The degree complexity of smooth surfaces of codimension 2

摘要

For a given term order, the degree complexity of a projective scheme is defined by the maximal degree of the reduced Grobner basis of its defining saturated ideal in generic coordinates (Bayer and Mumford, 1993). It is well known that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity (Bayer and Stillman, 1987). However, much less is known if one uses the graded lexicographic order (Ahn, 2008; Conca and Sidman, 2005). In this paper, we study the degree complexity of a smooth irreducible surface in P4 with respect to the graded lexicographic order and its geometric meaning. As in the case of a smooth curve (Ahn, 2008), we expect that 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 in a few cases, the degree complexity of a smooth surface S of degree d with h~0(ζ_s(2))≠ 0 in P~4 is given by 2 + (deg Y_1(S)-1/2) -g(Y_1 (S)), where Y_1(S) is a double curve of degree (d-1/2)-g(S∩H) under a generic projection of S. In particular, this complexity is actually obtained at the monomial where k[x_0, x_1, x_2, x_3, x_4] is a polynomial ring defining P~4. Exceptional cases are a rational normal scroll, a complete intersection surface of (2,2)-type, or a Castelnuovo surface of degree 5 in P~4 whose degree complexities are in fact equal to their degrees. This complexity can also be expressed in terms of degrees of defining equations of I_s in the same manner as the result of A. Conca and J. Sidman (Conca and Sidman, 2005). We also provide some illuminating examples of our results via calculations done with Macaulay 2 (Grayson and Stillman, 1997).