Birationally very ample line bundles on smooth curves

Abstract. For all integers r ≥ 2 and any smooth and connected projective curve X, let ρX(r) denote the minimal integer d such that there is a morphism φ: X → Pr birational onto its image and such that deg(φ(X)) = d and φ(X) spans Pr. Fix in-tegers d, g such that d ≥ 8 and d2/6 < g ≤ d2/4−d. Here we prove the existence of a smooth genus g curve X such that ρX(3) = d