Syzygies of projective bundles (Q2371809)

Let \(\mathcal E\) be a nef vector bundle of rank \(r\) on a smooth complex projective variety \(Y\) of dimension \(n\). Consider \(X = {\mathbb P}_Y (\mathcal{E})\) and the projection morphism \(\pi : X \rightarrow Y\). The author studies the property \(N_p\) for the tautological bundle \(\mathcal {O}_{{\mathbb P}_Y (\mathcal{E} \otimes B)} (1) \cong \mathcal{O}_X (1) \otimes \pi^* B\), where \(B\) is a sufficiently positive line bundle on \(Y\). The main result is the following. Theorem. For \(A,D \in \text{Pic}(Y)\), suppose that \(A\) is very ample and \(D\) is nef. Let \(e\) be an integer such that \(A^e \otimes \mathcal{E} \otimes \mathcal{E}^*\) is a nef vector bundle. Then (1) For \(f \geq er+n+1+p\), \(\mathcal{O}_X(1) \otimes \pi^* \mathcal{O}_Y(K_Y + f A + D)\) satisfies \(N_p\). (2) For \(f \geq er+d-1+p\), \(\mathcal{O}_X(1) \otimes \pi^* \mathcal{O}_Y(fA)\) satisfies \(N_p\), where \(d\) is the degree of \(A\). In the case \(\mathcal{E}\) is a line bundle (hence \(e = 0\)), the theorem reduces to a result of \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 111, 51--67 (1993; Zbl 0814.14040)]. By \textit{M. L. Green}'s work [J. Diff. Geom. 19, 125--171 (1984; Zbl 0559.14008)], the property \(N_p\) is controlled by the vanishing of certain cohomology groups. Now, the idea is to reduce the desired vanishing of cohomology groups on \(X\) to those on \(Y\) and then use a variation of the methods used by Ein-Lazarsfeld [loc. cit.]