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Archinard studied the curve $C$ over $\mathbb{C}$ associated to anAppell-Lauricella hypergeometric series and differential forms on itsdesingularization. In this paper, firstly as a generalization of Archinard'sresults, we describe a partial desingularization of $C$ over a field $K$ undera mild condition on its characteristic and the space of global sections of itsdualizing sheaf, especially we give an explicit basis of it. Secondly, when thecharacteristic is positive, we show that the Cartier operator on the space canbe defined and describe it in terms of Appell-Lauricella hypergeometric series.

Differential forms on the curves associated to Appell-Lauricella hypergeometric series and the Cartier operator on them