Plane curves of genus p and degree 2p as projections of space curves of the same degree
Let \(C\) denote an integral algebraic curve in \(\mathbb{P}^ 2\), of geometric genus \(p\) and degree \(n\), and write \(\tilde C\) for its normalization. If \(n>2p\), the pullback of \({\mathcal O}_{\mathbb{P}^ 2}(1)\) to \(\tilde C\) maps \(\tilde C\) to a smooth \(\overline C\) in \(\mathbb{P}^{n-p}\), of degree \(n\), which projects to \(C\). Now suppose \(n=2p\). Of course we can add a point of \(\tilde C\) to our linear system to obtain a smooth \(\overline C\) of degree \(n+1\) in \(\mathbb{P}^{p+1}\) which projects to \(C\), but to find a smooth space curve of degree \(n\) which projects to \(C\) is more difficult. For this, assume also that the only singularities of \(C\) are \(\delta\) ordinary nodes. Then the author shows that a smooth \(\overline C\) of degree \(n\) which projects to \(C\) exists exactly when no \(\delta-1\) of the nodes lie on a curve of degree \(2p-4\). (In any case, we can find a \(\overline C\) of degree \(n\) with at worst one node.) When \(C\) is hyperelliptic, then \(\overline C\) lies on a rational normal scroll, the Segre surface, ruled by the lines determined by the \(g^ 1_ 2\). Here the author shows more: with \(n=2p\), suppose \(C'\) is any space curve of degree \(n\) and geometric genus \(p\) which projects to \(C\). Then \(C\) is hyperelliptic if and only if either \(C'\) has one node \(N\) and lies on a rational cone of degree \(p-1\) with vertex \(N\), or \(C'\) is smooth and lies on a smooth rational scroll. In the second case the author gives a direct construction of the Segre surface as an image of \(\mathbb{P}^ 2\). As an application, suppose we have \(n=6\), \(p=3\), with 6 nodes on a conic. Then the 7th node lies on at least 2 bitangents if \(C\) is hyperelliptic, but lies on no bitangent otherwise.
scientific article; zbMATH DE number 10298