论文标题
在投影方案的Syzygies和条件$ {\ mathrm nd}（l）$的第一个非平地
On the first non-trivial strand of syzygies of projective schemes and Condition ${\mathrm ND}(l)$
论文作者
论文摘要
令$ x \ subset \ mathbb {p}^{n+e} $为任何$ n $ dimensional封闭的子Cheme。我们主要对与Syzygies相关的两个概念感兴趣：一个是属性$ \ MathBf {n} _ {d，p}〜（d \ ge 2，〜p \ geq 1）$，这意味着$ x $ as $ x $ as $ d $ d $ grigartiond $ p $ - 最小的免费分辨率，另一个是$ $ $ $} n new notion $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n n n n n n e n n e n n e n new New notion $ \ \ \ \ \ \经典的“非排定”的条件需要一般有限的线性部分，不得包含在$ \ ell $的任何高度表面中。 首先，我们引入条件$ \ mathrm {nd}（\ ell）$，并考虑从概念中得出的示例和基本属性。接下来，我们证明了第一个非平凡的syzygies的分级贝蒂（Betti）数量上的急剧上限，这将概括为二次病例，并为更高程度的病例提供了表征，并为极端情况提供了特征。此外，在将属性的某些后果$ \ mathbf {n} _ {d，p} $造成的后果之后，我们将$ x $的分辨率表征为$ d $ - 线性算术上的cohen-macaulay是具有属性$ \ mathbf {n} _ {n} _ {n} _ {d，e} $和条件$ \ mathrm {nd} $ {nd}（D）从此结果，我们获得了一个共氮化刚度定理，这表明由于Eisenbud-Green-Hulek-PopeScu而言，$ 2 $ d $ d $ regularity的syzygetic刚性对$ 2 $ regularity进行了自然概括。
Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\mathrm{ND}(\ell)$ which generalizes the classical "being nondegenerate" to the condition that requires a general finite linear section not to be contained in any hypersurface of degree $\ell$. First, we introduce condition $\mathrm{ND}(\ell)$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first non-trivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property $\mathbf{N}_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithmetically Cohen-Macaulay as having property $\mathbf{N}_{d,e}$ and condition $\mathrm{ND}(d-1)$ at the same time. From this result, we obtain a syzygetic rigidity theorem which suggests a natural generalization of syzygetic rigidity on $2$-regularity due to Eisenbud-Green-Hulek-Popescu to a general $d$-regularity.
