This text appears to be a summary of a lecture or discussion focused on algebraic geometry, specifically dealing with complex varieties, schemes, and toric degenerations. Here's a breakdown of the key points:
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Complex Varieties: The discussion primarily deals with algebraic varieties over complex numbers. However, it acknowledges that extending beyond the complex case is possible for those interested.
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General Schemes and Spec of a Ring: The lecture isn't focused on general schemes but rather on specific cases, like the spectrum (Spec) of some ring in variables defined by an ideal. This is related to the concept of vanishing sets of ideals, i.e., sets of points where functions in the ideal vanish.
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Vanishing Sets in (\mathbb{C}^n) and (\mathbb{C}^{n+1}): The speaker describes constructing varieties as vanishing sets of ideals within (\mathbb{C}^n) and (\mathbb{C}^{n+1}). They mention removing the zero vector from these sets and dividing by scaling, referring to a projective space