Is very well known that the only algebraic curve which admits no deformations is the projective line. 



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> **Q.** What are "rigid" smooth algebraic surfaces? Is there a sensible classification?
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Algebraic surfaces with no deformations
There are several different notions of "rigidity" (local rigidity, global rigidity, infinitesimal rigidity, étale rigidity and strong rigidity) and it is possible to provide examples for each of them.


This topic is discussed in the paper by I. Bauer and F. Catanese *[On rigid compact complex surfaces and manifolds](https://arxiv.org/abs/1609.08128)*, you can have a look at it for more details and references to the literature. 


