"Cuts deep" can certainly be a valid phrase. Without more context, I cannot give any opinion about its use in this particular piece of writing. But it might be worth recalling the traditional saying "still waters run deep". In that usage "deep" seems to work better than "deeply".

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“Cut deep” is a verb with a cognate accusative adjective here working as an adverb. “Cut deeply,” however, is a verb with an adverb. Both of these cases are valid grammatically.

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I'm editing an op-ed for a friend, and her title includes "the stigma cuts deeply". Sure, deeply can work adverbially here, but am I wrong to think that "deep" can also work? The former meaning that it's cutting deeply as a sort of continuous act and the latter as a kind of singular, in-the-instance example and therefore "deep"?

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The martensite reaction in steels normally occurs athermally, i.e. during cooling in a temperature range which can be precisely defined for a particular steel.

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A milling cutter is a primary cutting tool in the milling process, made from hard and wear-resistant materials such as alloy steel, tungsten carbide, or ...

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Jan 11, 2013 — Try to cut it with a ts55 or skil saw. When you are cutting plastic with a jigsaw, the width of the cut is such that it will melt back together behind the ...