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Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which signifies the number of holes in the Riemann surface.

In the minimal model program (which we may, for our purposes here restrict attention to surfaces) we look for a classification of projective varieties (over $\mathbb{C}$) up to birational isomorphism. A useful invariant is the Kodaira dimension of the projective surface (see, e.g., https://en.wikipedia.org/wiki/Kodaira_dimension). The genus of a complex projective surface does not tell us much and does not provide a useful invariant for projective varieties of dimension greater than one.

My question is:

Are there any concrete illustrations which show how bad the genus is as a birational invariant of projective varieties $X$, where $\dim_{\mathbb{C}} X \geq 2$?

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    $\begingroup$ That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ? $\endgroup$ – reuns Jan 23 at 0:45
  • $\begingroup$ @reuns What is your $P_g$ here? The Plurigenus? $\endgroup$ – Kyle Broder Jan 23 at 1:29
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    $\begingroup$ What definition of genus are you using for surfaces? $\endgroup$ – Mohan Jan 23 at 2:28

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