Geodesic
On the variety of linear series on a singular curve
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 631-639.

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Let $Y$ be an integral projective curve with $g := p_{a}(Y) \geq 2$. For all positive integers $d$, $r$ let $W^{r}_{d}(Y)(\text{}^{**})$ be the set of all $L \in \text{Pic}^{d}(Y)$ with $h^{0}(Y, L) \geq r+1$ and $L$ spanned. Here we prove that if $d \leq g-2$, then $\dim (W^{r}_{d}(Y) (\text{}^{**})) \leq d-3r$ except in a few cases (essentially if $Y$ is a double covering).
Sia $Y$ una curva proiettiva ridotta e irriducibile con $g:= p_{a}(Y) \geq 2$. Per ogni $d$, $r$ interi positivi sia $W^{r}_{d}(Y)(\text{}^{**})$ l'insieme di tutti gli $L \in \text{Pic}^{d}(Y)$ con $h^{0}(Y, L) \geq r+1$ e $L$ generato dalle sezioni globali. In questo lavoro si dimostra che se $d \leq g-2$, allora $\dim (W^{r}_{d}(Y) (\text{}^{**})) \leq d-3r$ fatte salve rare eccezioni (essenzialmente il caso in cui $Y$ sia un rivestimento doppio della retta proiettiva). 
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     title = {On the variety of linear series on a singular curve},
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Ballico, E.; Fontanari, C. On the variety of linear series on a singular curve. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 631-639. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a3/

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