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Jul 9, 2012 at 22:50 comment added Jack Huizenga Review the sections in Eisenbud-Harris, "The geometry of schemes," on flat limits for similar constructions. Alternately, this is clear from a topological standpoint, as for any neighborhood of the line being projected onto (in ordinary topology) the curves $M_t(C)$ will lie in that neighborhood for small enough $t$ (by a compactness argument, since $C$ is bounded away from the center of projection).
Jul 9, 2012 at 21:35 comment added Naga Venkata @Huizenga: Thanks for the answer. However, I do not quite understand why the limiting curve that you are talking about is supported on a line. Could you please explain this part or suggest me some kind of reference.
Jul 9, 2012 at 16:34 vote accept Naga Venkata
Jul 9, 2012 at 9:46 history answered Jack Huizenga CC BY-SA 3.0