Space models of degree \(g+1\) of curves of genus \(g\)
Let \(X\) be a smooth irreducible curve of genus \(g>1\) over the complex field. In the article under review, the authors consider the problem of the existence of a smooth linearly normal model of degree \(g+1\) of \(X\) in \(\mathbb P^3\), i.e., of a complete and very ample linear series \(g^3_{g+1}\) on \(X\). Note that, by Riemann-Roch, the degree \(d\) of a complete linear series of dimension \(3\) on \(X\) is at most \(g+3\), and that the condition \(d\geq 3/4g+3\) is required in order that the Brill-Noether number \(\rho(g,d,3)\) is non-negative. The existence of a complete and very ample \(g^3_{g+3}\) on any curve was proved by Halphen in 1882. The case \(d=g+2\) was treated by \textit{M. Coppens} and \textit{G. Martens} in [Math. Nachr. 213, 35--55 (2000; Zbl 0972.14021)]. Here the authors give a complete answer for \(d=g+1\) in the more general situation of a complete, base-point-free and simple linear series. The main theorem is the following: If \(g\geq 8\), then \(X\) has a complete base-point-free and simple \(g^3_{g+1}\) if and only if \(X\) is not hyperelliptic, not bielliptic and not a smooth plane sextic. The authors give an explicit example of a \(6\)-gonal curve of genus \(10\) not having any complete and very ample \(g^3_{11}\). As for complete and very ample \(g^3_{g+1}\)'s, such a linear series is proved to exist either if \(g\geq 8\), the gonality is different from \(3\) and \(\dim W^2_{k+5}\leq 2\), or if \(X\) is any smooth plane curve of degree \(d\geq 10\). Finally the following generalization of the main theorem is stated: if \(r\geq 2\) and \(\rho(g,g+r-2,r)\geq 0\), then there exists a complete base-point-free simple \(g^r_{g+r-2}\) if and only if \(X\) is not hyperelliptic, not bielliptic and not a smooth plane curve of degree at most \(10\).
scientific article; zbMATH DE number 6138666