On nondegeneracy of curves

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an alge-braically closed field is nondegenerate in the above sense. More generally, let Mndg be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimMndg = min(2g+1, 3g − 3), except for g = 7 where dimMnd 7 = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4