BOUND ON THE NUMBER OF RATIONAL POINTS ON CURVES ON HIRZEBRUCH SURFACES OVER FINITE FIELDS

This paper gives a bound on the number of rational points on an absolutely irreducible curve C lying on a minimal toric surface X. This upper bound improves pre-existing ones if C has large genus. The strategy consists in finding another curve that intersects C with good multiplicity at its rational points outside some well-handled closed set. Finding such a curve relies on an extension of K.O. StöhrSt¨Stöhr and F.J. Voloch's idea for plane curves to the toric framework based on homogenization