## Rational And Non-Rational Algebraic Varieties: Lectures Of János Kollár (1997)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Smith97rationaland,

author = {Karen E. Smith},

title = {Rational And Non-Rational Algebraic Varieties: Lectures Of János Kollár},

year = {1997}

}

### OpenURL

### Abstract

### Citations

1834 |
Algebraic Geometry
- Hartshorne
- 1977
(Show Context)
Citation Context ...es of varieties defined over Q that are Fano but not ruled. The notes have been written with the goal of making them accessible to students with a basic training in algebraic geometry at the level of =-=[H]-=-. The first lecture is to be relatively easy, with subsequent lectures requiring more of the reader. The first three lectures are in the realm of classical algebraic geometry, while a scheme theoretic... |

1338 |
Commutative algebra with a view toward algebraic geometry, Graduate Texts
- Eisenbud
- 1995
(Show Context)
Citation Context ...mension two subvarieties of projective space have resolutions of this form was proved by Hilbert in 1890 [Hil]; nowadays we recognize it as a special case of the well-known Hilbert-Burch theorem; see =-=[E, p502]-=-. Let Y0, Y1, Y2, Y3 be homogeneous coordinates for P 3 . The entries of the 3 × 1 matrix ⎡ A tr⎢ ⎣ Y0 Y1 Y2 Y3 ⎤ ⎥ ⎦ = ⎡ ⎣ H1 H1 are bihomogeneous of degree (1, 1) in each set of variables Xi and Yi,... |

426 |
Commutative ring theory
- Matsumura
- 1986
(Show Context)
Citation Context ...e discrete valuation ring V . 4 In fact, the base V can be any excellent discrete valuation ring, although we do not need to apply the theorem in such generality. For the definition of excellent, see =-=[M, p260]-=-. 35The generic fiber of ZS −→ S would clearly be ruled. However, the special fiber is a variety of prime characteristic p, defined in P by an equation of the form Y p − Fmp. We proved in Theorem 4.3... |

144 | The intermediate Jacobian of the cubic threefold
- Clemens, Griffiths
- 1972
(Show Context)
Citation Context ...three-folds had already been constructed by Segre [S60]. Around the same time, Clemens and Griffiths also resolved the Lüroth problem, by showing that there exist no smooth rational cubic three-folds =-=[CG]-=-. (It is not hard to see that every smooth cubic threefold is unirational, an elementary fact Clemens and Griffiths attribute to Max Noether; see [CG, p 352].) Clemens and Griffiths approach was entir... |

143 |
Rational curves on algebraic varieties
- Kollár
- 1996
(Show Context)
Citation Context ... a smooth projective variety X being rationally connected, at least in characteristic zero. A variety is said to be rationally connected if every pair of points can be joined by a rational curve. See =-=[K96, p202]-=- for a discussion of this conjecture, which has recently been proved in dimension three by Kollár, Miyaoka and Mori [KoMM]. Exercise 5. (1) Prove Corollary 1.11. (2) Show that the plurigenera of a smo... |

105 | Groupes Algébriques et Corps de Classes - Serre - 1959 |

102 |
Rationally connected varieties
- Kollár, Miyaoka, et al.
- 1992
(Show Context)
Citation Context ...connected if every pair of points can be joined by a rational curve. See [K96, p202] for a discussion of this conjecture, which has recently been proved in dimension three by Kollár, Miyaoka and Mori =-=[KoMM]-=-. Exercise 5. (1) Prove Corollary 1.11. (2) Show that the plurigenera of a smooth hypersurface of degree d in P n do not vanish when d > n. Conclude that no smooth hypersurface whose degree exceeds it... |

68 | Cubic forms - Manin - 1972 |

59 | Variétés de Prym et jacobiennes intermédiaires - Beauville - 1977 |

58 |
On the valuations centered in a local domain
- Abhyankar
- 1956
(Show Context)
Citation Context ...ot birational, then Γ0 must be an exceptional divisor for this map. But exceptional divisors of proper birational maps to regular schemes are always ruled, as the following theorem of Abhyankar shows =-=[Ab, p336]-=-. 5.4. Theorem. Let Y π −→ X be a proper birational morphism of irreducible schemes, with Y normal and X regular. 6 Then every exceptional divisor of π is ruled over its image. That is, if E is an int... |

