What is going on when pen stops working on a section of paper?
If in doubt, try it out.

I just spilled some olive oil on a sheet of paper and wiped it off again. 

Result: The ballpoint pen does not write on the oily spot. 

Did you fat-finger the paper?

EDIT

The experiment above confirmed the finding in the question but didn't give an explanation. 

Actually, two hours later it is still not possible to write on the oil-soaked and more transparent parts of the paper with a ballpoint pen, while a felt-tip pen (here: an Edding, ther German equivalent to a Sharpie) leaves perfect lines on every part of the paper.

Paper is fibrous material with a porous structure - without that and the capillary effects coming with it, fountain pens wouldn't be able to write on it.

For ballpoint pens, this probably isn't so important - the ink is very viscous.

But paper also has a rough surface - and this might be crucial, as already suggested by Lord Stryker in his comment. The rough paper surface provides friction, allows the tungsten carbide ball to roll, take up ink and spread it on the paper.

On a smooth or slippery surface - and this seems to be the case for the oily regions on the paper - the tip of a ballpoint pen only slides but does not roll. As a result, no ink is transfered to the paper.