The degree complexity of smooth surfaces of codimension 2

For a given term order, the degree complexity of a projective scheme is defined by the maximal degree of the reduced Gröbner basis of its defining saturated ideal in generic coordinates [2]. It is wellknown that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity [3]. However, much less is known if one uses the graded lexicographic order [1], [5]. In this paper, we study the degree complexity of a smooth irreducible surface in P with respect to the graded lexicographic order and its geometric meaning. Interestingly, this complexity is closely related to the invariants of the double curve of a surface under a generic projection. As results, we prove that except a few cases, the degree complexity of a smooth surface S of degree d with h(IS(2)) 6= 0 in P 4 is given by 2 + ( degY1(S)−1 2 ) − g(Y1(S)), where Y1(S) is a double curve of degree