The Degree Complexity of Smooth Surfaces of codimension 2

D.Bayer and D.Mumford introduced the degree complexity of a projective scheme for the given term order as the maximal degree of the reduced Gr\"{o}bner basis. It is well-known that the degree complexity with respect to the graded reverse lexicographic order is equal to the Castelnuovo-Mumford regularity (\cite{BS}). However, little is known about the degree complexity with respect to the graded lexicographic order (\cite{A}, \cite{CS}). In this paper, we study the degree complexity of a smooth irreducible surface in $\p^4$ 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 the generic projection. As results, we prove that except a few cases, the degree complexity of a smooth surface $S$ of degree $d$ with $h^0(\mathcal I_S(2))\neq 0$ in $\p^4$ is given by $2+\binom{\deg Y_1(S)-1}{2}-\rho_{a}(Y_{1}(S))$, where $Y_1(S)$ is a double curve of degree $\binom{d-1}{2}-\rho_{a}(S \cap H)$ under a generic projection of $S$ (Theorem \ref{mainthm2}). Exceptional cases are either a rational normal scroll or 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 only in terms of the maximal degree of defining equations of $I_S$ (Corollary \ref{cor:01} and \ref{cor:02}). We also provide some illuminating examples of our results via calculations done with {\it Macaulay 2} (Example \ref{Exam:01}).