A note on the Cohen-Macaulay type of lines in uniform position in $\textbf {A}^{n+1}$
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- by William C. Brown PDF
- Proc. Amer. Math. Soc. 87 (1983), 591-595 Request permission
Abstract:
Let ${\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}$ be $s$-distinct lines in ${\mathbf {A}}_k^{n + 1}$ passing through the origin. Assume $s = (_n^{n + d}) - \lambda$ where $n$, $d \geqslant 2$. If ${\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}$ are in generic $s$-position, and $\lambda = 0$. $1, \ldots ,n - 1$, then the Cohen-Macaulay type, $t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s})$, of ${\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}$ is given by the following formula: $t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{n - 1}^{n + d - 1}) - \lambda$. This formula is known to be false for $\lambda = n$. In this paper, we show that if ${\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}$ are in uniform position, and $\lambda = n$. then $t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{\;n - 1}^{n + d - 1}) - n$.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 591-595
- MSC: Primary 13H10; Secondary 13H15, 14B05
- DOI: https://doi.org/10.1090/S0002-9939-1983-0687623-X
- MathSciNet review: 687623