Birational geometry is a branch of algebraic geometry whose main task is to classify algebraic varieties up to birational equivalence [1] . This boils down to studying mappings that are defined by rational functions , rather than polynomials. A map may not be defined at some points that are poles of a rational function.
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Birational Mappings
The rational mapping of one ( ) variety X into another variety Y (written as the dashed arrow X ⇢ Y ) is defined as a morphism from a nonempty open subset U of the variety X into Y. By the definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always a complement of a subset X of smaller dimension. Specifically, a rational mapping 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 exists a rational map Y ⇢ X inverse to f . A birational map generates an isomorphism of a nonempty open subset X into a nonempty open subset Y. In this case, it is said that X and Y are birationally equivalent . In algebraic terms, two varieties over a field k are birationally equivalent if and only if their isomorphic as extensions of k .
A special case is a birational morphism f : X → Y , which means a morphism that is birational. Then f is defined on all of X , but its inverse may not be defined on all of Y. This usually happens when a birational morphism compresses some submanifolds of X into points in Y.
A variety X is said to be if it is rationally equivalent to an affine space (or, equivalently, a projective space ) of the same dimension. Rationality is a completely natural property - it means that X without some subset of lower dimension can be identified with an affine space without some subset of lower dimension. For example, the circle defined by the equation x 2 + y 2 - 1 = 0 is a rational curve because the formulas
determine the birational mapping of a line into a circle. (If we substitute rational numbers instead of t , we get Pythagorean triples .) The inverse mapping takes ( x , y ) to (1 - y ) / x .
More generally, a smooth quadratic (degree 2) hypersurface X of any dimension n is rational in view of the stereographic projection (for a quadratic variety X over a field k , it must be assumed that it has a . This is done automatically if k is algebraically closed. ) To define a stereographic projection, suppose p is a point in X. Then the birational map from X to the projective space P n of lines passing through p is defined by the map of the point q in X to the line passing through p and q . This map is birational equivalence, but not an isomorphism of varieties, since it is not defined for q = p (and the inverse map is not defined for lines passing through p and lying in X ).
Minimum Models and Feature Resolution
Any algebraic variety is birationally equivalent to a projective variety ( ). Thus, for birational classification, it is sufficient to work only with projective varieties, and this is the most common assumption.
Much deeper, according to Hironaki ’s theorem on the - over a field of characteristic 0 (such as complex numbers), any variety is birationally equivalent to a projective variety. With this in mind, it is sufficient to classify smooth projective varieties up to birational equivalence.
In dimension 1, if two smooth projective curves are birationally equivalent, they are isomorphic. However, this is not so in dimensions 2 and higher due to the construction of the bloat . When inflating, any smooth projective variety of dimension 2 and higher is birationally equivalent to an infinite number of “large” varieties, for example, with large Betti numbers .
This leads to the idea of minimal models - is there a single simple variety in every class of racial equivalence? The modern definition of a minimal model - a projective variety X is minimal if the K X has a non-negative degree on any curve in X. In other words, K X is a . It is easy to verify that swollen manifolds are never minimal.
This idea works well for algebraic surfaces (varieties of dimension 2). In modern terms, the central result of the Italian school of algebraic geometry in 1890-1910, part of the , was the fact that any surface X is birationally equivalent to either the product P 1 × C for some curve C or the minimal surface Y [2] . These two cases are mutually exclusive and Y is unique if exists. If Y exists, it is called a minimal model of the surface X.
Birational Invariants
First of all, it is not entirely clear how to show that there is any irrational algebraic surface. In order to prove this, it is necessary to use some invariants of algebraic varieties.
One useful set of birational invariants is . smooth manifold X of dimension n is the n- forms K X = Ω n , which is the nth external degree of the X. For an integer d , the dth tensor degree K X is again a linear bundle. For d ≥ 0, the vector space of global sections H 0 ( X , K X d ) has the remarkable property that the birational map f : X ⇢ Y between smooth projective varieties generates the isomorphism H 0 ( X , K X d ) ≅ H 0 ( Y , K Y d ) [3] .