50 |
Three-dimensional quartics and counterexamples to the Lüroth problem
- Manin
- 1971
(Show Context)
Citation Context ...y closed field of characteristic zero. Indeed, in 1971, Iskovskikh and Manin proved that any birational equivalence between smooth hypersurfaces of degree four in P 4 must be a projective equivalence =-=[IM]-=-. This implies that no smooth quartic threefold is rational: the birational automorphism group of the quartic threefold is the same as its group of projective automorphisms; since the latter is finite... |

47 |
Rational surfaces over perfect fields
- Manin
- 1967
(Show Context)
Citation Context ... not. 3. Third Lecture In this lecture we will prove Segre’s Theorem. Essentially the same argument, with minor modifications to be made afterwards, will prove the following stronger theorem of Manin =-=[M66]-=-. 3.1. Theorem. Two smooth cubic surfaces defined over a perfect field, each of Picard number one, are birationally equivalent if and only if they are projectively equivalent. Caution: Manin’s theorem... |

44 | Birational automorphisms of three-dimensional algebraic varieties”, Itogi Nauki Tekh - Iskovskikh - 1979 |

34 |
On the Kodaira dimension of minimal threefolds
- Miyaoka
- 1988
(Show Context)
Citation Context ...rogram for the birational classification of algebraic varieties; see [KaMM]. Miyaoka proved that uniruledness is equivalent to negative Kodaira dimension for smooth three-folds of characteristic zero =-=[Mi]-=-; see also [S-B]. The conjecture remains open in higher dimension. A similar conjecture, attributed to Mumford, predicts that the vanishing of H0 (X, Ω ⊗m X ) for all m ≥ 0 is the only obstruction to ... |

31 | Local Rings - Nagata - 1962 |

18 |
Birational isomorphisms of four-dimensional quintics
- Pukhlikov
- 1987
(Show Context)
Citation Context ...s of [IM] to show that a birational equivalence between two smooth quintics in P 5 is actually a projective equivalence. Again the corollary follows: there exist no smooth rational quintic four-folds =-=[P]-=-. The short paper of Tregub [Tr] gives some nice examples of rational cubic hypersurfaces in P 5 . But there is not a single known example of a smooth rational quartic hypersurface of dimension four o... |

17 | On the rationality problem for conic bundles - Iskovskikh - 1991 |

17 | Birational automorphisms of conic bundles - Sarkisov - 1981 |

16 | Basic Algebra I. Freeman - Jacobson - 1974 |

16 | Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a suo punti tripli apparenti - Severi - 1901 |

14 |
Nonrational hypersurfaces
- Kollár
- 1995
(Show Context)
Citation Context ...ties that are “close to rational.” Recent work of Kollár has established the existence of abundant examples of non-rational varieties with various other nice properties, such as rational connectivity =-=[K95]-=-, [K97]. A smooth projective variety is rationally connected if every two points are joined by a rational curve. We have seen that a smooth hypersurface whose degree is no greater than its embedding d... |

13 |
Über the Theorie von algebraischen Formen
- Hilbert
- 1978
(Show Context)
Citation Context ...a 4×3 matrix whose 3-minors are precisely the generators si for I. That defining ideals of codimension two subvarieties of projective space have resolutions of this form was proved by Hilbert in 1890 =-=[Hil]-=-; nowadays we recognize it as a special case of the well-known Hilbert-Burch theorem; see [E, p502]. Let Y0, Y1, Y2, Y3 be homogeneous coordinates for P 3 . The entries of the 3 × 1 matrix ⎡ A tr⎢ ⎣ Y... |

12 |
A note on arithmetical properties of cubic surfaces
- Segre
- 1943
(Show Context)
Citation Context ...ane to this point intersects the surface in a singular cubic, giving rise to a rational curve on X. This curve contains plenty of k-points, and so we can apply 1.8.1. This argument is due to B. Segre =-=[S43]-=-. In fact, Segre later showed that if a smooth cubic surface (over an infinite field k) contains a k-point, then it contains infinitely many k-points [S51]. It follows from the argument described abov... |

11 | On the structure of conic bundles - Sarkisov - 1982 |

10 |
Arithmétique des variétés rationnelles et problèmes birationnels
- Colliot-Thélène
- 1986
(Show Context)
Citation Context ...this idea to problems posed by Mordell regarding the representation of rational numbers by ternary cubic forms. A nice survey of recent developments in this direction is offered by Colliot-Thélène in =-=[C86]-=-. Exercise 11. (Swinnerton-Dyer [S-D]) Consider the cubic surface defined by Prove that t(x 2 + y 2 ) = (4z − 7t)(z 2 − 2t 2 ). (1) The real points of this surface, considered as a real two-manifold, ... |

