(with A. Harris and T. Keller) will appear on 2022. Submitted.
The relative parameter is non-negative integer associated to a pair (G,N), with G a finite group and N a normal subgroup that generalizes the parameter e which is associated to G and satisfies |G|=d(d+e) where d is the square root of a conjugacy class. Harrison proved several results concerning the non-relative parameter e and bounded |G| above in terms of e and characterized all groups G satisfying equality. In Section (3) of this paper we study the relative parameter corresponding to certain pairs (G,N), and bound |G:N| above by the relative parameter and partially characterize those groups achieving equality. In section (4) we explain why complete characterization is not completely possible.
A graded k-algebra is said to be Koszul if the minimal R-free graded resolution of k is linear. In this paper we study the Koszul property of the homogeneous coordinate ring R of a set of m lines in the complex projective space. Kempf proved that R is Koszul if there is s points in general linear position and s is less than or equal to 2n. Further, Conca, Trung and Valla showed that if the points are algebraically independent over Q, then the coordinate ring is Koszul if and only if s less than or equal to 1+n+n^2/4. We expand on Kempfs Theorem with the exception we consider lines in projective space
Note: In this project, I used Macaulay2 to demonstrate that a generic collection of 5 lines in projective 6-space does not have a nice filtration. Here is some small code you use to check that the required colon ideals are equal.