Birational geometry

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In algebraic geometry, the goal of birational geometry is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

Birational maps[edit]

A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow X – → Y, is defined as a morphism from a nonempty open subset U of X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always the complement of a lower-dimensional subset of X. Concretely, a rational map can be written in coordinates using rational functions.

A birational map from X to Y is a rational map f: X – → Y such that there is a rational map Y – → X inverse to f. A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y. In this case, we say that X and Y are birational, or birationally equivalent. In algebraic terms, two varieties over a field k are birational if and only if their function fields are isomorphic as extension fields of k.

A special case is a birational morphism f: XY, meaning a morphism which is birational. That is, f is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of X to points in Y.

We say that a variety X is rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset. For example, the circle with equation x2 + y2 − 1 = 0 is a rational curve, because the formulas

x=\frac{2\,t}{1+t^2}
y=\frac{1-t^2}{1+t^2}\,,

define a birational map from the affine line to the circle. (Explicitly, the inverse map sends (x,y) to (1 − y)/x.)

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, we have to assume that X has a k-rational point; this is automatic if k is algebraically closed.) To define stereographic projection, let p be a point in X. Then we define a birational map from X to the projective space Pn of lines through p by sending a point q in X to the line through p and q. This is a birational equivalence but not an isomorphism of varieties, because it fails to be defined where q = p (and the inverse map fails to be defined at those lines through p which are contained in X).

Minimal models and resolution of singularities[edit]

Every algebraic variety is birational to a projective variety. So, for the purposes of birational classification, we can work only with projective varieties, and this is usually the most convenient setting.

Much deeper is Hironaka's 1964 theorem on resolution of singularities: over a field of characteristic 0 (such as the complex numbers), every variety is birational to a smooth projective variety. Given that, we can concentrate on classifying smooth projective varieties up to birational equivalence.

In dimension 1, if two smooth projective curves are birational, then they are isomorphic. But that fails in dimension at least 2, by the blowing up construction. By blowing up, every smooth projective variety of dimension at least 2 is birational to infinitely many "bigger" varieties, for example with bigger Betti numbers.

This leads to the idea of minimal models: can we find a unique simplest variety in each birational equivalence class? The modern definition is that a projective variety X is minimal if the canonical line bundle KX has nonnegative degree on every curve in X; in other words, KX is nef. It is easy to check that blown-up varieties are never minimal.

This notion works perfectly for algebraic surfaces (varieties of dimension 2). In modern terms, one central result of the Italian school of algebraic geometry from 1890-1910, part of the classification of surfaces, is that every surface X is birational either to a product P1 × C for some curve C or to a minimal surface Y. The two cases are mutually exclusive, and Y is unique if it exists. When Y exists, it is called the minimal model of X.

Birational invariants[edit]

Main article: Kodaira dimension

At first, it is not clear how to show that there are any algebraic varieties which are not rational. In order to prove this, we need to build up some birational invariants of algebraic varieties.

One useful set of birational invariants are the plurigenera. The canonical bundle of a smooth variety X of dimension n means the line bundle of n-forms,

\,\!K_X = \Omega^n_X,

which is the nth exterior power of the cotangent bundle of X. For an integer d, the dth tensor power of KX is again a line bundle. For d ≥ 0, the vector space of global sections H0(X,KXd) has the remarkable property that a birational map f: X – → Y between smooth projective varieties induces an isomorphism H0(X,KXd) ≅ H0(Y,KYd).

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. In particular, if any plurigenus Pd with d > 0 is not zero, then X is not rational.

A fundamental birational invariant is the Kodaira dimension, which measures the growth of the plurigenera Pd as d goes to infinity. The Kodaira dimension divides all varieties of dimension n into n + 1 types, with Kodaira dimension −∞, 0, 1, ..., or n. This is a measure of the complexity of a variety, with projective space having Kodaira dimension −∞. The most complicated varieties are those with Kodaira dimension equal to their dimension n, called varieties of general type.

More generally, for any natural summand E1) of the rth tensor power of the cotangent bundle Ω1 with r ≥ 0, the vector space of global sections H0(X,E1)) is a birational invariant for smooth projective varieties. In particular, the Hodge numbers hr0 = dim H0(Xr) are birational invariants of X. (Most other Hodge numbers hpq are not birational invariants, as we see by blowing up.)

The fundamental group π1(X) is a birational invariant for smooth complex projective varieties.

The "Weak factorization theorem", proved in 2002 by Abramovich, Karu, Matsuki, and Włodarczyk, 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. This is important to know, but it can still be very hard to determine whether two smooth projective varieties are birational.

Minimal models in higher dimensions[edit]

Main article: Minimal model program

A projective variety X is called minimal if the canonical bundle KX is nef. For X of dimension 2, it is enough to consider smooth varieties in this definition. In dimensions at least 3, we have to allow minimal varieties to have certain mild singularities, for which KX is still well-behaved; these are called terminal singularities.

That being said, the minimal model conjecture would imply that every variety X is either covered by rational curves or birational to a minimal variety Y. When it exists, Y is called a minimal model of X.

Minimal models are not unique in dimensions at least 3, but any two minimal varieties which are birational are very close. For example, they are isomorphic outside subsets of codimension at least 2, and more precisely they are related by a sequence of flops. So the minimal model conjecture would give strong information about the birational classification of algebraic varieties.

The conjecture was proved in dimension 3 by Mori (1988). There has been great progress in higher dimensions, although the general problem remains open. In particular, Birkar, Cascini, Hacon, and McKernan (2010) proved that every variety of general type over a field of characteristic zero has a minimal model.

Uniruled varieties[edit]

Main article: Ruled variety

Another part of the minimal model conjecture would say a lot about the birational classification of varieties covered by rational curves, known as uniruled varieties. These do not have a minimal model, but the conjecture is that every uniruled variety is birational to a Fano fiber space. This leads to the problem of the birational classification of Fano fiber spaces and (as the most interesting special case) Fano varieties. By definition, a projective variety X is Fano if the anticanonical bundle KX* is ample. Fano varieties can be considered the algebraic varieties which are most similar to projective 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. In particular, smooth cubic 3-folds are not rational by Clemens-Griffiths (1972), and smooth quartic 3-folds are not rational by Iskovskikh-Manin (1971). Nonetheless, the problem of determining exactly which Fano varieties are rational is far from solved. For example, it is not known whether there is any smooth cubic hypersurface in Pn+1 with n ≥ 4 which is not rational.

Birational automorphism groups[edit]

Algebraic varieties differ widely in how many birational automorphisms they have. Every variety of general type is extremely rigid, in the sense that its birational automorphism group is finite. At the other extreme, the birational automorphism group of projective space Pn over a field k, known as the Cremona group Crn(k), is large (in a sense, infinite-dimensional) for n ≥ 2. For n = 2, we know at least that the complex Cremona group Cr2(C) is generated by the "quadratic transformation"

[x,y,z] ↦ [1/x, 1/y, 1/z]

together with the group PGL(3,C) of automorphisms of P2, by Max Noether and Castelnuovo. By contrast, the Cremona group in dimensions n ≥ 3 is very much a mystery: no explicit set of generators is known.

Iskovskikh-Manin (1971) showed that the birational automorphism group of a smooth quartic 3-fold is equal to its automorphism group, which is finite. In this sense, quartic 3-folds are far from being rational, since the birational automorphism group of a rational variety is enormous. This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces.

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