In algebraic geometry, singularities refer to points on a variety where the mathematical object fails to be well-behaved in some way, such as having undefined tangents or differing dimensions. These points can significantly impact the study of a variety, as they may indicate where the structure behaves differently from the regular or smooth parts of the variety. Understanding singularities is crucial in analyzing the properties and characteristics of varieties, especially when considering projective closures and the process of homogenization.
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