On strengthening of the Kleiman-Chevalley projectivity criterion (Q2846928)

In the fifties of the last century Chevalley [\textit{M. Nagata}, Ill. J. Math. 2, 490-498 (1958; Zbl 0081.37503)] posed the following problem: assume that a normal variety \(X\) satisfies the condition that for any finite number of points of \(X\) there exists an affine open subvariety containing them. Can then \(X\) be imbedded in a projective variety? Kleiman gave a positive answer for smooth varieties, proving that a complete non-singular variety is projective if and only if any finite set of points is contained in some open affine subset \textit{S. L. Kleiman} [Ann. Math. (2) 84, 293--344 (1966; Zbl 0146.17001)]. In the present article the author studies the relations of Chevalley conjecture with the Picard number \(\rho(X)\), the cardinality \(\text{mqos}(X)\) of the set of maximal quasiprojective open subsets of \(X\), and \(a(X)\), the \(\sup\) of the integer numbers \(n\) such that every set of \(n\) points in \(X\) is contained in some affine open subset. Kleiman result can then be stated saying that if \(X\) is complete and \(\mathbb Q\)-factorial and \(a(X)\geq 2\rho(X)\), then \(X\) is projective. This bound has been improved by \textit{J. Kollár} [Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag (1995; Zbl 0877.14012)], who showed that it is enough to assume \(a(X)\geq \rho(X)+1\). Here the author gives examples of smooth complete and non projective varieties having \(a(X)=\rho(X)-2\). He also gives examples of smooth complete varieties with \(\text{mqos}(X)=(\rho(X)-1)!\) and of normal complete toric varieties with \(\rho(X)=0\) and \(a(X)\) arbitrarily large.NEWLINENEWLINEWe signal that the conjecture of Chevalley has been recently proved by \textit{O. Benoist} [``Quasi-projectivity of normal varieties'', Int. Math. Res. Not. 17, 3878--3885 (2013)].