Birationally Equivalent
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, birational geometry is a field of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
in which the goal is to determine when two
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
outside lower-dimensional subsets. This amounts to studying mappings that are given by
rational functions In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rati ...
rather than
polynomials In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
; the map may fail to be defined where the rational functions have poles.


Birational maps


Rational maps

A
rational map In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal de ...
from one variety (understood to be
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
) X to another variety Y, written as a dashed arrow , is defined as a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
from a nonempty open subset U \subset X to Y. By definition of the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.


Birational maps

A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map 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, ''X'' and ''Y'' are said to be 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 , 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''.


Birational equivalence and rationality

A variety ''X'' is said to be
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
if it is birational to affine space (or equivalently, to
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
) 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.


Birational equivalence of a plane conic

For example, the circle X with equation x^2 + y^2 - 1 = 0 in the affine plane is a rational curve, because there is a rational map given by :f(t) = \left( \frac, \frac\right), which has a rational inverse ''g'': ''X'' ⇢ \mathbb^1 given by :g(x,y) = \frac. Applying the map ''f'' with ''t'' a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
gives a systematic construction of
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s. The rational map f is not defined on the locus where 1 + t^2 = 0. So, on the complex affine line \mathbb^1_, f is a morphism on the open subset U = \mathbb^1_-\, f: U \to X. Likewise, the rational map is not defined at the point (0,−1) in X.


Birational equivalence of smooth quadrics and Pn

More generally, a smooth
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
(degree 2) hypersurface ''X'' of any dimension ''n'' is rational, by
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. (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.) To define stereographic projection, let ''p'' be a point in ''X''. Then a birational map from ''X'' to the projective space \mathbb^n of lines through ''p'' is given 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 (and the inverse map fails to be defined at those lines through ''p'' which are contained in ''X'').


= Birational equivalence of quadric surface

= The
Segre embedding In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. Definition The Segre map may be defined as the map ...
gives an embedding \mathbb^1\times\mathbb^1 \to \mathbb^3 given by :( ,y ,w \mapsto z,xw,yz,yw The image is the quadric surface x_0x_3=x_1x_2 in \mathbb^3. That gives another proof that this quadric surface is rational, since \mathbb^1\times\mathbb^1 is obviously rational, having an open subset isomorphic to \mathbb^2.


Minimal models and resolution of singularities

Every algebraic variety is birational to a
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
(
Chow's lemma Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If ...
). So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting. Much deeper is Hironaka's 1964 theorem on
resolution of singularities In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characterist ...
: over a field of characteristic 0 (such as the complex numbers), every variety is birational to a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
projective variety. Given that, it is enough to classify 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 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 th ...
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 number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s. This leads to the idea of minimal models: is there 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 In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
''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 In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
from 1890–1910, part of the
classification of surfaces In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
, is that every surface ''X'' is birational either to a product \mathbb^1\times 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

At first, it is not clear how to show that there are any algebraic varieties which are not rational. 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.


Plurigenera

One useful set of birational invariants are the
plurigenera In mathematics, the pluricanonical ring of an algebraic variety ''V'' (which is non-singular), or of a complex manifold, is the graded ring :R(V,K)=R(V,K_V) \, of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (for ...
. The
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
of a smooth variety ''X'' of dimension ''n'' means the
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
of ''n''-forms , which is the ''n''th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
of ''X''. For an integer ''d'', the ''d''th tensor power of ''KX'' is again a line bundle. For , the vector space of global sections has the remarkable property that a birational map between smooth projective varieties induces an isomorphism . For , define the ''d''th plurigenus ''P''''d'' as the dimension of the vector space ; then the plurigenera are birational invariants for smooth projective varieties. In particular, if any plurigenus ''P''''d'' with is not zero, then ''X'' is not rational.


Kodaira dimension

A fundamental birational invariant is the
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
, which measures the growth of the plurigenera ''P''''d'' as ''d'' goes to infinity. The Kodaira dimension divides all varieties of dimension ''n'' into 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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
.


Summands of ⊗''k''Ω1 and some Hodge numbers

More generally, for any natural summand :E(\Omega^1) = \bigotimes^k \Omega^1 of the ''r-''th tensor power of the cotangent bundle Ω1 with , the vector space of global sections is a birational invariant for smooth projective varieties. In particular, the Hodge numbers :h^ = H^0(X,\Omega^p) are birational invariants of ''X''. (Most other Hodge numbers ''h''''p'',''q'' are not birational invariants, as shown by blowing up.)


Fundamental group of smooth projective varieties

The
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
''π''1(''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. 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

A projective variety ''X'' is called minimal if the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
''KX'' is nef. For ''X'' of dimension 2, it is enough to consider smooth varieties in this definition. In dimensions at least 3, minimal varieties must be allowed to have certain mild singularities, for which ''KX'' is still well-behaved; these are called
terminal singularities In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singula ...
. That being said, the minimal model conjecture would imply that every variety ''X'' is either covered by
rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s 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 In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate meas ...
. 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. 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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
over a field of characteristic zero has a minimal model.


Uniruled varieties

A variety is called uniruled if it is covered by rational curves. 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. This leads to the problem of the birational classification of Fano fiber spaces and (as the most interesting special case)
Fano varieties In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has ...
. By definition, a projective variety ''X'' is Fano if the anticanonical bundle K_X^* is
ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
. 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 In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...
) 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 Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
. 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 \mathbb^ with which is not rational.


Birational automorphism groups

Algebraic varieties differ widely in how many birational automorphisms they have. Every variety of
general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
is extremely rigid, in the sense that its birational automorphism group is finite. At the other extreme, the birational automorphism group of projective space \mathbb^n over a field ''k'', known as the
Cremona group In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the n-dimensional projective space over a field It is denoted by Cr(\mathbb^n(k)) or Bir(\mathbb^n(k)) or Cr_n(k). The Cremona group is natura ...
''Cr''''n''(''k''), is large (in a sense, infinite-dimensional) for . For , the complex Cremona group Cr_2(\Complex) is generated by the "quadratic transformation" : 'x'',''y'',''z''/''x'', 1/''y'', 1/''z'' together with the group PGL(3,\Complex) of automorphisms of \mathbb^2, by
Max Noether Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the ...
and Castelnuovo. By contrast, the Cremona group in dimensions 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 In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), the ...
is enormous. This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces.


Applications

Birational geometry has found applications in other areas of geometry, but especially in traditional problems in algebraic geometry. Famously the minimal model program was used to construct
moduli spaces In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of varieties of general type by
János Kollár János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry. Professional career Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 ...
and
Nicholas Shepherd-Barron Nicholas Ian Shepherd-Barron, Fellow of the Royal Society, FRS (born 17 March 1955), is a British mathematician working in algebraic geometry. He has been, since 2013, professor of mathematics at King's College London, having moved there from hi ...
, now known as KSB moduli spaces.J. Kollar. Moduli of varieties of general type, Handbook of moduli. Vol. II, 2013, pp. 131–157. Birational geometry has recently found important applications in the study of
K-stability of Fano varieties In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where i ...
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.Xu, C., 2020. K-stability of Fano varieties: an algebro-geometric approach. ''arXiv preprint arXiv:2011.10477''. Important results in birational geometry such as Birkar's proof of boundedness of Fano varieties have been used to prove existence results for moduli spaces.


See also

*
Abundance conjecture In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonic ...


Citations


Notes


References

* * * * * * * * * {{DEFAULTSORT:Birational Geometry