Birational invariant
Encyclopedia
In algebraic geometry
, 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.
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a birational invariant is a quantity or object that is well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
on a birational equivalence class of algebraic varieties. In other words, it depends only on the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of the variety.
The first example is given by the grounding work of Riemann himself: in his thesis, he shows that one can define a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
to each algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
; 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
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
is a birational invariant.
A more complicated example is given by Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
: in the case of an algebraic surface
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 and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
, 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
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 projectivized tangent space at that point...
a point to a curve on the surface can augment it.