I have been reading the paper "Heegner points and derivatives of $L$-series" by Gross and Zagier.
In section III of the paper, they use intersection theory to express a local height at a non-archimedean place in terms of an intersection product. I have been having trouble understanding the algebraic geometry in this section.
So far I have been reading chapters 7 and 9 of Qing Liu's book "Algebraic Geometry and Arithmetic Curves", but it seems as though I will need to learn scheme theory in more technical detail in order to get anything out of it.
I have read through the non-archimedean argument in Gross-Zagier's paper "On singular moduli", which can be regarded as a special case of the "Heegner points and derivatives of $L$-series". This has helped motivate the use of Hom-sets between points of $X_0(N)$, but doesn't seem to require the intersection theory arguments in the latter paper (maybe this is present in the form of Deuring's lifting theorem).
Finally I will state my question. Where should I look in order to get the necessary background to understand these calculations?