Index of speciality and arithmetically Gorenstein subschemes

Aim of this note is to prove that if $D \neq 0$ is an arithmetically Gorenstein ({\bf aG} for short) effective divisor on an {\bf aG} smooth connected closed subscheme $V \subset {\bf P}^N$, with $D$ not linearly equivalent to any $tH$, $t \in {\bf Z}$, then $-v \leq e(C) \leq 2 e(V) +v$, where $v=dim(V)$, $H$ is a general hyperplane section of $V$, $e(D)$ and $e(V)$ are the {\it indeces of speciality} of $D$ and $V$. Moreover we show that divisors whose index of speciality is in the lower set $\{-v, ..., e(V)-1 \}$ and divisors whose index of speciality is in the upper set $\{e(V)+1, ..., 2 e(V)+v \}$ are, up to linear equivalence, in correspondence (a symmetry with respect to the central value $e(V)$): the first divisors are {\it minimal} and {\it lone}, while the second ones are obtained, up to linear equivalence, from the the first by adding a suitable number of hyperplane sections. In the central value $e(V)$ we find minimal, not lone, divisors (see Thm.6, Rem.7 and 1, 2 for the definitions ). This allows in particular to classify the ``extremal'' cases $2 e(V) +v-1 \leq e(D) \leq 2 e(V) +v$, which are those, whose minimal divisor is a reduced quadric subvariety or a linear subvariety (see Cor.8). As an application we find all possible {\bf aG} integral curves on {\bf aG} smooth (connected) surfaces $S$ in case of $e(S)=1$ (see Prop.9).