“Direct” calculation of $K_0$ for surfaces, 3-folds

I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-dimensional varieties (besides nonsingular curves, which is an exercise in Hartshorne).

To try to explain further, here is an example of what I am not looking for: in the case of toric surfaces, one can obtain the $T$-equivariant $K$-theory by localization, and then mod out the equivariance to obtain regular $K$-theory. This does not give me enough understanding about which relations between sheaves are really at play.

For smooth projective surfaces (over $\mathbb{C}$ say) one has filtration $F^1K_0(X)\subset K_0(X)$ with the quotient being $\mathbb{Z}$, $F^2K_0(X)\subset F^1K_0(X)$, with the quotient being Picard group of $X$ and $F^2K_0(X)$ isomorphic to the the Chow group of zero cycles mod rational equivalences, $A_0(X)$. The latter can be calculated in some instances, for example, if $X$ is rational, it is $\mathbb{Z}$.
• More generally, for a smooth projective variety the Chern character gives an isomorphism of $K_0(X)\otimes _{\mathbb{Z}}\mathbb{Q}$ with the Chow ring with rational coefficients. So modulo torsion this is the same problem as computing the Chow ring -- not so easy, but doable in some cases. – abx Jul 30 '17 at 6:14
• When two sheaves are equal in $K$-theory, their difference in the free abelian group generated by isomorphism classes of sheaves is in the subgroup generated by relations coming from exact sequences. I guess I want to see what the difference actually is, in this subgroup. – A. S. Jul 30 '17 at 15:18