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While reading a paper by Ciliberto and Lazarsfeld ("On the uniqueness of certain linear series on some classes of curves") I came across a lemma of Castelnuovo used in the paper. The lemma says that if a divisor $D$ (on a smooth projective curve) is in $t_i$-uniform position with respect to a $g^{r_i}_{d_i}$ (i.e. degree $t_i$ divisors in $D$ impose independent conditions on the $g^{r_i}_{d_i}$), then for $t := \sum_i(t_i-1) + 1$ and $V:= \sum_i g^{r_i}_{d_i}$ we have:

  • if $t < \text{deg}(D)$ then $D$ is in $t$-uniform position with respect to $V$
  • if $t \geq \text{deg}(D)$ then $D$ imposes independent conditions on $V$

where the sum $V$ is the minimal sum of the linear series defined as the linear series associated to the image (in the appropriate global sections space) under the product map on sections.

The authors indicate that a proof of the lemma can be found in the paper "Alcune applicazioni di un classico procedimento di Castelnuovo" by Ciliberto. Unfortunately, after extensive searching online, I cannot find this paper. Nor can I find a proof of the lemma anywhere else.

Does anyone know where I might find some version of this paper or at least a proof of the lemma?

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  • $\begingroup$ Here at least is a little more complete reference. Alcune applicazioni di un classico procedimento di Castelnuovo, Seminario di Geometria, Universita di Bologna (1982-83), 17-43. Maybe you could email Ciro. [email protected] $\endgroup$
    – roy smith
    Jan 12, 2017 at 0:30

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