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

Abstract

Let I be the defining ideal of a non-degenerate smooth integral curve of degree d and of genus g in P-n where n >= 3. The degree-complexity of I with respect to a term order tau is the maximum degree in a reduced Grobner basis of I, and is exactly the highest degree of a minimal generator of in. (I). For the degree lexicographic order, we show that the degree-complexity of I in generic coordinates is 1 + ((d-1)(2)) - g with the exception of two cases: (1) a rational normal curve in P-3 and (2) an elliptic curve of degree 4 in P-3, where the degree-complexities are 3 and 4 respectively. Additionally if X subset of P-n 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. (C) 2007 Elsevier Ltd. All rights reserved.