$K_0$ ring of del-pezzo surface

Question

Is there book or paper where described structure of $K_0$ ring of a del-pezzo surfaces? Especially in case of the blowup of projective line in three points. I know method for computing $K_0$ ring of blowup but it's computationally heavy and i'm afraid of making a mistake.

Answer 1

Let $X$ be the blowup of a plane at points $P_1$, \dots, $P_n$ with the corresponding exceptional divisors $E_1$, \dots, $E_n$. Let $H$ be the pullback of the line class from the plane to $X$. Then $$ O, O(H), O(2H), O_{E_1}, \dots, O_{E_n} $$ is a full exceptional collection, hence the corresponding classes $$ [O], [O(H)], [O(2H)], [O_{E_1}], \dots, [O_{E_n}] $$ form a basis in $K_0$.

To understand the multiplication note that there are exact sequences $$ 0 \to O((t-3)H) \to O((t-2)H)^{\oplus 3} \to O((t-1)H)^{\oplus 3} \to O(tH) \to 0 $$ for each $t$ (this is a twist of the pullback of the Koszul complex), that allows to express inductively all $[O(tH)]$ from $[O]$, $[O(H)]$, $[O(2H)]$, and $$ 0 \to O \to O(H)^{\oplus 2} \to O(2H) \to O_P \to 0 $$ (again the pullback from the plane) that allows to express $[O_P]$.

After that you can write $$ [O(aH)] \cdot [O(bH)] = [O((a+b)H)],\ [O(aH)] \cdot [O_{E_i}] = [O_{E_i}],\ [O_{E_i}] \cdot [O_{E_j}] = -\delta_{ij}[O_P]. $$