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The dual of a line (a curve of degree 1) is an exception to this and is taken to be a point in the dual space (namely the original line). The dual of a single point is taken to be the collection of lines though the point; this forms a line in the dual space which corresponds to the original point.
If {{mvar|X}} is smooth, i.e. there are no [[Singular point of a curve|singular points]] then the dual of {{mvar|X}} has the maximum degree {{math|''d''(''d'' − 1)}}. If {{mvar|X}} is a conic this implies its dual is also a conic. This can also be seen geometrically: the map from a conic to its dual is [[one-to-one correspondence|one-to-one]] (since no line is tangent to two points of a conic, as that requires degree 4), and tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires degree 3).
For curves with singular points, these points will also lie on the intersection of the curve and its polar and this reduces the number of possible tangent lines. The degree of the dual given in terms of the ''d'' and the number and types of singular points of {{mvar|X}} is one of the [[Plücker formula]]s.
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