algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a birational invariant is a property that is preserved under birational equivalence.

Formal definition

A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties. In other words, it depends only on the function field of the variety.

Examples

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 Geometric genus is a birational invariant. A more complicated example is given by

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

: in the case of an

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

, the

Hodge number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

s ''h''^0,1 and ''h''^0,2 of a non-singular projective complex surface are birational invariants. The Hodge number ''h''^1,1 is not, since the process of blowing up a point to a curve on the surface can augment it.

References

*{{citation , last1 = Reichstein , first1 = Z. , last2 = Youssin , first2 = B. , doi = 10.2140/pjm.2002.204.223 , issue = 1 , journal = Pacific Journal of Mathematics , mr = 1905199 , pages = 223–246 , title = A birational invariant for algebraic group actions , volume = 204 , year = 2002, arxiv = math/0007181 . Birational geometry