Recently I read some results about derived categories of coherent sheaves, and see one use Fourier-Mukai transforms to prove that the derived categories of coherent sheaves of a scheme, under some conditions, determine the scheme. More precisely, I have seen

(Bondal and Orlov) Let $X$ and $Y$ be smooth projective varieties. If there is an exact equivalence $\text{D}^{\text{b}}(X)\simeq\text{D}^{\text{b}}(Y)$ and the canonical sheaf of $X$ is ample or anti-ample, then $X\simeq Y.$

(corollary of above plus some discussions on elliptic curves) Let $C$ be a smooth projective curve and $Y$ be a smooth projective variety. Then $\text{D}^{\text{b}}(X)\simeq\text{D}^{\text{b}}(Y)$ if and only if $X\simeq Y.$

(Gabriel) Let $X$ and $Y$ be smooth projective varieties. Then $\text{Coh}(X)\simeq\text{Coh}(Y)$ if and only if $X\simeq Y.$

My question is, do these results really help determine a variety? I mean, passing the question to the level of derived version will make it easier? Is there any concrete applications?

Thanks!!