For d ≥ 0, we define the dth pluriod P d as the dimension of the vector space H 0 ( X , K X d ). Then plurirodes are birational invariants of smooth projective varieties. In particular, if some plurirod P d for d > 0 is not equal to zero, then X is not a rational variety.
The fundamental birational invariant is the , which measures the growth of pluriods P d as d tends to infinity. The Kodaira dimension divides all manifolds of dimension n into n + 2 types with the Kodaira dimension −∞, 0, 1, ..., n . This invariant shows the complexity of the manifold, and the projective space has the Kodaira dimension −∞. The most complex manifolds are those for which the Kodaira dimension coincides with the dimension of the space n , and these manifolds are called manifolds of the .
More generally, any natural direct summand E (Ω 1 ) of the rth tensor degree of the tangent sheaf Ω 1 with r ≥ 0, the vector space of global sections H 0 ( X , E (Ω 1 )) is a birational invariant for smooth projective varieties. In particular, the Hodge numbers h r , 0 = dim H 0 ( X , Ω r ) are birational invariants of X. (Most of the other Hodge numbers h p, q are not birational invariants, which is shown by bloating .)
The fundamental group π 1 ( X ) is a birational invariant for smooth complex projective varieties.
The “weak factorization theorem”, which was proved by Abramovich, Karu, Matsuki, and Vlodarchik [4] , states that any birational map between two smooth complex projective varieties can be decomposed into a finite number of blow-ups or blow-offs of smooth submanifolds. This is important to know, but it remains a difficult task to determine whether two smooth projective varieties are birationally equivalent.
Minimal models in high dimensions.
A projective variety X is said to be minimal if the K X is a . For X of dimension 2, it suffices to consider smooth manifolds. In dimensions 3 and above, minimal manifolds should be allowed to have some weak singularities for which K X remains having good behavior. They are called .
Nevertheless, it would follow from the validity of the minimal model hypothesis that any variety X is either covered by rational curves or is birationally equivalent to the minimal variety Y. If one exists, Y is called the minimal model of X.
Minimal models are not unique in dimensions 3 and higher, but any two minimal birational varieties are very close. For example, they are isomorphic outside subsets of codimension 2 and above, and, more precisely, they are connected by a sequence of . So the hypothesis of a minimal model would provide essential information on the birational classification of algebraic varieties.
Mori proved the hypothesis for dimension 3 [5] . There is great progress in higher dimensions, although the main problem remains open. In particular, Birkar, Cassini, Hakon and McKernan [6] proved that any variety of over a field of characteristic 0 has a minimal model.
Unilinear manifolds
A manifold is called unilinear if it is covered by rational curves. A unilinear manifold does not have a minimal model, but there is a good substitute - Birkar, Cassini, Hakon and Mackernan have shown that any unilinear manifold over a field with zero characteristic is a birational Fano bundle [7] . This leads to the problem of birational classification of Fano bundles and (as the most interesting case) . By definition, a projective variety X is a Fano variety if the anticanonical sheaf K X * is ample . Fano varieties can be regarded as closest to projective spaces.
In dimension 2, any Fano variety (known as ) over an algebraically closed field is rational. The main discovery of the 1970s was that, starting from dimension 3, there are many Fano varieties that are not . In particular, smooth cubic three-dimensional manifolds, according to Clemens and Griffiths [8] , are not rational, and smooth three-dimensional manifolds of the fourth degree are not rational according to Iskovskikh and Manin [9] . Nevertheless, the problem of precisely determining which Fano varieties are rational is far from being solved. For example, it is not known whether an irrational smooth cubic hypersurface exists in P n +1 with n ≥ 4.
Birational automorphism groups
Algebraic varieties differ significantly in the number of their birational automorphisms. Any variety of very rigid in the sense that its group of birational automorphisms is finite. At the other extreme, the group of birational automorphisms of the projective space P n over the field k , known as Cr n ( k ), is large (has infinite dimension) for n ≥ 2. For n = 2, the complex Cremona group Cr 2 ( C ) generated by "quadratic transformation"
- [ x , y , z ] ↦ [1 / x , 1 / y , 1 / z ]
together with the group PGL (3, C ) of automorphisms P 2 , according to Max Noether and Guido Castelnuovo . In contrast, the Cremona group in dimension n ≥ 3 is very mysterious, for it is not known an explicit set of generators.
