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This might be a very simple question but I can't seem to find a precise definition of the degree of an algebraic curve, if such can even be defined. In the plane, the degree of an algebraic curve is clear; it is simply the degree of the defining polynomial (a curve in the plane is defined by the equation $f(x,y) = 0$, for some polynomial $f$). In higher dimensions, say $n$, an algebraic curve is defined by $n-1$ polynomial equations $f_1 =0, f_2 =0, \cdots, f_{n-1} = 0$. However, the degree of the curve is no longer clear.

Can the degree of an algebraic curve be defined in higher dimensions? If so, how is it defined?

Thanks

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    I'd go with the degree of whichever of the $f_i$ has the largest degree.2011-08-18
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    It can be shown that the degree of a projective plane curve is equal to the degree of its hyperplane divisor; in simpler terms, the degree is equal to the number of intersections with a generic line. This has an obvious generalisation to higher dimensions, but I don't know whether this is the accepted definition.2011-08-18
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    For a purely algebraic definition, see I.7 of "Hartshorne - Algebraic Geometry" for the definition of *degree* of a projective variety.2011-08-18

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