  Given a polarized variety $(X,L)$ we use an elementary observation to produce
projections of low degree $ X\subset \mathbb{P}(H^0(L^\vee)) \dashrightarrow
\mathbb P ^r $ via vector bundles. For surfaces this observation can be used to
show that maps of the degree at most $d$ move in families. In the case of
polarized $K3$ surfaces we study projections of the lowest possible degree up
to genus $6$ and we give a new upper bound for higher genera. As a different
application of our construction we exhibit new rational maps of low degree for
some hyper-K\"ahler varieties, abelian varieties and Gushel-Mukai threefolds.

Algebraic Geometry