Let $X_4$ be a nonsingular quartic surface in $\mathbb{P}^3$. Let $C_1, C_2 \subset X_4$ be two nonsingular curves on $X_4$, both having degree 6 and genus 3. Suppose that $C_1 \cup C_2$ is the complete intersection of $X_4$ and a smooth cubic surface $X_3$. What is the intersection product $C_1 \cdot C_2$?
I am not familiar with intersection theory. I have been told that this is a simple calculation using the Adjunction formula and that the answer should be 14. I have looked up the Adjunction formula in Hartshorne, but cannot figure out how to derive the answer from that.
Best Answer
Here is one way such calculations are formally done. $C_1+C_2$ as a divisor is $X_3\cdot X_4$. So, $C_1\cdot(C_1+C_2)=18$, since $\deg C_1=6$. The canonical bundle of $X_4$ is trivial, being a quartic in 3-space. So, adjunction says $2g(C_1)-2=4=C_1^2$. Thus, $C_1\cdot C_2=18-C_1^2=14.$