On a conjecture of Beltrametti and Sommese (Q2913222)

Consider a projective manifold \(X\) of dimension \(n\). A conjecture by Beltrametti and Sommese states that if \(A\) is an ample divisor on \(X\) and the adjoint line bundle \(K_X+(n-1)A\) is nef then it has global sections (cf. Conjecture 7.2.7 in [\textit{M. C. Beltrametti} and \textit{A. J. Sommese}, The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. Berlin: de Gruyter (1995; Zbl 0845.14003)]).NEWLINENEWLINEThe paper under review proves a weak version of this conjecture. In fact (see Theorem 1.2) for \(X\) a normal projective variety of dimension \(n \geq 2\) with at most rational singularities, \(A\) a nef and big Cartier divisor such that \(K_X+(n-1)A\) is generically nef (the restriction to a general complete intersection curve is nef) the non-vanishing \(H^0(X, {\mathcal O}_X(K_X+jA)) \neq 0\) holds for some \(0 \leq j \leq n-1\) and in particular if \(A\) is effective it holds for \(j=n-1\). Also the theorem is true if \(X\) has irrational singularities with some exceptions (\((X,A)\) is birationally a scroll over a positive genus curve). The proof consists on the study of the Hilbert polynomial \(\chi(X, {\mathcal O}_X(K_X+tA))\) which by the usual vanishing theorems coincides with \(h^0(X, {\mathcal O}_X(K_X+tA))\) when \(t\) is a positive integer. Riemann-Roch theorem and considerations about the main invariants of \(X\) lead to the proof of the theorem, being the most difficult case when \(X\) is uniruled but not rationally connected.NEWLINENEWLINEThe techniques of this proof provide a stronger result on threefolds with \(\mathbb{Q}\)-factorial canonical singularities for which, if \(A\) is a nef and big Cartier divisor, \(H^0(X, {\mathcal O}_X(K_X+A))\neq 0\). An application to the computation of Seshadri constants is also given.