9 | Su alcune varietá algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche - Fano - 1942 |

9 |
On the rational solutions of homogeneous cubic equations in four variables
- Segre
- 1951
(Show Context)
Citation Context ...pply 1.8.1. This argument is due to B. Segre [S43]. In fact, Segre later showed that if a smooth cubic surface (over an infinite field k) contains a k-point, then it contains infinitely many k-points =-=[S51]-=-. It follows from the argument described above that any smooth cubic surface containing a k-point is unirational. 1.8.3. An interesting variation on the map discussed in 1.7 is when we allow U = V . F... |

9 |
Two special cubic surfaces’, Mathematika 9
- SWINNERTON-DYER
- 1962
(Show Context)
Citation Context ...l regarding the representation of rational numbers by ternary cubic forms. A nice survey of recent developments in this direction is offered by Colliot-Thélène in [C86]. Exercise 11. (Swinnerton-Dyer =-=[S-D]-=-) Consider the cubic surface defined by Prove that t(x 2 + y 2 ) = (4z − 7t)(z 2 − 2t 2 ). (1) The real points of this surface, considered as a real two-manifold, consist of two connected components. ... |

8 | On the fundamental group of a unirational variety - Serre - 1959 |

7 |
Die Geometrie auf den Flachen dritter Ordnung
- Clebsch
(Show Context)
Citation Context ...nts of the matrix of linear forms. 50(3) We now show that every smooth cubic surface is determinantal. The earliest proof of this fact appears to be in an 1866 paper of Clebsch, who credits Schröter =-=[Cl]-=-. We give here two different proofs. 9 First Proof: This proof uses the configuration of the twenty seven lines on the cubic surface S. We claim that there are nine lines on the surface that can be re... |

7 | Nuove ricerche sulle varietà algebriche a tre dimensioni a curvesezionicanoniche, Pontificia Acad - Fano - 1947 |

6 |
Algebraic deformations of polarized varieties
- Matsusaka
- 1968
(Show Context)
Citation Context ...al fiber is non-ruled: we need only choose Fmp so that its reduction modulo p has only non-degenerate critical points in each affine patch. Now the idea is to apply the following theorem of Matsusaka =-=[Mats]-=- about the behavior of ruledness in families. 345.3. Theorem. Let V be a discrete valuation ring that is a localization of a finitely generated algebra over a field or over the integers. 4 Let K (res... |

6 |
Über Flächen welche Schaaren rationaler Curven besitzen
- Noether
(Show Context)
Citation Context ...)(f 1/pe), a subfield of the purely transcendental extension k(x 1/pe 1 , . . ., x 1/pe n ). Exercise 7. This is sometimes called Tsen’s theorem [Ts]. The case d = n = 2 is due to Max Noether in 1871 =-=[No]-=-. We seek solutions xi = ∑m j=0 aijtj , where the aij are unknown elements of C, to the degree d polynomial F(X0, . . .Xn). Plugging in Xi = xi, and gathering up all terms tr , we see that the coeffic... |

5 |
Variazione continua ed omotopia in geometria algebrica
- Segre
- 1960
(Show Context)
Citation Context ...anin on the non-existence of rational quartic three-folds settled— negatively— the Lüroth problem, because undisputed examples of unirational quartic three-folds had already been constructed by Segre =-=[S60]-=-. Around the same time, Clemens and Griffiths also resolved the Lüroth problem, by showing that there exist no smooth rational cubic three-folds [CG]. (It is not hard to see that every smooth cubic th... |

5 |
Two remarks on four dimensional cubics
- Tregub
(Show Context)
Citation Context ...onal equivalence between two smooth quintics in P 5 is actually a projective equivalence. Again the corollary follows: there exist no smooth rational quintic four-folds [P]. The short paper of Tregub =-=[Tr]-=- gives some nice examples of rational cubic hypersurfaces in P 5 . But there is not a single known example of a smooth rational quartic hypersurface of dimension four or higher; nor is there any proof... |

4 |
Lectures on the non-singular cubic surface
- Geramita
- 1989
(Show Context)
Citation Context ... Let S/Q be the cubic surface it defines. If S is unirational, then the 1 For this, and other basic properties of cubic surfaces, the reader is referred to [H, V 4], or the more elementary account in =-=[Ger]-=-. The discussion in [R] is also quite fun and informative, although lacking in proofs. 10rational map A2 φ ��� S can be used to find rational, and hence integer, solutions of the equation: for each (... |

