$K$-theory of union of smooth curves. Is there a method that one can calculate algebraic $K$ theory of a number of smooth curves with some singularity points corresponding to the intersection points of two different curves, given we know the $K$ groups of each curve? (For example bunch of projective lines with some intersecting points)
 A: Question: "Is there a method that one can calculate algebraic K theory of a number of smooth curves with some singularity points corresponding to the intersection points of two different curves, given we know the K groups of each curve?"
Answer. Let $C:=C_1 \cup C_2$ and let $Z:=C_1 \cap C_2$ with $U:=C-Z$ the open complement. There is a long exact localization sequence
$$ \cdots \rightarrow K_{i+1}(U) \rightarrow K_i(Z) \rightarrow K_i(C) \rightarrow K_i(U) \rightarrow K_{i-1}(Z) \rightarrow \cdots $$
hence if you know the K-theory of the complement $U$ and $Z$ then you can calculate $K_i(C)$. If $Z:=\{p_1,..,p_l\}$ is the $l$ singular points on $C$, let  $C'_1:=C_1-\{p_1,..,p_l\}$ and $C'_2:=C_2-\{p_1,..,p_l\}$.
It follows $U = C'_1 \cup C'_2$ is the disjoint union of the two smooth curves $C_1',C_2'$. Hence $K_i(U) \cong K_i(C_1') \oplus K_i(C'_2)$.
You get the long exact sequence
$$ \cdots \rightarrow K_{i+1}(C_1')\oplus K_{i+1}(C_2') \rightarrow K_i(Z) \rightarrow K_i(C) \rightarrow K_i(C_1')\oplus K_i(C_2') \rightarrow K_{i-1}(Z) \rightarrow \cdots $$
For algebraic K-theory there is the long exact localization sequence and the Mayer-Vietoris sequence. A systematic use of the MV sequence sequences can be used to calculate the algebraic K-theory $K_i(X)$ of $X$ in terms of an open cover $U_i$ of $X$. I believe this holds for singular curves but do not have a precise reference.
