On blowup of secant varieties of curves

In this paper, we show that for a nonsingular projective curve and a positive integer \begin{document}$ k $\end{document} , the \begin{document}$ k $\end{document} -th secant bundle is the blowup of the \begin{document}$ k $\end{document} -th secant variety along the \begin{document}$ (k-1) $\end{document} -th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.