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Nonrational covers of \(\mathbb{C}\mathbb{P}^m\times \mathbb{C}\mathbb{P}^n\). (English) Zbl 0961.14033

Corti, Alessio (ed.) et al., Explicit birational geometry of 3-folds. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 281, 51-71 (2000).
This paper aims to provide further examples of higher dimensional varieties that are rationally connected, but not rational, and not even ruled. The original methods of V. A. Iskovskikh and Yu. I. Manin [Math. USSR, Sb. 15 (1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)], and of C. H. Clemens and Ph. A. Griffiths [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.48302)] have been further developed by many authors [see for example A. Béauville, Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 309-391 (1977; Zbl 0368.14018), V. Iskovskikh, J. Sov. Math. 13, 815-868 (1980); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 159-236 (1979; Zbl 0415.14025), F. Bardelli, Ann. Mat. Pura Appl., IV. Ser. 137, 287-369 (1984; Zbl 0579.14033)], and give a fairly complete picture in dimension three. On the other hand, in higher dimension, only special examples were known until recently [see M. Artin and D. Mumford, Proc. Lond. Math. Soc., III. Ser. 25, 75-95 (1972; Zbl 0244.14017), V. G. Sarkisov, Math. USSR, Izv. 17, 177-202 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 918-945 (1980; Zbl 0453.14017) and Math. USSR, Izv. 20, 355-390 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371-408 (1982; Zbl 0593.14034), A. V. Pukhlikov, Invent. Math. 87, 303-329 (1987; Zbl 0613.14011), J.-L. Colliot-Thélène and M. Ojanguren, Invent. Math. 97, No. 1, 141-158 (1989; Zbl 0686.14050)].
In his previous paper [J. Kollár, J. Am. Math. Soc. 8, No. 1, 241-249 (1995; Zbl 0839.14031)], the author proposes a new approach to the problem of non-rationality, proving that a “very general” hypersurface \(X_d \subset \mathbb{P}^n\) of degree \(d\) is not rational for \(2n+9 \leq 3d \leq 3n+3\); very general means that \(X_d\) is a point in the complement of countably many closed proper subsets in the space of all hypersurfaces. This technique involves reduction to characteristic \(p\), and a rather clever and surprising analysis of the stability of the tangent bundle in characteristic \(p\).
The same method applies to hypersurfaces \(X_{c,d} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^{n+1}\) of bidegree \((c,d)\). The more interesting is the study of some cases when the fibers of \(X_{c,d} \rightarrow \mathbb{P}^m\) are rational – for example: (1) conic bundles, or (2) families of cubic surfaces. Especially for these two cases the result is (see theorem 1.4):
1. The very general conic bundle \(X_{c,2} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^2\), \(m \geq 2\), is not rational for \(c \geq m+3\);
2. The very general family of cubic surfaces \(X_{c,3} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^3\), \(m \geq 1\), is not rational for \(c \geq m+4\).
Especially the above result for conic bundles makes a progress after the works by V. G. Sarkisov (see above). A similar result is obtained for cyclic covers \(X_{ap,bp} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^n\) (of prime degree \(p\), ramified over a hypersurface \(B\) of bidegree \((ap,bp)\)). The author, in his 1995 paper cited above, showed that if \(ap > m+1\) and \(bp > n+1\) then \(X_{pa,pb}\) is not rational, and not even ruled, for a very general \(B\).
This paper studies the cases when \(bp = n+1\) and \(n = 1,2\) (which are the analogs of cases (1) and (2) avove), and gets (see theorems 1.2 and 1.3):
1. The very general double cover \(X_{2a,2} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^1\), \(m \geq 2\), is non-rational for \(2a > m+1\);
2. The very general cyclic triple cover \(X_{3a,3} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^2\), \(m \geq 1\), is non-rational, and not even ruled, for \(3a > m+1\).
For the entire collection see [Zbl 0942.00009].
Reviewer: A.Iliev (Sofia)

MSC:

14M20 Rational and unirational varieties
14E20 Coverings in algebraic geometry
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