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Migliore, Juan; Patnott, Megan

doi:10.1016/j.jpaa.2010.10.009

Cited by 8 publications

(1 citation statement)

References 17 publications

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“…For example, if one has a graded Cohen–Macaulay -algebra, then one can take a quotient by generic linear forms to produce an Artinian ring. A reduction of this kind is called an Artinian reduction and provides many useful tools to work with, and almost all homological invariants of the ring are preserved [18]. Unfortunately, we will not be able to use these tools or reductions as the coordinate ring of a generic collection of lines is almost never Cohen–Macaulay, whereas the coordinate ring of a generic collection of points is always Cohen–Macaulay.…”

Section: Introductionmentioning

confidence: 99%

Generic Lines in Projective Space and the Koszul Property

RICE

2023

Nagoya Math. J.

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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. Furthermore, 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 that if $\mathcal {M}$ is a generic collection of m lines such that $$ \begin{align*} m> \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$ 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.

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Section: Introductionmentioning

confidence: 99%