Some remarks on plane curves related to freeness

TL;DR

This paper investigates the properties of plane curves related to freeness, providing formulas, bounds, and characterizations that distinguish free arrangements from non-free ones, with implications for the exponents and topological invariants.

Contribution

It extends known formulas for free curves to non-free cases, characterizes the difference between Poincaré and Betti polynomials, and improves bounds on exponents of line arrangements.

Findings

For free curves, a simple relation between degree, exponents, and Tjurina number is established.

The polynomial difference P(t)-B(t) is quadratic with non-negative coefficients, indicating freeness.

New bounds for the second exponent d_2 of line arrangements are derived, improving previous results.

Abstract

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 Poincar\'e polynomial coincides with the Betti polynomial $B(t)$ and with the product $P(t)=(1+d_1t)(1+d_2t)$. Our second result shows that for any curve $C$, the difference $P(t)-B(t)$ is a polynomial $a t +bt^2$, 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 $\mathcal A$.