On deformation of nef values
Let \(L\) be an ample line bundle over a smooth complex projective variety \(X\). Assume that the canonical divisor \(K_ X\) is not nef. Then there exists a positive rational number \(\tau\) such that the \(\mathbb{Q}\)-divisor \(K_ X+\tau L\) is nef but not ample; this number is called the nef value of \(L\). The author proves that the nef value is invariant under deformation. The proof uses the following result: Let \(X\to S\) be a smooth projective family, with \(S\) smooth irreducible, and let \(NE(X/S)\) be the cone of effective relative 1-cycles. Then, if \(R\) is an extremal ray of \(NE(X/S)\), the locus of curves whose class is in \(R\) dominates \(S\). The main theorem has nice geometrical consequences which improve results of \textit{L. Ein} [Duke Math. J. 52, 895-907 (1985; Zbl 0603.14026)]. For example, if \(E\) is a rank-\(r\) vector bundle on \(\mathbb{P}^ m\) which is a specialization of the trivial bundle, and if \(E(1)\) is nef and \(r\leq m\), then \(E\) is trivial (note that the Euler sequence gives a nontrivial specialization of \({\mathcal O}^{\otimes(m+1)})\). Moreover, if \(X\) is a smooth subvariety of \(\mathbb{P}^ N\) of dimension \(n\) which contains an \(m\)- plane with \(m\geq n/2\) and trivial normal bundle, then \(X\) is a projective bundle over a smooth projective variety of dimension \(n-m\).
scientific article; zbMATH DE number 58549
scientific article