The first example is given by the grounding work of Riemann himself: in his thesis, he shows that one can define a Riemann surface to each algebraic curve; every Riemann surface comes from an algebraic curve, well defined up to birational equivalence and two birational equivalent curves give the same surface. Therefore, the Riemann surface, or more simply its genus is a birational invariant.
A more complicated example is given by Hodge theory: in the case of an algebraic surface, the Hodge numbers h0,1 and h0,2 of a non-singular projective complex surface are birational invariants. The Hodge number h1,1 is not, since the process of blowing up a point to a curve on the surface can augment it.
- Reichstein, Z.; Youssin, B. (2002), "A birational invariant for algebraic group actions", Pacific Journal of Mathematics, 204 (1): 223–246, MR 1905199, doi:10.2140/pjm.2002.204.223.
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