On the minimal value of global Tjurina numbers for line arrangements

Abstract

We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by A. A. du Plessis and C.T.C. Wall, can be improved when the curve is a line arrangement.

Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves

“…In this setting, some results and some open problems are stated in [14]. It is a remarkable fact in our opinion that this lower bound τ (d, r) min is not optimal if we restrict our attention to line arrangements, as shown in [13]. When C is a line arrangement, its global Tjurina number, coincides with its global Milnor number µ(C), and is given by [17,Proposition 4.7].…”

“…When C is a line arrangement, its global Tjurina number, coincides with its global Milnor number µ(C), and is given by [17,Proposition 4.7]. With this notation we have the following result, see [13]. The line arrangements such that r = mdr(f ) ∈ {0, 1, 2} are classified, see [55] or [17,Theorem 4.11] for the case r = 2, which is the only difficult case.…”