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New free divisors from old. (English) Zbl 1280.32016

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.

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
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