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}).Comment: 18 pages. Some theorems and examples are added. The case of singular space curves is delete

Generic Initial ideals of Singular Curves in Graded Lexicographic Order

In this paper, we are interested in the generic initial ideals of \textit{singular} projective curves with respect to the graded lexicographic order. Let $C$ be a \textit{singular} irreducible projective curve of degree $d\geq 5$ with the arithmetic genus $\rho_a(C)$ in \p^r where $r\ge 3$. If $M(I_C)$ is the regularity of the lexicographic generic initial ideal of $I_C$ in a polynomial ring $k[x_0,..., x_r]$ then we prove that $M(I_C)$ is $1+\binom{d-1}{2}-\rho_a(C)$ which is obtained from the monomial x_{r-3} x_{r-1}\,^{\binom{d-1}{2}-\rho_a(C)}, provided that \dim\Tan_p(C)=2 for every singular point $p \in C$. This number is equal to one plus the number of non-isomorphic points under a generic projection of $C$ into \p^2. %if $\deg(C)=3,4$ then $M(I_C)= \deg(C)$ by the direct computation. Our result generalizes the work of J. Ahn for \textit{smooth} projective curves and that of A. Conca and J. Sidman \cite{CS} for \textit{smooth} complete intersection curves in \p^3. The case of singular curves was motivated by \cite[Example 4.3]{CS} due to A. Conca and J. Sidman. We also provide some illuminating examples of our results via calculations done with {\it Macaulay 2} and \texttt {Singular} \cite{DGPS, GS}.Comment: 11 page

Sharp bounds for higher linear syzygies and classifications of projective varieties

In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X$ and those of its inner projection $X_q$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).Comment: 22 pages, 7 figures, comments are welcome