arXiv 1402.31002014-02-13math.AGmath.AC

具有$\textbf{N}_{3,p}$性质的射影概形的合点、度和几何性质

On Syzygies, degree, and geometric properties of projective schemes with property $\textbf{N}_{3,p}$

Jeaman Ahn, Sijong Kwak

备注：: 14 pages

摘要：

对于嵌入射影空间的代数集$X$（union of varieties），如果齐次坐标环的$i$次合势是由$0\le i\le p$的次数$< d+i$的元素生成的，则称$X$满足性质$\textbf{N}_{d,p}$，$(d\ge 2)$（详见\cite{EGHP2}）.许多注意力已经支付给线性syzygies的二次计划$(d=2)$和他们的几何解释（参见。\cite{AK}、\cite{EGHP1}、\cite{HK}、\cite{GL2}、\cite{KP}）。然而，对于满足性质$\textbf{N}_{3,p}$的情况，实际上知之甚少。当$X$满足性质$\textbf{N}_{3,p}$时，本文利用分次Betti数给出了$X$的零维线性段的最大长度的一个精确上界（定理1.2（a））.特别地，如果$p$是$X$的余维$e$，则$X$的次数小于或等于$\binom{e+2}{2}$，并且等式成立当且仅当$X$是具有$3$ -线性分辨率的算术Cohen-Maucalay（定理1.2（b））。这是Eisenbud等人（\cite{EGHP1,EGHP2}）的结果推广到$\textbf{N}_{3,p}$、$(p\leq e)$的情况。

英文摘要：

For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $< d+i$ for $0\le i\le p$ (see \cite{EGHP2} for details). Much attention has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}). However, not very much is actually known about the case satisfying property $\textbf{N}_{3,p}$. In this paper, we give a sharp upper bound on the maximal length of a zero-dimensional linear section of $X$ in terms of graded Betti numbers (Theorem 1.2 (a)) when $X$ satisfies property $\textbf{N}_{3,p}$. In particular, if $p$ is the codimension $e$ of $X$ then the degree of $X$ is less than or equal to $\binom{e+2}{2}$, and equality holds if and only if $X$ is arithmetically Cohen-Maucalay with $3$-linear resolution (Theorem 1.2 (b)). This is a generalization of the results of Eisenbud et al. (\cite{EGHP1,EGHP2}) to the case of $\textbf{N}_{3,p}$, $(p\leq e)$.

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