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Rationally connected varieties. (English) Zbl 0780.14026

A smooth projective variety \(X\) (over an uncountable algebraically closed field \(k)\) is said to be rationally connected (R. C. for short) if any two generic points of \(X\) are contained in some connected curve all irreducible components of which are rational (possibly singular). Unirational varieties, as well as Fano varieties, are R. C. A surface is R. C. iff it is rational. Not all R. C. threefolds (for example: the general quartic hypersurface of \(\mathbb{P}_ 4)\) are expected to be unirational. The difficulty is that all birational invariants (such as \(\pi_ 1\) if \(k=\mathbb{C}\), or \(h^ 0(X,\otimes^ \nu\Omega^ 1_ x))\) which are known to vanish for unirational varieties also vanish for \(X\) which are R.C. By contrast, rational connectedness is deformation invariant. This easily follows from the main result of this paper, which is of fundamental importance for the theory of R. C. varieties: if \(X\) is R. C., then any finite subset of \(X\) is contained in some irreducible rational curve. The argument rests on refined techniques of deformation theory, which also provide the following result (of independent interest, from which the above follows easily): Let \(g:D\to X\) be a nonconstant morphism from a smooth proper curve \(D\) to \(X\), and let \(\overline b_ 1,\ldots,\overline b_ q\) belong to \(g(D)\). Let \(C_ 1,\ldots,C_ N\) be semipositive rational curves on \(X\) (i.e.: \(C_ i=f_ i(\mathbb{P}_ 1)\), with \(f^*_ i(TX)\) semipositive for all \(i\)’s). Assume no \(C_ i\) contains some \(\overline b_ j\). Then: if \(N\) is sufficiently large (depending on \(g^*(TX))\), there exists \(I\subset\{1,\ldots,N\}\) such that the curve \((g(D)+\sum_{i\in I}C_ i)=D_ 0\) moves in an algebraic family of curves \((D_ s)_{s\in S}\), all \(D_ s\) going through \((b_ j)\), and \(D_ s\) being irreducible of the same genus as \(D\) for \(s\) generic in \(S\).
An effective bound for \(N\) above when \(D=\mathbb{P}_ 1\) permitted the authors to show that the family of Fano \(n\)-folds is bounded [cf. Classification of irregular varieties minimal models and abelian varieties, Proc. Conf., Trento/Italy 1990; Lect. Notes Math. 1515, 100- 105 (1992; Zbl 0776.14012)]. The last section contains (among others) the following characterisation of R. C. threefolds: \(X\) is R. C. iff \[ h^ 0(X,\Omega^ 1_ X)=h^ 0(X,S^ 2(\Omega^ 1_ X))=h^ 0(X,{\mathcal O}_ X(\nu K_ x))=0\quad\text{(for all } \nu>0). \] This result needs Mori’s minimal model program, as well as Miyaoka’s characterization of uniruledness in dimension 3.

MSC:

14M20 Rational and unirational varieties
14D15 Formal methods and deformations in algebraic geometry
54D05 Connected and locally connected spaces (general aspects)
14H45 Special algebraic curves and curves of low genus

Citations:

Zbl 0776.14012
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