Birationaalisesti

Birationaalisesti, or birationally in English, is a concept in algebraic geometry describing a relationship between algebraic varieties that are indistinguishable from the standpoint of rational functions. Two varieties X and Y over a field K are birationally equivalent if there exist rational maps f: X → Y and g: Y → X whose compositions are the identities on dense open subsets of X and Y. Equivalently, their function fields K(X) and K(Y) are isomorphic as extensions of K.

A rational map is given by ratios of polynomial functions, and a birational map is a rational map that has a rational inverse. If a morphism (an everywhere defined map) induces a birational equivalence, it is called a birational morphism. Birational equivalence is an equivalence relation and partitions varieties into birational classes.

Birational invariants include the dimension of the variety and its function field. For curves, birational equivalence is the same as having isomorphic function fields, and hence smooth projective curves are birationally classified by their genus. In higher dimensions, birational geometry studies which properties are preserved under birational equivalence; many geometric features, such as singularities or certain differential forms, may not be preserved.

Common examples illustrate the concept: any nonsingular projective conic in the plane is birational to the projective line P^1, and P^1 itself is birational to the affine line A^1. The quadratic Cremona transformation provides a classic birational self-map of the projective plane P^2.

Birationaalisesti is central to the study of rationality questions and the classification of varieties up to birational equivalence, forming a core part of the broader field of birational geometry.