The inspiration for this paper comes in a theorem proven in [Sch] that implies for a geometric generic hypersurface X_(|C) of degree d in P~(n+1), with n+ 2 ≤ d ≤ 2n-2, there exist two lines on X_(|C) whose difference has infinite order in C H_1 (X_(|C))_(alg). (This follows from [Sch, Thm 0.7.] and a connectedness result in [Bo, Thm 4.1.].) The argument involves a deformation of lines to a singular fiber, where some information is known. A different proof of this result, based on Roitman's theorem on zero cycles on varieties of non-zero genus, can be found in [P]. Alberto Collino [Co] has also indicated another proof, in a similar spirit to [P]. We would like to arrive at a general result which will have a broader scope of application. The proof will involve a combination of a deformation argument, together with some of Roitman's results on dimensions of orbits.
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