On Terao's conjecture for at most nine planes

Source abstract

In this note we show that Terao's freeness conjecture holds for arrangements of at most nine planes in the complex projective $3$-space. Moreover, we show that every essential and irreducible free arrangement of at most nine planes in $\mathbb{P}^{3}_{\mathbb{C}}$ is divisionally free, and we list all intersection lattices of such arrangements. Our approach is based on Ziegler's restriction theorem and a simple count involving the numbers of multiple lines. As a by-product, we show that Cynk--Szemberg octic arrangements are never free. Finally, motivated by a recent work of Dimca and Pokora, we present examples of weak Ziegler pairs of plane arrangements, including pairs of octic arrangements leading to rigid double octic Calabi--Yau threefolds.