论文标题
投影空间和Koszul属性中的通用线条
Generic Lines in Projective Space and the Koszul Property
论文作者
论文摘要
在本文中，我们研究了$ \ mathbb {p}^n $的一般线的均匀坐标环的Koszul属性，以及在$ \ mathbb {p}^n中以一般线性位置的一系列线性位置的均匀坐标环。 $ \ mathbb {p}^n $带有$ 200M \ leq n+1 $和$ r $是$ \ nathcal {m}的坐标环，然后$ r $是koszul。 Further, if $\mathcal{M}$ is a generic collection of $m$ lines in $\mathbb{P}^n$ and $R$ is the coordinate ring of $\mathcal{M}$ with $m$ even and $m +1\leq n$ or $m$ is odd and $m +2\leq n,$ then $R$ is Koszul.最后，我们显示$ \ Mathcal {M} $是$ M $行的一般集合，因此\ [m> \ frac {1} {72} {72} {72} \ left（3（n^2+10n+13）+\ sqrt {3（n-1）^3（n-1）^3（3n+5）}}}}}} \ right）我们为$ n \ leq 6 $或$ m \ leq 6 $的通用行集合的Koszul属性进行了完整的表征。我们还确定了坐标环的Castelnuovo-Mumford规则性，用于一般的线路集合以及一般线性位置的线收集的坐标环的射击尺寸。
In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb{P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb{P}^n.$ We show that if $\mathcal{M}$ is a collection of $m$ lines in general linear position in $\mathbb{P}^n$ with $2m \leq n+1$ and $R$ is the coordinate ring of $\mathcal{M},$ then $R$ is Koszul. Further, if $\mathcal{M}$ is a generic collection of $m$ lines in $\mathbb{P}^n$ and $R$ is the coordinate ring of $\mathcal{M}$ with $m$ even and $m +1\leq n$ or $m$ is odd and $m +2\leq n,$ then $R$ is Koszul. Lastly, we show if $\mathcal{M}$ is a generic collection of $m$ lines such that \[ m > \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\] then $R$ is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n \leq 6$ or $m \leq 6$. We also determine the Castelnuovo-Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.
