Archinard studied the curve C over 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 perfect 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.
In this paper, we consider the finite groups which act on the 2-sphere S 2 and the projective plane P 2 , and show how to visualize these actions which are explicitly defined. We obtain their quotient types by distinguishing a fundamental domain for each action and identifying its boundary. If G is an action on P 2 , then G is isomorphic to one of the following groups: S 4 , A 5 , A 4 , Z m or Dih(Z m ). For each group, there is only one equivalence class (conjugation), and G leaves an orientation reversing loop invariant if and only if G is isomorphic to either Z m or Dih(Z m ). Using these preliminary results, we classify and enumerate the finite groups, up to equivalence, which act on P 2 ×I and the twisted I-bundle over P 2 . As an example, if m > 2 is an even integer and m/2 is odd, there are three equivalence classes of orientation reversing Dih(Z m )-actions on the twisted I-bundle over P 2 . However if m/2 is even, then there are two equivalence classes.
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