On the space of projective curves of maximal regularity
Let \(r,d\) be integers such that \(r\geq 3\), \(d\geq r+2\). Let \(C\subseteq {\mathbb P}^r\) be a nondegenerate, irreducible, projective curve of degree \(d\); \(C\) is called \(m\)-regular if \(H^i({\mathbb P}^r, {\mathcal I}_C(m-i))=0\) for \(i\geq 1\). The \textit{regularity} \(\text{reg}(C)\) of \(C\) is the least integer \(m\) such that \(C\) is \(m\)-regular. We have \(\text{reg}(C)\leq d-r+2\) [\textit{L. Gruson} et al., Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)], and if equality holds, \(C\) is named of \textit{maximal regularity}. In this case, \(C\) is a nonsingular rational curve admiting a \((d-r+2)\)-secant line, the so-called \textit{extremal secant line} [loc. cit.]. In this paper the authors are interested in spaces that parametrize projective curves with a fixed regularity condition; in particular, they show that \(R=R^{d-r+2}_{r,d}\), the set of all maximal regularity curves in \({\mathbb P}^r\) of degree \(d\), is an irreducible variety of dimension \(3d+r^2-r-1\) (cf. Lemma 2.4 in [\textit{T. Johnsen} and \textit{S. L. Kleiman}, Commun. Algebra 24, No. 8, 2721--2753 (1996; Zbl 0860.14038)]. Moreover, as the automorphism group \(A=\mathrm{PGL}(r+1)\) of \({\mathbb P}^r\) acts in a natural way on \(R\), we can define in particular the stabilizer group \(A_C\) for each \(C\in R\); they show that \(A_C\) is finite and \(C\) cuts out the extremal secant line in at least 4 points.
scientific article; zbMATH DE number 6666256
scientific article