The degree-complexity of the defining ideal of a smooth integral curve

Let I be the defining ideal of a non-degenerate smooth integral curve of degree d and of genus g in where n≥3. The degree-complexity of I with respect to a term order τ is the maximum degree in a reduced Gröbner basis of I, and is exactly the highest degree of a minimal generator of . For the degree lexicographic order, we show that the degree-complexity of I in generic coordinates is with the exception of two cases: (1) a rational normal curve in and (2) an elliptic curve of degree 4 in , where the degree-complexities are 3 and 4 respectively. Additionally if is a non-degenerate integral scheme then we show that, for the degree lexicographic order, the degree-complexity of X in generic coordinates is not changed by an isomorphic projection of X from a general point.