Ordinary plane models and completely split divisors

Let C be a smooth, non-hyperelliptic curve over an algebraically closed field of genus g ≥ 4. We show that the projection from the canonical model of C through (g−3) generic points on C is a birational morphism to a plane curve which has only finitely many non-ordinary tangents, that is, flexor bitangents. For smooth, non-hyperelliptic curves of a fixed genus g ≥ 4 over finite fields, we show that the probability that an effective divisor of degree (g−3) defines such an “ordinary (birational) plane model” converges to 1. This result has an application to the solution of the discrete logarithm problem for smooth, non-hyperelliptic curves curves of a fixed genus g over finite fields Fq: By first changing the representation to such an ordinary plane model and then using an algorithm by the first author, the problem can be solved in an expected time of Õ(q2− 2 g−1 ). Another consequence is that for smooth, non-hyperelliptic curves of a fixed genus g over finite fields Fq, the number of completely split divisors in the canonical system is ∼ 1 (2g+2)! ⋅ q q−1.