$m$-blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces

Abstract

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on $n$-dimensional smooth projective varieties $X$ with an $n$-block collection $\mathcal{B}$ which generates the bounded derived category $\mathcal{D}^{b}({\mathcal{O}}_{X}{\textit -mod})$. To this end, we use the theory of $n$-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf $F$ on $X$ with respect to the $n$-block collection $\mathcal{B}$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on $\mathbb{P}^{n}$ and for the $n$-block collection $\mathcal{B} = (\mathcal{O}_{\mathbb{P}^{n}}, \mathcal{O}_{\mathbb{P}^{n}}(1), \dots, \mathcal{O}_{\mathbb{P}^{n}}(n))$ on $\mathbb{P}^{n}$ Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space $\mathbb{P}^{n_{1}} \times \dots \times \mathbb{P}^{n_{r}}$ with respect to a suitable $n_{1}+\dots+n_{r}$-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].