Plane curves of genus p and degree 2p as projections of space curves of the same degree
type: Item
instance of: scholarly article
title: Plane curves of genus p and degree 2p as projections of space curves of the same degree
zbMATH Open document ID: 0758.14020
DOI: 10.1007/BF02568412
author: Luisella Caire
published in: Manuscripta Mathematica
publication date: 1992-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/155621
review text: 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.
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 10298
zbMATH Keywords: projection of space curve
zbMATH Keywords: plane curve
zbMATH Keywords: Segre surface
reviewed by: Robert Speiser
MaRDI profile type: Publication
cites work: Q3347989
cites work: Plane curves as projections of non singular space curves
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description: scientific article; zbMATH DE number 10298