Families of rationally connected varieties.(English)Zbl 1092.14063

A complex algebraic variety $$X$$ is said to be rationally connected if two general points $$p,q\in X$$ are contained in the image of a map $${\mathbb P^1} \to X$$. In this beautiful paper the authors prove an important property of the rational connectivity (not shared by the rationality):
(T1) Let $$f: X\to Y$$ be a dominant morphism of complex varieties, if $$Y$$ and the general fiber of $$f$$ are rationally connected, then $$X$$ is rationally connected.
The proof is given in several steps.
1) (T1) is deduced from the following result:
(T2) Let $$\pi: X\to B$$ a proper morphism of complex varieties, with $$B$$ a smooth curve. If the general fiber of $$\pi$$ is rationally connected, then $$\pi$$ has a section.
2) The case $$B\simeq {\mathbb P^1}$$ is considered. The notions of flexible and prefexible stable map $$f: C\to X$$ are introduced and it is proved that:
(T3) if $$X$$ admits a prefexible map, then $$\pi$$ has a section.
3) A prefexible stable map is obtained, starting from a 1-dimensional linear section $$f:C \to X$$ , by two ingenious constructions which consist in attaching rational curves to the fibres of $$\pi$$ and in performing small deformations of the branch divisor of $$\pi f: C\to B$$ (in the case of multiple fibers of $$\pi$$).
4) Finally the result (T2) is extended to the case of an arbitrary curve $$B$$.

MSC:

 14M20 Rational and unirational varieties 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
Full Text:

References:

 [1] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601 – 617. · Zbl 0909.14007 [2] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45 – 88. · Zbl 0909.14006 [3] K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), no. 1, 1 – 60. · Zbl 0872.14019 [4] F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 539 – 545 (French). · Zbl 0783.14022 [5] A. Clebsch, Zur Theorie der Riemann’schen Flachen, Math Ann. 6 (1872), 216-230 Springer-Verlag, Berlin, 1996. · JFM 05.0227.01 [6] B. Fantechi, R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), 345-364. · Zbl 1054.14033 [7] William Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542 – 575. · Zbl 0194.21901 [8] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45 – 96. · Zbl 0898.14018 [9] T. Graber, J. Harris, J. Starr, A note on Hurwitz schemes of covers of a positive genus curve, preprint alg-geom/0205056. [10] A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891) 1-61. · JFM 23.0429.01 [11] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. · Zbl 0877.14012 [12] János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429 – 448. · Zbl 0780.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.