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Birational equivalence is a relationship between algebraic varieties that indicates they are isomorphic outside of lower-dimensional subsets. This implies that the varieties have the same function field, allowing for the study of their geometric properties through rational functions.
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A canonical divisor on an algebraic curve is a divisor class that is associated with the differential forms on the curve, reflecting its geometric properties. It plays a crucial role in the Riemann-Roch theorem, which connects the geometry of the curve with the algebraic structure of its function field.
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Rational equivalence is a fundamental concept in algebraic geometry, establishing a relationship between two algebraic varieties that share the same function field, implying they are birationally equivalent. This means that while they may differ in their specific geometric structures, they can be transformed into one another through a series of rational maps, preserving their essential algebraic properties.
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