On syzygies of Calabi-Yau varieties and varieties of general type
In the paper under review, the author studies projective normality, \(N_{p}\) properties, and effective very ampleness for line bundles on a Calabi-Yau variety, as well as for pluricanonical bundles of a variety of general type. Let us recall that for \(X\) a nonsingular projective complex variety of dimension \(n \geq 1\), Fujita's conjecture predicts that the adjoint line bundle \(L_{d} = K_{X}+ dB\) is globally generated for \(d \geq n + 1\) and is very ample for \(d \geq n + 2\), where \(K_{X}\) is the canonical line bundle of \(X\) and \(B\) is an ample line bundle. It was suggested through years that that the conjecture could be extended to higher syzygies through \(N_{p}\) property. Specifically, suppose that \(L_d\) is globally generated and write \(S = \mathrm{Sym}(H^{0}(X, L_{d}))\). The section ring \[ R(L_{d}) = \bigoplus_{\ell \geq 0}H^{0}(X,\ell L_{d}) \] is naturally a finitely generated graded \(S\)-module and therefore admits a minimal graded free resolution \[ \dots \rightarrow E_{p} \rightarrow \dots \rightarrow E_{2} \rightarrow E_{1} \rightarrow E_{0} \rightarrow R(L_{d}) \rightarrow 0. \] Then \(L_{d}\) is said to satisfy (Property) \(N_{p}\) for \(p \geq 0\) if \(E_{0} = S\) and \(E_{i} = \oplus S(-i - 1)\) for \(1 \leq i \leq p\). The first main result of the paper is devoted to \(N_p\) properties for Calabi-Yau varieties. Recall that \(X\) is called Calabi-Yau if \(K_X \cong \mathcal{O}_{X}\) and \(h^1(\mathcal{O}_{X}) = 0\). Assume additionally that \(B\) is globally generated and denote by \(\phi_{B}\) the map associated to the complete linear system \(|B|\). A curve \(C\) cut out by \(n-1\) general global sections of \(B\) is called a curve section of \(B\). {Theorem A}. Let \(X\) be a Calabi-Yau variety of dimension \(n\geq 2\). Let \(B\) be a globally generated line bundle on \(X\) such that \(\dim \phi_{B}(X) = n\). Let \(C\) be a curve section of \(B\). When \(n\geq 3\), assume further that \(h^{2}(\mathcal{O}_{X}) = 0\). {\parindent=6mm \begin{itemize}\item[1)] \(B^{n}\) satisfies \(N_{0}\) except when \(h^{0}(B^{2}_{C})=3\). \item[2)] \(B^{n-1}\) satisfied \(N_{0}\) except when \(h^{0}(B_{C}^{2})=3\) or \(\phi_{B}(X)\) is variety of minimal degree \(\geq 2\). \item[3)] If \(h^{0}(B) \geq n+2\), then \(B^{n-1+p}\) satisfies \(N_{p}\) for \(p\geq 1\). \end{itemize}} {Theorem B}. Let \(X\) be a projective nonsingular variety of dimension \(n \geq 2\) such that the canonical bundle \(K_{X}\) is globally generated and \(\text{dim} \, \phi K_{X}(X) = n\). Let \(C\) be a curve section of \(K_{X}\). Assume that \(h^{1}(\mathcal{O}_{X}) = h^{1}(K_{X}) = 0\) and when \(n \geq 3\), \(h^{2}(\mathcal{O}_{X}) = 0\). {\parindent=6mm \begin{itemize}\item[1)] \(K_{X}^{n+1}\) satisfies \(N_{0}\) except when \(h^{0}(K_{X}^{2}|_{C}) = 3\). \item[2)] \(K^{n}_{X}\) satisfies \(N_{0}\) except when \(h^{0}(K_{X}^{2}|_{C}) = 3\) or \(\phi_{K_{X}}(X)\) is a variety of minimal degree \(\geq 2\). \item[3)] If \(h^{0}(K_{X}) \geq n+2\), then \(K_{X}^{n+p}\) satisfies \(N_{p}\) for \(p\geq 1\). \end{itemize}} Finally, in the last part of the paper, the author studies the Arbarello-Sernesi module associated to the line bundle \(B\) from the viewpoint of generating degrees.
scientific article; zbMATH DE number 7003605
scientific article