Monday 24 October 2022
Takuro Abe (Kyushu University) — Addition-restriction theorems and projective dimensions of logarithmic vector fields of hyperplane arrangements
In the theory of hyperplane arrangements, as their associated algebraic structure, the logarithmic vector fields have played important roles. In particular, we say that an arrangement is free if its logarithmic vector field is a free module, and the multi-set of degrees of a free basis is called the exponents. When an arrangement is free, it is natural to ask whether the arrangement to which one new hyperplane is added (addition), and the restriction onto that hyperplane (restriction) is free or not. Terao's famous addition theorem says that the addition is free if the restriction is free and there is an inclusion between two exponents.
However in many cases it occurs that the original and restriction are free, but there is no inclusion between exponents. We study the logarithmic vector field of the addition in this case, and determine the algebraic structure of the addition under certain conditions.
* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.