I was reading a novel. And in the novel I read a sentence where the writer wrote scrubbed at the paper but as per my opinion it should have been scrubbed on the paper. Because the preposition at shows a location. So, if the writer wanted to show the location then it sounds fine. But if the writer wanted to tell that something he scrubbed and he did on a piece of paper then I think the preposition on should have been used instead at.
At does not always signify location; it may also signify direction.
In object/complement position, a directional at phrase is often used instead of a simple noun phrase or a more conventional preposition to express a higher degree of energy and effort. It represents its object as in some sense a target of that effort.
He worked on finishing his essay tells you only what the object of his work was, but
He worked at finishing his essay suggests that finishing the essay was a matter of some urgency.
With verbs of motion an at phrase usually implies intention to collide:
He drove into the wall could be accidental, but
He drove at the wall can only be deliberate.
The child ran to her mother suggests only desire to get close to her mother, but
The child ran at her mother implies a desire to attack her mother.
The police charged the crowd implies merely that the police advanced swiftly toward the crowd, but The police charged at the crowd implies that they intended violent action.
In your example,
He scrubbed the paper suggests only an attempt to clean it, and
He scrubbed on the paper merely tells you where his scrubbing took place, but
He scrubbed at the paper suggests a strenuous effort; depending on the context, it may imply a strong emotion behind the effort, such as anger or frustration.
OALD describes the usage of scrub (at something) The example there is
The woman scrubbed at her face with a tissue.
She cursed, scrubbed at the paper, trying to dry it. And then it tore, nearly in half. This made tears creep to her eyes, she was not sure why.
The OALD page does not describe any usage such as scrub (on something). Interestingly, the Cambridge Dictionary says using at is an optional in its example.
But when I searched, I found that scrubbed on is possible but then it is in a different way.
This week I was wiped and rubbed. A little cotton pad was scrubbed on my hands, then put into a machine, which checks it for terrorist-type chemicals.
So, the correct expression will be using scrubbed at as the author wrote it.