Multicusps

For a given multicusp $f=c_{({\theta }_0,\mathrm{.}\mathrm{.}\mathrm{.},{\theta }_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\bar{\omega }f$, where ${}_i\bar{\omega }f$ is a higher version of the reduced Kodaira-Spencer-Mather map $\bar{\omega }f$. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathbb{N}$ and any $f=c_{({\theta }_0,\mathrm{.}\mathrm{.}\mathrm{.},{\theta }_i)}$, ${}_i\bar{\omega }f$ is bijective. The corollary is an affirmative answer to the question raised by M. A. S. Ruas during the 11th International Workshop on Real and Complex Singularities at the University of S$\tilde{\mathrm{a}}$o Paulo in S$\tilde{\mathrm{a}}$o Carlos (2010).