Castelnuovo-Mumford Regularity over Scrolls and Splitting Criteria
F. Malaspina, G. Sankaran
TL;DR
The paper defines a new Castelnuovo-Mumford regularity for scrolls X = Proj(Sym V) over P^m, where V = ⊕ O_{P^m}(a_i), and proves Horrocks-type splitting criteria for vector bundles on these scrolls. The authors introduce (p,q)-regularity via two families of cohomology vanishings, analyze positivity (a_0>0) effects, and compare their notion with Maclagan–Smith multigraded regularity, noting that top-degree requirements differ and yield a weaker condition in this setting. They show that regular coherent sheaves are globally generated and that regularity behaves well under restriction to divisors, then specialize to rational normal scrolls (m=1) where the vanishings simplify and hyperplane sections remain scrolls with preserved regularity. The results provide explicit splitting criteria in terms of cohomology vanishings, recovering known Horrocks-type results in special cases (e.g., X = P^n × P^m) and yielding a precise description of indecomposable regular bundles on scrolls. Overall, the work extends regularity theory to scrolls, offering practical tools for decomposing vector bundles on these toric varieties and highlighting distinctions from existing multigraded reg notions.
Abstract
We introduce and study a notion of Castelnuovo-Mumford regularity suitable for scrolls obtained as projectivisations of sums of line bundles on $\mathbb P^m$. We show that this is a natural generalisation of the well known regularity on projective and multiprojective spaces and we prove Horrocks-type splitting criteria for vector bundles.