Based on the work of Harada, Nowroozi, and Van Tuyl, who provided particular length two virtual resolutions for finite sets of points in F1 & times; F1, we prove that the vast majority of virtual resolutions of a pair for minimal elements of the multigraded regularity in this setting are of Hilbert-Burch type. We give explicit descriptions of these short virtual resolutions that depend only on the number of points. Moreover, despite initial evidence, we show that these virtual resolutions are not always short and give sufficient conditions for their length to be three.
Let (X,d,μ) be a metric space with a metric d and a doubling measure μ. Assume that L is a nonnegative self-adjoint operator on L2(X) satisfying the Davies–Gaffney estimates. In this paper, we prove the sharp estimates for the imaginary power operator Liα, α∈R on the Hardy space associated with the operator L. In addition, if L satisfies an additional Lp0−L2 estimate for p0∈[1,2), we prove the sharp weak type (p0,p0) estimate for Liα.
Let X_0 be an irreducible smooth projective curve defined over ℚ and f_0 : X_0 →ℙ^1_ℚ a nonconstant morphism whose branch locus is contained in the subset {0,1, ∞}⊂ℙ^1_ℚ. For any vector bundle E on X = X_0×_ Spec ℚ Specℂ, consider the direct image f_*E on ℙ^1_ℂ, where f= (f_0)_ℂ. It decomposes into a direct sum of line bundles and also it has a natural parabolic structure. We prove that E is the base change, to ℂ, of a vector bundle on X_0 if and only if there is an isomorphism f_*E ∼→⊕_i=1^r 𝒪_ℙ^1_ℂ(m_i), where r = rank(f_*E), that takes the parabolic structure on f_*E to a parabolic structure on ⊕_i=1^r 𝒪_ℙ^1_ℂ(m_i) defined over ℚ.