4 | of Algebraic Geometry, American Mathematical Society Colloquium Publications 29 - Weil - 1946 |

4 |
Non-rational covers of CP n
- Kollár
- 2000
(Show Context)
Citation Context ...at are “close to rational.” Recent work of Kollár has established the existence of abundant examples of non-rational varieties with various other nice properties, such as rational connectivity [K95], =-=[K97]-=-. A smooth projective variety is rationally connected if every two points are joined by a rational curve. We have seen that a smooth hypersurface whose degree is no greater than its embedding dimensio... |

3 |
Some elementary examples of uniruled varieties which are not rational
- Artin
- 1972
(Show Context)
Citation Context ...n all dimensions three or higher, using the observation that the torsion subgroup of the third integral singular cohomology group of a non-singular projective variety over C is a birational invariant =-=[AM]-=-. We now have a reasonably complete understanding of rationality for smooth three-folds; see the papers of Sarkisov, Iskovskikh, Bardelli and Beauville listed in the bibliography. Meanwhile, Colliot-T... |

3 | Polarized mixed Hodge structures, Annali di Math. pura e appl - Bardelli - 1984 |

3 | Algebraic threefolds with special regard to the problem of rationality - Iskovskikh - 1983 |

3 |
Some remarks on rational points, Mem
- Nishimura
- 1955
(Show Context)
Citation Context ... Q, which is smooth for generic choices of G. In fact, it is smooth when G = 0, giving a truly explicit example. 45Solutions for Exercises Exercise 1. This was originally proved by Nishimura in 1955 =-=[Ni]-=-. The following proof is due to Endre Szabó. Use induction on the dimension of Y . If Y has dimension one, rational maps are morphisms defined everywhere, and the result is obvious. If Y is a smooth v... |

3 | Algebraic groups and class fields, [translation - Serre - 1997 |

2 |
L'arithm'etique des vari'et'es rationnelles
- Colliot-Th'el`ene
- 1992
(Show Context)
Citation Context ...e bibliography. Segre was motivated, at least at first, by arithmetic questions of Mordell on representing integers by ternary forms. The arithmetic applications are still important today; see [C86], =-=[C92]-=-, [K96a]. Manin’s theorem— that any two birationally equivalent smooth cubic surfaces of Picard number one are actually projectively equivalent— has a stronger analog for three-folds, at least over an... |

2 | Variétés unirationelles non rationelles: au-delà de l’exemple d’Artin et - Colliot-Thélène, Ojanguren - 1989 |

2 | Ueber diejenigen Flachen dritter Grades, auf denen sich drei gerade Linien in einem Punkte schneiden - Eckardt |

2 | Cremona transformations - Hudson - 1927 |

2 |
Quasi-algebraische-abgeschlossene funktionenkorper
- Tsen
- 1936
(Show Context)
Citation Context ...ed, its function field is isomorphic to k(x1, . . ., xn)(f 1/pe), a subfield of the purely transcendental extension k(x 1/pe 1 , . . ., x 1/pe n ). Exercise 7. This is sometimes called Tsen’s theorem =-=[Ts]-=-. The case d = n = 2 is due to Max Noether in 1871 [No]. We seek solutions xi = ∑m j=0 aijtj , where the aij are unknown elements of C, to the degree d polynomial F(X0, . . .Xn). Plugging in Xi = xi, ... |

1 |
The uniform bound problem for local birational non-singular morphisms
- Johnston
- 1989
(Show Context)
Citation Context ...ated by the pullbacks of generators of J Y/X; because J Y/X is invertible, this is the same as π ∗ J Y/X. We can now give an easy proof of Abhyankar’s theorem; the idea is from a paper of B. Johnston =-=[Jo]-=-. Proof of Theorem 5.4 assuming Y is of finite type over X. Because Y is normal, it is regular in codimension one. So we are free to replace Y by an open set containing the generic point of E so as to... |

1 |
Introduction to the minimal model program, Alg
- Kawamata, Matsuda, et al.
(Show Context)
Citation Context ... Kodaira dimension (or Kodaira dimension −∞). An understanding of these varieties is an essential feature of Mori’s Minimal Model Program for the birational classification of algebraic varieties; see =-=[KaMM]-=-. Miyaoka proved that uniruledness is equivalent to negative Kodaira dimension for smooth three-folds of characteristic zero [Mi]; see also [S-B]. The conjecture remains open in higher dimension. A si... |