# Sulle sezioni di un fascio riflessivo di rango 2 su $P^3$: casi estremi per la prima sezione

Roggero, Margherita and Valabrega, Paolo (1995) Sulle sezioni di un fascio riflessivo di rango 2 su $P^3$: casi estremi per la prima sezione. Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXXIII. pp. 103-111.

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If $F$ is a rank two normalized reflexive sheaf on $P^3$ with first chern class 0 or -1, it is possible to define the integers $\alpha \$ =smallest number such that $F(\alpha \ )$ has a non-vanishing section and $\beta \$ =smallest number such that $F( \beta \ )$ has a new sectiot, not multiple of one section of $F( \alpha \ )$. Then it is well known that the zero-locus of a non-vanishing general section of $F(t)$ gives rise to a locally Choen-Macaulay, almost everywhere complete intersection curve if and only if either $t= \alpha \ or \t \ge \ \beta \$. Moreover in $[H_2]$ it is proved that $\alpha\le\sqrt{3c_2+1+3c_1 / 4}-1-{c_1 / 2}$ (the $c_i$'s being the chern classes of $F$). In this paper it is shown that, ig $\alpha\$ is as high as possible, i.e. $\alpha\$ =integral part of $\alpha\le\sqrt{3c_2+1+3c_1 / 4}-1-{c_1 / 2}$ , then alpha=beta and moreover $h^2$ F(alpha)=0