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Maybe the question is too general, but nevertheless:

under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence?

What are typical classes of varieties which have this property? As far as know, it is true for $\mathbb{P}^2$, but what makes this variety special? (maybe rationality?)

Let's say we're over $\mathbb{C}$.

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    $\begingroup$ Isn't it just $H^1$? $\endgroup$
    – Alex Degtyarev
    Commented Dec 9, 2014 at 20:42
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    $\begingroup$ For smooth projective complex varieties, this is the same as the Picard variery (or equivalently the Albanese) is zero, almost by definition. As the dimension of the Picard variety is the first Betti number, here is your criterion: $b_1=0$. $\endgroup$
    – Felipe Voloch
    Commented Dec 9, 2014 at 20:43
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    $\begingroup$ And the dimension of the Picard variety is a birational invariant. $\endgroup$
    – David Lampert
    Commented Dec 10, 2014 at 0:42

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CW answer from the comment by Felipe Voloch:

For smooth projective complex varieties, this is the same as the Picard variety (or equivalently the Albanese) is zero, almost by definition. As the dimension of the Picard variety is the first Betti number, here is your criterion: $b_1=0$. — Felipe Voloch Dec 9 at 20:43

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