  Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$
over a field $\k$ of characteristic~0. We calculate the $K$-theory of $R$ in
terms of the geometry of the projective embedding of $X$. In particular, if $X$
is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that
$K_{-1}(R)=\oplus H^1(C,\cO(n))$. The formula for $K_0(R)$ involves the Zariski
cohomology of twisted K\"ahler differentials on the variety.
