分野別セミナー

A hyperplane arrangement A is a finite set of linear hyperplanes in a vector space K^l, where K is a field. The graded module D(A) of the logarithmic vector fields consists of polynomial vector fields over K^l tangent to A. An arrangement A is said to be free if D(A) is a free module. The degrees of a homogeneous basis of the free module D(A) are called the exponents of A. The famous factorization theorem asserts that the characteristic polynomial of the intersection lattice of a free arrangement A completely factors into linear polynomials over the integers, and the roots of the polynomial are the exponents of A. This motivates us to study the degrees of minimal homogeneous generators of the module D(A) and their connections with combinatorics. In the first part of the talk, we will discuss a complete classification of the free arrangements in the three-dimensional real vector space with exponents of the form (1,3,d) for some d≥3. To analyze the relationship between algebraic and combinatorial structures, we often consider the interplay between adding and deleting hyperplanes in an arrangement. The arrangement in which one hyperplane is deleted from a free arrangement has been extensively studied by Takuro Abe. In the second part of the talk, we will turn our attention to the algebraic structure of a new class of hyperplane arrangements obtained by deleting two hyperplanes from a free arrangement. We compute the degrees of minimal homogeneous generators of D(A). In particular, the degrees are determined solely combinatorially for three-dimensional arrangements.
We present illustrative examples that show our result’s strength and provide insights into the relation between algebraic and combinatorial properties for close-to-free arrangements.


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