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

TL;DR

This paper generalizes previous work on curves linked to Appell-Lauricella hypergeometric series by describing a partial desingularization over fields with certain characteristics and analyzing the Cartier operator in positive characteristic.

Contribution

It provides an explicit basis for the dualizing sheaf on a desingularized curve and characterizes the Cartier operator using hypergeometric series in positive characteristic.

Findings

Explicit basis for the dualizing sheaf on the desingularized curve

Description of the Cartier operator in terms of hypergeometric series

Partial desingularization under mild characteristic conditions

Abstract

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