If "V" is now taken to be of dimension 1, we get a picture of a typical algebraic curve "C" presented 'over' P1("K"). Assuming "C" is non-singular (which is no loss of generality starting with "K"("C")), it can be shown that such a rational map from "C" to P1("K") will in fact be everywhere defined. (That is not the case if there are singularities, since for example a "double point" where a curve "crosses itself" may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is ramification.