Double covers and extensions (Q6185382)

The authors characterize hypersurfaces \(W'\) of degree \(2k\) in \(\mathbb P^n\) that lift to a divisor in the class \(|k\pi^*(H)|\) (\(H\)=hyperplane class) in a suitable double cover \(\pi:X\to \mathbb P^n\). It is well known that such \(W'\) has an ordinary double point along a complete intersection subvariety \(Z\) of type \(k,k-d\), where \(2d\) is the degree of the branch locus \(B\) of \(\pi\). The authors prove that, conversely, if \(W'\) is a hypersurface with a an ordinary double point along a complete intersection of type \(k,k-d\), then there exists a double cover \(\pi:X\to \mathbb P^n\), branched along a hypersurface of degree \(2d\), such that \(W'\) lifts to \(W\in |k\pi^*(H)|\). The authors use their characterization to describe sections of varieties that arise as double covers of some projective spaces. For instance, they determine smooth degree-\(k\) sections of \(K3\) surfaces \(S\) of genus \(2\), by considering that \(S\) arises as a sextic double plane, and study their extensions. Also for \(K3\) surfaces of genus \(3\) the authors are able to study extensions of degree-\(2\) sections. Indeed, a general \(K3\) surface of genus 3 can be realized as an anticanonical divisor of some double cover of \(\mathbb P^3\). Finally, the authors use their main result to analyze the extensions of plane quintic curves in their canonical model.