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Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon X\hookrightarrow\...

Consider the following fact from algebraic geometry: Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set. I know how to prove this fact ...

Background Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$ ...