Buchweitz, Ragnar-Olaf; Conca, Aldo New free divisors from old. (English) Zbl 1280.32016 J. Commut. Algebra 5, No. 1, 17-47 (2013). The paper under review provides several methods for constructing new free divisors from given ones and characterizations of free divisors for certain families of divisors. There is a determinantal characterization due to K. Saito: a reduced polynomial \(f\in K[x]:=K[x_1, \dots, x_n]\) over a field \(K\) is a free divisor if and only if there exists an \(n\times n\) matrix \(A\) over \(K[x]\) such that \(\det A=f\) and \(\mathrm{grad} (f)A\equiv 0\pmod f\).The results obtained in this paper are as follows. For polynomials \(f\in K[x]\) annihilated by \(n-2\) linearly independent Euler vector fields, a criterion of freeness in terms of the Buchsbaum-Rim complex is given; an explicit classification is proved for the case \(n=3\). Under certain condition, if \(f=f_1\cdots f_k\in K[x]\) and \(H=y_1\cdots y_k H_1\in K[y_1, \dots, y_k]\) are free divisors, then so is \(H(f_1, \dots,f_k)\in K[x]\). For a free divisor \(f_0\in K[y]=K[y_1, \dots, y_n]\), one can construct a free divisor \(f_0\cdots f_i\in K[y][x_1, \dots, x_i]\) inductively for \(i>0\). For binomials of type \(L(M+N)\), where \(L\) is a product of variables and \(M\), \(N\) are coprime monomials, conditions for freeness are given. It is shown that these conditions are also necessary for homogeneous binomials. From a given homogeneous free divisor, a new free divisor on the tangent bundle can be constructed explicitly.The authors provide various explicit examples for each result. Reviewer: Tomohiro Okuma (Yamagata) Cited in 1 ReviewCited in 8 Documents MSC: 32S25 Complex surface and hypersurface singularities 14J17 Singularities of surfaces or higher-dimensional varieties 14B05 Singularities in algebraic geometry 14J70 Hypersurfaces and algebraic geometry Keywords:free divisor; discriminant; Saito matrix; Euler vector field PDFBibTeX XMLCite \textit{R.-O. Buchweitz} and \textit{A. Conca}, J. Commut. Algebra 5, No. 1, 17--47 (2013; Zbl 1280.32016) Full Text: DOI arXiv Euclid