Special linear systems and syzygies
The author considers linear systems \(\Lambda\) of hypersurfaces on a projective space \(\mathbb{P}^r(\mathbb{C})\) and the associated rational map \(\Phi:=\Phi_\Lambda :\mathbb{P}^r\to\mathbb{P}^\alpha .\) In Theorem 1 it is shown that the existence of linear syzygies for the homogeneuos ideal of the base locus of \(\Lambda\) has strong consequences for the fibers of \(\Phi.\) The result reminds a well-known fact according to which the \(K_d\) property of the base locus implies the linearity of the fibers of \(\Phi.\) The next main result, Theorem 3, concerns the special case of linear systems of quadrics. It states that, when the Gauss map of \(\Phi(\mathbb{P}^r)\) is degenerate, the inverse image of its non-trivial fibers is a linear space in \(\mathbb{P}^r.\) A number of interesting applications and examples are given.
scientific article; zbMATH DE number 5677586