Abstract
Given a $0$-dimensional scheme $\mathbb{X}$ in an $n$-dimensional projective space $\mathbb{P}^n_K$ over an arbitrary field $K$, we use liaison theory to characterize the Cayley-Bacharach property of $\mathbb{X}$. Our result extends the result for sets of $K$-rational points given in [8]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of $\mathbb{X}$ when $\mathbb{X}$ has the Cayley-Bacharach property.
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