Large gains in the rate of cache-aided broadcast communication are obtained using coded caching, but to obtain this most existing centralized coded caching schemes require that the files at the server be divisible into a large number of parts (this number is called subpacketization). In fact, most schemes require the subpacketization to be growing asymptotically as exponential in $\sqrt[\leftroot{-1}\uproot{1}r]{K}$ for some positive integer $r$ and $K$ being the number of users. On the other extreme, few schemes having subpacketization linear in $K$ are known; however, they require large number of users to exist, or they offer only little gain in the rate. In this work, we propose two new centralized coded caching schemes with low subpacketization and moderate rate gains utilizing projective geometries over finite fields. Both the schemes achieve the same asymptotic subpacketization, which is exponential in $O((\log K)^2)$ (thus improving on the $\sqrt[\leftroot{-1}\uproot{1}r]{K}$ exponent). The first scheme has a larger cache requirement but has at most a constant rate (with increasing $K$), while the second has small cache requirement but has a larger rate. As a special case of our second scheme, we get a new linear subpacketization scheme, which has a more flexible range of parameters than the existing linear subpacketization schemes. Extending our techniques, we also obtain low subpacketization schemes for other multi-receiver settings such as distributed computing and the cache-aided interference channel. We validate the performance of all our schemes via extensive numerical comparisons. For a special class of symmetric caching schemes with a given subpacketization level, we propose two new information theoretic lower bounds on the optimal rate of coded caching.