On sections of a reflexive sheaf of rank 2 on $\bbfP\sp 3$: Extremal cases for the first section

If ${\cal F}$ is a rank two normalized reflexive sheaf on $\bbfP^3$ with Chern classes $c_1=0$ or $-1$, $c_2$, $c_3$, it is possible to define the integers $\alpha=$ smallest number such that ${\cal F} (\alpha)$ has a non vanishing section and $\beta=$ smallest number such that ${\cal F} (\beta)$ has a new section, not multiple of one section of ${\cal F} (\alpha)$. Then it is well known that the zero-locus of a non-vanishing general section of ${\cal F} (t)$ gives rise to a locally Cohen-Macaulay, almost everywhere complete intersection curve if and only if either $t=\alpha$ or $t\ge \beta$. Moreover $\alpha \le\sqrt {3c_2+1 +{3\over 4} c_1} -1- {1\over 2} c_1$ [{\it R. Hartshorne}, Invent. Math. 66, 165-190 (1982; Zbl 0519.14008)].\par In the paper under review it is shown that, if $\alpha$ is as high as possible, i.e. $\alpha=$ integral part of $\sqrt {3c_2+ 1+ {3\over 4} c_1}- 1-{1\over 2} c_1$, then $\alpha= \beta$ and moreover $h^2 {\cal F} (\alpha) =0$.