Gaussian maps for double coverings
Given a line bundle \(L\) on a smooth projective variety \(X\), the Gauss map \[ \bigwedge^ 2 H^ 0(X,L) \to H^ 0 (X,L^{\otimes 2}) \] is defined by sending \(\sum s_ i \otimes t_ i\) to \(\sum(ds_ i\otimes t_ i- s_ i\otimes dt_ i)\). If \(X \to Y\) is a cyclic covering, with \(Y\) and \(X\) smooth and projective, and \(L\) is a pull-back from \(Y\), then, using projection formulas, it is possible to express the Gauss map for \(L\) in terms of similar maps on \(Y\). After having described this general setting, the author focuses on the case of double covers of \(\mathbb{P}^ m\) \((m>1)\) and computes the cokernel of the Gauss map for the pull-back of any effective line bundle on \(\mathbb{P}^ m\). The paper also contains several references on the general theory and applications of the Gauss maps.
scientific article; zbMATH DE number 554378
scientific article