On the geometric genus of projective curves Original Research Article

We determine all the possible geometric genera of curves of degree d in image which are not contained in any surface of degree < r + α − 1 with α ≤ r − 2. We show that all geometric genera between 0 and πα(d, r) (the Harris-Eisenbud bound) do in fact occur by simplifying the nodes of rational curves of large arithmetic genus lying on ruled rational surfaces. We also give a simple proof of the explicit simplification theorem that we need.