Gorenstein liaison and special linear configurations

Abstract

Liaison theory has been extensively studied during the past decades. In codimension 2, the theory has reached a very satisfactory state, but in higher codimensions there are still many open problems. In this paper we prove that two unions $V= \bigcup_{i=1}^k L_i$ and $V'= \bigcup_{i=1}^{k'} L'_i$ of independent linear varieties of dimension $d \geq 1$ in $\mathbb{P}^n$ are in the same G-liaison class if and only if $k=k'$ or, equivalently, if $V$ and $V'$ have isomorphic deficiency modules $M^i(V) \cong M^i (V')$, $i=1, \dots, d$. We also describe the G-liaison classes of arithmetically Buchsbaum divisors on rational normal scrolls.