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Let $X\subset CP^N$ be a homogeneous (or perhaps just smooth) complex subvariety and let $S^r(X)$ denote its abstract $r$-th secant variety (the incidence variety in $X\times \cdots \times X\times CP^N$), and assume the map to the actual secant variety in $CP^N$ (the projection onto the last factor) is generically finite to one. Is it possible that there is some point in the actual secant variety whose fiber has a component that is an isolated point and another component that is positive dimensional? (I just care about the case where $X$ is a triple Segre product if that makes any difference.)

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