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Weil bounds for singular curves. (English) Zbl 0877.11038

The paper under review contains three theorems. The first one is a generalization to the singular case of the Weil bound for the number of rational points of a smooth irreducible curve defined over a finite field. Note that this was also (and independently) obtained by Y. Aubry and M. Perret in [Manuscr. Math. 88, 467-478 (1995; Zbl 0862.11042)]. This generalization states that the Weil bound remains true if one replaces in the formula the geometric genus of the curve \(C\) by its arithmetic genus (this statement can be slightly sharpened; see the complete statement of the theorem). This is proved by desingularization of the curve: one has only to evaluate the number of points of the smooth model \(\widetilde C\) of \(C\) lying over a given singular point of \(C\), which is the object of lemma 4.1. Note that this lemma is not the best one: the given bound is twice the correct one, which is given in [op. cit.] (in particular, the sharpened version of the theorem is not the best one…).
The second theorem states that the arithmetic genus of a curve of degree \(d\) in projective \(n\)-space is less than \({d(d-1) \over 2}\). This is done by induction, asserting that the projection from a generic point on the hyperplane at infinity can only make an increase in the arithmetic genus and leaves the degree unchanged. Note a misprint in the proof of theorem 3.2: one should read “\(p_a(C) \leq p_a(D)\)”.
Finally, the third theorem gives another upper bound for the arithmetic genus of a curve in projective \(n\)-space in terms of the degrees of the \((n-1)\) first polynomials defining it (note that the curve need not be a complete intersection). One has to prove that the arithmetic genus of the curve is less than the arithmetic genus of any complete intersection containing it.
Reviewer: M.Perret (Lyon)

MSC:

11G20 Curves over finite and local fields
14H20 Singularities of curves, local rings

Citations:

Zbl 0862.11042
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References:

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