mathcal{A}$, the corresponding lower bound being an improvement of a result by H. Schenck concerning the relation between the maximal exponent of $\mathcal{A}$ and the maximal multiplicity of points in $\mathcal{A}$.