On higher order embeddings of Calabi-Yau threefolds (Q1977801)

The authors prove the following: (a) Let \(X\) be a smooth Calabi-Yau threefold and let \(L\) be an ample and spanned line bundle on \(X\). Then \(nL\) is k-jet spanned provided \( n \geq k + 3 \). (b) Let \(X\) be a smooth Calabi-Yau threefold and \(L\) an ample and spanned line bundle on \(X\) with \(L^3 \geq \text{max} \{8, k+ 4 \} \) and \(k\) a positive integer. Then there is a Zariski open subset \(U \subset X^r \) such that for \( n \geq k+ 3 \) the evaluation map: \[ H^0(L) \rightarrow H^0( L \otimes{\mathcal O}_X / m_{x_1}^{k_1} \otimes \cdots \otimes m_{x_r}^{k_r}) \] is surjective for \( ( x_1, \cdots, x_r) \in U \) and \( \sum_{i=1}^r {k_i} = k+ 1 \). The previous results should parallel the results of \textit{Th. Brauer, S. Di Rocco}, and \textit{T. Szemberg} [J. Pure Appl. Algebra 146, 17-27 (2000; Zbl 0956.14002)] for the case of \(K3\) surfaces.