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Non-projective compactifications of $$\mathbb{C}^ 3$$. I. (English) Zbl 0886.32021
The author determines the singular Fano compactifications $$(V,A)$$ of $${\mathbb{C}}^{3}$$ with index $$r=2$$. In fact, it is shown that $$(V,A) \cong (V_{4},H_{4})$$, where $$V_{4}$$ is a complete intersection of two quadrics in $${\mathbb{P}}^{5}$$ with exactly one small Gorestein singularity $$p$$, and $$H_{4}$$ is a non-normal hyperplane section of $$V_{4}$$ which is a non-normal cone over a nodal curve in $${\mathbb{P}}^{3}$$ of degree $$4$$.

MSC:
 32J05 Compactification of analytic spaces
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