A note on the Cohen-Macaulay type of lines in uniform position in 𝐴ⁿ⁺¹

Abstract

Let L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} be s s -distinct lines in A k n + 1 {\mathbf {A}}_k^{n + 1} passing through the origin. Assume s = ( n n + d ) − λ s = (_n^{n + d}) - \lambda where n n , d ⩾ 2 d \geqslant 2 . If L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} are in generic s s -position, and λ = 0 \lambda = 0 . 1 , … , n − 1 1, \ldots ,n - 1 , then the Cohen-Macaulay type, t ( L 1 , … , L s ) t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) , of L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} is given by the following formula: t ( L 1 , … , L s ) = ( n − 1 n + d − 1 ) − λ t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{n - 1}^{n + d - 1}) - \lambda . This formula is known to be false for λ = n \lambda = n . In this paper, we show that if L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} are in uniform position, and λ = n \lambda = n . then t ( L 1 , … , L s ) = ( n − 1 n + d − 1 ) − n t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{\;n - 1}^{n + d - 1}) - n .