The fundamental group of the complement of a resultant hypersurface.
Let \(X\) be a smooth irreducible complex projective variety of dimension \(n\), and consider \(n+1\) very ample line bundles \(L_0, \ldots, L_n\) on \(X\). Let \(V:=\bigoplus_i H^0(L_i)\). The resultant variety \(R \subset V\) is the locus of the elements \((f_0,\ldots,f_n)\) such that the \(f_i\)' s have common zeroes. It is known [\textit{I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky}, ``Discriminants, resultants, and multidimensional determinants'' (1994; Zbl 0827.14036)] that \(R\) is an irreducible hypersurface (called the resultant hypersurface). In this paper it is proved that the fundamental group of the complement of a resultant hypersurface is isomorphic to the group of the integers \(\mathbb{Z}\).
scientific article; zbMATH DE number 2101093