Birational geometry
, written as a dashed arrow X ⇢Y, is defined as a morphism from a nonempty open subset
By definition of the Zariski topology used in algebraic geometry, a nonempty open subset
given by Applying the map f with t a rational number gives a systematic construction of Pythagorean triples.
More generally, a smooth quadric (degree 2) hypersurface X of any dimension n is rational, by stereographic projection.
(For X a quadric over a field k, X must be assumed to have a k-rational point; this is automatic if k is algebraically closed.)
So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting.
This leads to the idea of minimal models: is there a unique simplest variety in each birational equivalence class?
This notion works perfectly for algebraic surfaces (varieties of dimension 2).
In order to prove this, some birational invariants of algebraic varieties are needed.
A birational invariant is any kind of number, ring, etc which is the same, or isomorphic, for all varieties that are birationally equivalent.
For an integer d, the dth tensor power of KX is again a line bundle.
[2] For d ≥ 0, define the dth plurigenus Pd as the dimension of the vector space H0(X, KXd); then the plurigenera are birational invariants for smooth projective varieties.
A fundamental birational invariant is the Kodaira dimension, which measures the growth of the plurigenera Pd as d goes to infinity.
More generally, for any natural summand of the r-th tensor power of the cotangent bundle Ω1 with r ≥ 0, the vector space of global sections H0(X, E(Ω1)) is a birational invariant for smooth projective varieties.
(Most other Hodge numbers hp,q are not birational invariants, as shown by blowing up.)
The fundamental group π1(X) is a birational invariant for smooth complex projective varieties.
The "Weak factorization theorem", proved by Abramovich, Karu, Matsuki, and Włodarczyk (2002), says that any birational map between two smooth complex projective varieties can be decomposed into finitely many blow-ups or blow-downs of smooth subvarieties.
A projective variety X is called minimal if the canonical bundle KX is nef.
So the minimal model conjecture would give strong information about the birational classification of algebraic varieties.
[3] There has been great progress in higher dimensions, although the general problem remains open.
In particular, Birkar, Cascini, Hacon, and McKernan (2010)[4] proved that every variety of general type over a field of characteristic zero has a minimal model.
A uniruled variety does not have a minimal model, but there is a good substitute: Birkar, Cascini, Hacon, and McKernan showed that every uniruled variety over a field of characteristic zero is birational to a Fano fiber space.
In dimension 2, every Fano variety (known as a Del Pezzo surface) over an algebraically closed field is rational.
A major discovery in the 1970s was that starting in dimension 3, there are many Fano varieties which are not rational.
Every variety of general type is extremely rigid, in the sense that its birational automorphism group is finite.
over a field k, known as the Cremona group Crn(k), is large (in a sense, infinite-dimensional) for n ≥ 2.
By contrast, the Cremona group in dimensions n ≥ 3 is very much a mystery: no explicit set of generators is known.
This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces.
Famously the minimal model program was used to construct moduli spaces of varieties of general type by János Kollár and Nicholas Shepherd-Barron, now known as KSB moduli spaces.
[5] Birational geometry has recently found important applications in the study of K-stability of Fano varieties through general existence results for Kähler–Einstein metrics, in the development of explicit invariants of Fano varieties to test K-stability by computing on birational models, and in the construction of moduli spaces of Fano varieties.
