Original sentence: (While it would be possible to take  even if R were not finitely generated, the result would not be a variety.) The canonical model is the image of the canonical map, and therefore the homogeneous coordinate ring of the image of the canonical map is the canonical ring.In fact, this is true even more generally: For instance, if V is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map.Its dimension is called the Kodaira dimension of V.The Proj of the canonical ring is a projective variety called the canonical model of R.The canonical ring is a graded ring whose zeroth-degree component is the underlying field.The canonical ring of V is the direct sum:
Edited sentence: For instance, if V is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map.In all cases the canonical ring of V is the direct sum and graded ring.Its dimension measure by transcendence degree is called the Kodaira dimension of V.The canonical model is the image of the canonical map, and therefore its homogeneous coordinate ring is the canonical ring.The zeroth-degree component is by definition the underlying field.To take in more examples, the Proj of the canonical ring is also called the canonical model; it is possible to take  even if R is not finitely generated, bur the result would not be a variety