Classification of bi-polarized 3-folds \((X, L_{1}, L_{2})\) with \(h^{0}(K_{X}+L_{1}+L_{2})=1\)
Consider \(X\) a smooth complex projective variety of dimension \(n\) and \(L\) an ample line bundle on \(X\). Several questions on the adjoint bundles \(K_X+tL\), \(t\) a positive integer, are of interest, in particular those on the interplay between their effectivity and nefness (see the Introduction of this paper and references therein). More generally, one can consider multipolarized manifolds \((X,L_1, \dots, L_{n-1})\) (\(L_i\) ample), and it is conjectured (see Conj. 3) that for \(n \geq 3\), if \(K_X+L_1+\dots+L_{n-1}\) is nef then it is effective, i.e., \(h^0(K_X+L_1+\dots+L_{n-1})>0\). This conjecture has been proved when \(n=3\) leading to a classification of \((X,L_1,L_2)\) such that \(K_X+L_1+L_2\) is not effective (\(X\) is either a linear space, or a quadric, or a \(\mathbb{P}^2\)-bundle over a smooth projective curve, the \(L_i\)'s are precisely described, proper references in Conj. 3). As a natural continuation of this research, the paper under review presents a complete classification of multipolarized threefolds \((X,L_1,L_2)\) such that \(h^0(K_X+L_1+L_2)=1\). The complete list of possibilities is collected in Section 3, Main result. There are seven cases: a Del Pezzo variety, a linear space, a quadric, a product of a line and a plane, a scroll or a quadric fibration over an elliptic curve, and a scroll over a smooth projective surface; the possible \(L_i\)'s are also presented.
scientific article; zbMATH DE number 6965539