In algebraic geometry, a birational invariant is a property that is preserved under

birational equivalence In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...

Formal definition

A birational invariant is a quantity or object that is

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...

on a

class of

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

. 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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...

himself: in his thesis, he shows that one can define a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

to each

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

; 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 In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...

is a birational invariant. A more complicated example is given by Hodge theory: in the case of an algebraic surface, the Hodge numbers ''h''^0,1 and ''h''^0,2 of a

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

projective complex surface are birational invariants. The Hodge number ''h''^1,1 is not, since the process of

blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the ...

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