On Euler-homogeneity for free divisors
Abraham del Valle Rodríguez
Journal of Singularities
volume 27 (2024), 112-127
Received: 16 September 2023. In revised form: 25 January 2024
Abstract:
In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem must be strongly Euler-homogeneous and it was proved for the two-dimensional case. Later, in 2006, it was shown that the conjecture is also true in dimension three, but, today, the answer for the general case remains unknown. In this paper, we use the decomposition of a singular derivation as the sum of a semisimple and a topologically nilpotent derivation that commute in order to deal with this problem. By showing that this decomposition preserves the property of being logarithmic, we are able to give alternative proofs for the low-dimensional known cases.
2020 Mathematical Subject Classification:
32C38, 14F10, 14F40
Key words and phrases:
free divisor, Logarithmic Comparison Theorem, logarithmic vector field, Euler-homogeneity
Author(s) information:
Abraham del Valle Rodríguez
Departamento de Álgebra
Universidad de Sevilla
C/Tarfia s/n
41012 Sevilla (Spain)
email: adelvalle2@us.es