Birational GeometryEdit
Birational geometry is a branch of algebraic geometry that studies the properties of algebraic varieties up to birational equivalence. In plain terms, it asks when two spaces look the same if we ignore lower-dimensional glitches and focus on their fields of rational functions. Two varieties X and Y are birational if there is a rational map between them that has a rational inverse on dense open sets; equivalently, their function fields K(X) and K(Y) are isomorphic. This line of inquiry emphasizes the intrinsic geometry captured by the function field rather than the particular presentation of a variety. birational map birational equivalence Function field Algebraic variety
The field has its roots in the classical study of plane Cremona transformations and the geometry of algebraic surfaces, but it has grown into a broad program aimed at higher dimensions. A central engine of modern birational geometry is the Minimal Model Program, a systematic procedure to modify a given variety by birational transformations so that its canonical class acquires a controlled positivity property. The canonical divisor, commonly denoted K_X, guides which birational moves are allowed and what the endpoint should look like: a minimal model with nef K_X, or a Mori fiber space that reveals a structured fibration. Cremona group Algebraic surface Minimal model program Canonical divisor Mori fiber space
Core ideas
Birational maps and equivalence
A birational map f: X -> Y is a rational map with a rational inverse, defined on dense open subsets U ⊂ X and V ⊂ Y with f|_U: U → V an isomorphism. This perspective replaces detailed, presentation-dependent features with global, function-field-driven properties. The notion of birational equivalence concentrates on the behavior of rational functions rather than every geometric detail. Birational map Birational equivalence
Function fields and invariants
Birational geometry rests on the equivalence of function fields: X and Y are birational exactly when K(X) ≅ K(Y). This viewpoint explains why certain numerical and structural invariants survive birational transformations; for example, the Kodaira dimension and many plurigenera are meaningful in the birational category. The study of invariants helps distinguish when a given variety is, in a birational sense, simple or highly intricate. Function field Kodaira dimension plurigenera
Singularities and resolutions
Real progress in higher dimensions depends on controlling singularities. Resolution of singularities, proved in characteristic zero by Hironaka, allows one to work with nice models while preserving birational type. The development of singularity categories—such as canonical and terminal singularities, and more generally Kawamata log terminal (klt) or divisorial log terminal (dlt) structures—provides a precise language for measuring how far a model is from being smooth and how birational modifications affect the geometry. Resolution of singularities Canonical singularities Kawamata–Viehweg Kawamata log terminal Divisorial log terminal
The Minimal Model Program and birational surgery
The Minimal Model Program (MMP) seeks canonical representatives in a birational class. It proceeds by birational operations that either improve the positivity of the canonical class or reveal a fibration structure. The main operations include divisorial contractions (removing a divisor), flips (exchanging one problematic locus for another in a way that improves K_X), and flops (birational maps that preserve K_X). The goal is either a minimal model (where K_X is nef) or a Mori fiber space (a controlled fibration whose fibers are Fano-like). This program provides a unifying framework for understanding birational geometry in higher dimensions, and it intertwines with the study of singularities, log pairs, and positivity theory. Flips Flop (birational geometry) Divisorial contraction Mori theory Minimal model program Fano variety
Mori theory, Fano geometry, and birational rigidity
Within the MMP, Mori theory analyzes contractions in the presence of negative K_X and organizes birational maps through sequences of elementary links. A particularly fruitful arena is Fano geometry, where one studies varieties with ample anti-canonical class and their fibrations. In several cases, birational rigidity phenomena occur: certain varieties admit very few birational self-maps or birationally equivalent models, constraining their birational classification. These rigidity results illuminate how sometimes the birational class is tightly controlled, despite the space’s complexity. Mori fiber space Fano variety Birational rigidity
The Sarkisov program and birational decompositions
To understand maps between Mori fiber spaces, the Sarkisov program provides a framework to decompose birational maps into elementary links, each of which is a more tractable birational operation. This modular viewpoint helps organize the global birational zoo and clarifies how different models relate to one another within the same birational class. Sarkisov program Mori fiber space
Invariants, questions, and methods
Kodaira dimension and birational invariants
The Kodaira dimension measures the growth of sections of powers of the canonical bundle and is a central birational invariant in many settings. It helps distinguish varieties of general type from those with more restrictive canonical positivity. Other birational invariants, such as certain plurigenera and volumes, contribute to a robust birational toolkit that remains meaningful across many models within a class. Kodaira dimension plurigenera
Rationality and birational classification
A core, long-standing question is rationality: when is a given variety birational to projective space? While many surfaces are rational, higher-dimensional cases reveal a spectrum of complexity, with some varieties showing deep obstructions to rationality. Birational rigidity and related phenomena show that even when a variety is not rational, its birational behavior can be severely constrained. Rationality Birational rigidity
Singularities, resolutions, and log pairs
Modern birational geometry uses singularities strategically. Log pairs (X, Δ) and their singularity types (such as klt or dlt) are essential in the MMP, guiding where and how to run flips and contractions. Discrepancies quantify the failure of a divisor to be canonical and control convergence of the program. Kawamata log terminal Divisorial log terminal Discrepancy (algebraic geometry)
Positive characteristic, open problems, and controversy
Many foundational results in characteristic zero have not yet fully transferred to positive characteristic, where resolution of singularities and parts of the MMP remain incomplete. The field debates how best to extend these tools, and which conjectures (such as the abundance conjecture) will hold in greater generality. The tension between developing broad, powerful frameworks and pursuing explicit, constructive classifications continues to shape priorities in research programs. Resolution of singularities Minimal model program Abundance (algebraic geometry)
Controversies and debates
The balance between modern machinery and explicit constructions. Proponents of the minimal model program emphasize a unified, systematic approach to classification in higher dimensions, while critics argue that the machinery can become overly abstract and computationally opaque. They advocate complementary, hands-on methods emphasizing concrete birational maps, explicit blow-ups, and constructive descriptions of models when feasible. Minimal model program Blow up (algebraic geometry)
Abundance and termination questions. The belief that certain positivity and termination statements (e.g., abundance: if K_X is nef, then it is semi-ample) hold in general is widely supported but not fully proven in all dimensions and characteristics. The status of these conjectures shapes both expectations and methods in research programs. Abundance (algebraic geometry)
Extensions to positive characteristic. While many results are settled over the complex numbers, extending the framework to positive characteristic raises subtle obstacles, spurring ongoing work and occasional controversy over the correct formulation of the theory in that setting. Positive characteristic Resolution of singularities
Rationality versus rigidity. The study of when a variety is rational versus when its birational class is tightly constrained leads to different research programs. Some results show surprising rigidity in certain families, while others demonstrate surprising flexibility in different contexts. Rationality Birational rigidity