Some remarks on plane curves related to freeness
Alexandru Dimca
TL;DR
The paper investigates how the exponents $(d_1,d_2)$ of a plane curve $C$ relate to classical invariants, extending freeness criteria beyond the classical case. It characterizes when the Tjurina number satisfies $\tau(C)=(d-1)^2-d_1d_2$, showing this equality occurs exactly for 3-syzygy curves with $d_3=d-1$; otherwise the inequality $\tau(C)>(d-1)^2-d_1d_2$ holds. A key result connects the Poincaré polynomial of a free line arrangement to the Betti polynomial of the complement, and for general curves shows $(1+d_1t)(1+d_2t)-B(M(C))(t)$ equals $a(C)t+b(C)t^2$ with $a(C),b(C)\ge0$, where equality cases characterize freeness. The work also provides new bounds for the second exponent $d_2$ of line arrangements and derives constraints on potential counterexamples to Terao’s conjecture, advancing understanding of freeness, curve exponents, and topology of curve complements.
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 Poincaré 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 $\mathcal A$.