Frobenius non-classical curves

Abstract

The proof of Proposition 5 of [3] is incomplete. With notation as in the paper, the possibility that the polynomial X~/q'P o + X~/~' P1 + X~/q" Pz in (11) could be identically zero was overlooked. We will sketch here a proof that in this case X does not have controlled singularities so this case can indeed be discarded in the proof of Proposition 5. and P1 is a zero of X~/q' P2 and, since we are in the generic case, is a zero of PE and gives a singular point of F = 0 with Jacobian ideal of multiplicity at least q' since c~F/~X i = P~'. rather than ~ ee < 2d. P~XWe would like to thank M. Homma who pointed out to us the case overlooked and suggested part of the above argument 1o show that any counterexample would be singular. References[1] E. BALLICO and A. HEwz, Non-reflexive projective curves of low degree. Manuscripta Math. 70, 385-396 (1991).[2] A. HEFEZ, Non reflexive curves. Compositio Math. 69, 3-35 (1989).[3] A. HEFEZ and J. E VOLOCH, Frobenius non-classical curves.

On multi-Frobenius non-classical plane curves

“…The curve F is called q-Frobenius non-classical if the image Fr(P ) of each simple point P of F under the Frobenius map lies on the tangent line at P . Based on [8], Hefez and Voloch extended the study of the q-Frobenius non-classical curves in [6] where some interesting arithmetic and geometric properties of such curves were first pointed out. For instance, if F is smooth of degree d, Hefez and Voloch proved that the number of F q -rational points of F is given by…”

The separability of the Gauss map and the reflexivity for a projective surface

“…Is the converse also true? The case of dim X = 1 has been studied by several authors (see [4, (3.5)], [5,Proposition 4], [6, Remark below Corollary 2.3], [7,Theorem], [10, Theorem 1]), and it is well known that the answer is affirmative. Then how about higher dimensional cases?…”

The separability of the Gauss map versus the reflexivity

“…F. Voloch [14,Proposition 4], the author [6, Remark following Corollary 2.3], the inseparable degree of the Gauss map γ is also equal to q, where we note that i(X, T P X ; P) = i(X, H ; P) for a general hyperplane H tangent to X at P. Thus, as is observed in Voloch [15, Theorem 1], we have the following (see [16][17][18] for a generalization to higher order cases): Theorem 2.1 For a projective curve X , the Gauss map γ and the conormal map π have the same inseparable degree.…”