Subextremal curves
\textit{M. Martin-Deschamps} and \textit{D. Perrin} obtained sharp bounds for the Rao function \(h^1(I_C(n))\) of a locally Cohen-Macaulay purely 1-dimensional curve \(C\subset\mathbb{P}^3\) [C. R. Acad. Sci., Paris Sér. I, 317, No. 12, 1159-1162 (1993; Zbl 0796.14029)] and characterized the extremal curves (i.e. the non-arithmetically Cohen-Macaulay curves achieving equality for their bound) in terms of certain Koszul modules in Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005). In Boll. Unione Mat. Ital. (7), A 9, No. 3, 593-607 (1995; Zbl 0866.14020), \textit{P. Ellia} recovered the bounds of Martin-Deschamps and Perrin and characterized extremal curves as the schematic union of a line and of a plane curve. Moreover he proved that a curve \(C\) of degree \(d\leq 5\) and genus \(g\), which is neither arithmetically Cohen-Macaulay, nor extremal, verifies the stronger bound: \(h^1(I_C(n))\leq 1+(d-3)(d-4)/2-g\). All previous papers assumed the characteristic 0 hypothesis. In this article, without any particular assumption on the characteristic of the ground field, Scott Nollet improves Ellia's results, by proving, for neither arithmetically Cohen-Macaulay, nor extremal curves \(C\), the bounds: \[ h^1\bigl( I_C(n)\bigr)\leq\begin{cases} 0\quad & \text{if }n<1+g-(d-3)(d-4)/2\\ (d-3)(d-4)/2-g+n \quad & \text{if }1+g- (d-3)(d-4)/2\leq n<1\\ 1+(d-3)(d-4)-g\quad & \text{if }1\leq n\leq d-3\\ 1+(d-2)(d-3)/2-g-n\quad & \text{if }d-3<n\leq(d-2)(d-3)/2-g\\ 0\quad & \text{if }n>(d-2)(d-3)/2-g\end{cases} \] (theorem 2.11), proves that the above bounds are sharp and classifies subextremal curves (i.e. neither arithmetically Cohen-Macaulay, nor extremal curves achieving their bound) as the curves obtained from extremal ones by elementary biliaison of height 1 on a quadric surface (theorem 2.14). -- A key-tool to get the previous results is the notion of spectrum of a curve (see section 1), studied by \textit{Enrico Schlesinger} in his Ph. D. thesis [see also J. Pure Appl. Algebra 136, 267-283 (1999)]. In particular Nollet proves that a subextremal curve has the spectrum: \(\{1+g-(d-3)(d-4)/2\}\cup\{0,1^2,2,\dots,d-3\}\) (theorem 2.14). Some examples conclude the paper in order to observe that, unlike the extremal case, subextremal curves are not determined by their spectrum (see 2.15) and to suggest that there are no natural stronger bounds on the Rao function for curves, which are neither arithmetically Cohen-Macaulay, nor extremal, nor subextremal (see 2.16, 2.17, 2.18).
scientific article; zbMATH DE number 1135241
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