For , a basis of the space of regular differentials defines the canonical map . Equivalently, after choosing a differential with divisor , one may use a basis of .
For a nonhyperelliptic smooth projective curve of geometric genus four, the canonical map is an embedding of degree six. Dimension counts and supply a quadric and a cubic not divisible by it. The quadric is irreducible because the curve is nondegenerate. Degree six and unmixedness of a complete intersection then identify the intersection with the curve as a scheme.
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In mathematics, particularly in the fields of topology and algebra, a **canonical map** refers to a specific type of structure-preserving function that is considered "natural" in a given context. It often arises in various mathematical settings and can have different interpretations depending on the area of mathematics in which it is used.