Advances on a conjecture about free divisors

Abstract

In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Today, it is known to be true only in ambient dimension less or equal than three or assuming Koszul-freeness. Thanks to our advances in the comprehension of strong Euler-homogeneity, we are able to prove the conjecture in the following new cases: assuming strong Euler-homogeneity on a punctured neighbourhood of a point; assuming the divisor is weakly Koszul-free; for ambient dimension $n=4$; for linear free divisors in ambient dimension $n=5$. We also refute a conjecture that states that all linear free divisors satisfy LCT and are strongly Euler-homogeneous.