Ok, answering my own question. The user Nefertiti pointed in the comments to the following picture in Hartshorne, and it seems indeed to be a clearer version of the same idea.
![enter image description here](
Ok, answering my own question. The user Nefertiti pointed in the comments to the following picture in Hartshorne, and it seems indeed to be a clearer version of the same idea.
![enter image description here](