Characterizations of spaces curves containing a planar subcurve (Q972114)

For a curve \(C\) in \(\mathbb P^3\), the Rao function is defined by \(\rho_C(j) = \dim H^1(\mathcal I_C(j))\), where \(\mathcal I_C\) denotes the ideal sheaf of \(C\). Given \(C\) of degree \(d\) and (arithmetic) genus \(g\), it is natural to ask what the largest possible Rao function can be. This was answered by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [C. R. Acad. Sci., Paris, Sér. I 317, No.12, 1159--1162 (1993; Zbl 0796.14029)]. The curves that they described are called \textit{extremal curves}. They also showed that an extremal curve of degree \(d\) and genus \(g\) contains a planar subcurve of degree \(d-1\). Excluding the extremal curves, the curves with maximal Rao function among the remaining curves, called \textit{subextremal curves}, were described by \textit{S. Nollet} [Manuscr. Math. 94, No.3, 303--317 (1997; Zbl 0918.14014)]. \textit{N. Chiarli, S. Greco} and \textit{U. Nagel} [J. Algebra 307, No. 2, 704--726 (2007; Zbl 1108.14027)] showed that a subextremal curve of degree \(d\) and genus \(g\) contains a planar subcurve of degree \(d-2\). After excluding also the subextremal curves, the situation becomes more delicate. This situation was analyzed in several papers. In particular, Notari and Sabadini introduced the \textit{\(h\)-extremal} curves, which are curves of degree \(d\) and genus \(g\) containing a planar subcurve of degree \(d-h\) and having a certain precisely defined Rao function in terms of \(d\), \(g\) and \(h\). They showed that their curves have maximal Rao function among curves containing planar subcurves of degree \(d-h\). In the current paper, the authors define a curve to be of \textit{\(h\)-extremal type} if its Rao function agrees with that introduced by Notari and Sabadini, but only in a certain range of degrees. They give a structure theorem with some geometric characterizations of such curves. In particular, if \(d\) is sufficiently large with respect to \(h\) then a curve of \(h\)-extremal type contains a planar subcurve of degree \(d-h\) and lies on a non-integral quadric surface. It is not true, however, that a curve of \(h\)-extremal type is necessarily \(h\)-extremal.