Iskovskikh and Manin [9] showed that the group of birational automorphisms of smooth fourth-order hypersurfaces (quartics) of three-dimensional manifolds is equal to its automorphism group, which is finite. In this sense, fourth-order three-dimensional manifolds are far from rationality, since the group of birational automorphisms of a is enormous. This phenomenon of “birational rigidity” has since been discovered for many layered Fano spaces.
Notes
- ↑ Dolgachev, Iskovskikh, 1977 , p. 463.
- ↑ Kollár, Mori, 1998 , p. Theorem 1.29.
- ↑ Hartshorne, 1977 , p. Exercise II.8.8.
- ↑ Abramovich, Karu, Matsuki, Włodarczyk, 2002 .
- ↑ Mori, 1988 .
- ↑ Birkar, Cascini, Hacon, McKernan, 2010 .
- ↑ ( Birkar, Cascini, Hacon, McKernan 2010 ); Corollary 1.3.3 implies that any unilinear manifold in characteristic zero is a birational Fano bundle if we use the simple fact that the unilinear manifold X is covered by a family of curves for which K X has a negative degree. This statement can be found in the book of Debarre ( Debarre 2001 ), Corollary 4.11 and Example 4.7 (1).
- ↑ Clemens, Griffiths, 1972 .
- ↑ 1 2 Iskovskikh, Manin, 1971 , p. 140-166.
Literature
- I.V.Dolgachev, V.A. Iskovskikh. Mathematical Encyclopedia / I. M. Vinogradov. - M .: "Soviet Encyclopedia", 1977. - T. 1 (A - D).
- Dan Abramovich, Kalle Karu, Kenji Matsuki, Jarosław Włodarczyk. Torification and factorization of birational maps // Journal of the American Mathematical Society. - 2002. - T. 15 , no. 3 . - P. 531-572. - DOI : 10.1090 / S0894-0347-02-00396-X .
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan. Existence of minimal models for varieties of log general type // Journal of the American Mathematical Society. - 2010 .-- T. 23 , no. 2 . - P. 405-468. - DOI : 10.1090 / S0894-0347-09-00649-3 . - arXiv : math.AG/0610203 .
- C. Herbert Clemens, Phillip A. Griffiths . The intermediate Jacobian of the cubic threefold // Annals of Mathematics. Second Series . - The Annals of Mathematics, Vol. 95, No. 2, 1972. - T. 95 , no. 2 . - P. 281.356. - ISSN 0003-486X . - DOI : 10.2307 / 1970801 .
- Olivier Debarre. Higher-Dimensional Algebraic Geometry. - Springer-Verlag , 2001. - ISBN 0-387-95227-6 .
- Phillip A. Griffiths , Joseph Harris. Principles of Algebraic Geometry. - John Wiley & Sons, 1978. - ISBN 0-471-32792-1 .
- Robin_Hartshorne Algebraic Geometry. - Springer-Verlag, 1977 .-- ISBN 0-387-90244-9 .
- V. A. Iskovskikh , Yu. I. Manin . Three-dimensional quartics and counterexamples to the Lurot problem // Mat. Sat .. - 1971. - T. 86 (128) , No. 1 (9) . - S. 140–166 . - DOI : 10.1070 / SM1971v015n01ABEH001536 .
- János Kollár, Shigefumi Mori . Birational Geometry of Algebraic Varieties. - Cambridge University Press , 1998. - ISBN 0-521-63277-3 .
- Shigefumi Mori . Flip theorem and the existence of minimal models for 3-folds // Journal of the American Mathematical Society. - American Mathematical Society, 1988.- T. 1 , no. 1 . - P. 117–253. - ISSN 0894-0347 . - DOI : 10.2307 / 1990969 .