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I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical problems. For example: moduli spaces problems, automorphism groups of varieties, birational classification of varieties, minimal model program and so on.

Furthermore, I would also be interested to know about theorems that are more aesthetically pleasing being stated in the derived category language. For instance, I think Serre duality can be generalised to singular varieties through derived categories.

Any reference about results is very welcome.

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I think that a good example of the usefulness of the Derived Category of coherent sheaves for studying classical questions is the recent preprint by Soheyla Feyzbakhsh

Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing,

where the author uses wall-crossing with respect to Bridgeland stability conditions in order to solve the classical Mukai problem of reconstructing a K3 surface from its hyperplane section.

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    $\begingroup$ As pointed out by dhy, this is not what is called Derived Algebraic Geometry. $\endgroup$ – abx Jan 28 '19 at 14:04
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    $\begingroup$ I agree. This is intended as an answer to the second part of the question, since "we show how to reconstruct the K3 surface containing the curve C as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on C" is a pleasant statement expressed by using the derived category of coherent sheaves (at least in my opinion). $\endgroup$ – Francesco Polizzi Jan 28 '19 at 15:18
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    $\begingroup$ Sure, in mine too. In fact I think the derived category of coherent sheaves has many nice applications to classical geometry (see e.g. Kuznetsov's work). I am not sure about Derived Algebraic Geometry, I'd like to see some interesting answers to the question. $\endgroup$ – abx Jan 28 '19 at 19:25

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