The hidden smoothness principle refers to the conjectural picture envisioned in 1980s by Deligne, Drinfeld, Beilinson, Kontsevich that moduli spaces in algebraic geometry which are often singular, should be just truncations of a moduli spaces in some derived sense (precursors seen often at the level of derived categories of coherent sheaves and conjecturally these moduli spaces would locally involve commutative differential graded algebras); furthermore these moduli spaces should be smooth, and this property is lost due truncation. The derived moduli spaces were indeed realized in derived algebraic geometry and often indeed smoothness fails for truncation reasons. However, the smoothness is, even for the derived moduli spaces, not always present, hence the conjecture resulted in a heuristic rather than in a general rule.

One of the principal concrete motivations was the idea to give the intrinsic geometric presentation and proof of the Serre intersection formula.

- Maxim Kontsevich,
*Enumeration of rational curves via torus actions*, in: The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math.**129**, Birkhäuser 1995. MR1363062 (97d:14077), hep-th/9405035 - Clark Barwick,
*Why derived algebraic geometry*, 2007, pdf

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