GENERIC LINES IN PROJECTIVE SPACE AND THE KOSZUL PROPERTY

Abstract

AbstractIn 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 ofmlines in general linear position in$\mathbb {P}^n$with$2m \leq n+1$andRis the coordinate ring of$\mathcal {M},$thenRis Koszul. Furthermore, if$\mathcal {M}$is a generic collection ofmlines in$\mathbb {P}^n$andRis the coordinate ring of$\mathcal {M}$withmeven and$m +1\leq n$ormis odd and$m +2\leq n,$thenRis Koszul. Lastly, we show that if$\mathcal {M}$is a generic collection ofmlines such that$$ \begin{align*} m> \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$thenRis 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.