No abstract

Let C ⊂ P g −1 be a general curve of genus g and let k be a positive integer such that the Brill-Noether number ρ(g , k, 1) ≥ 0 and g > k + 1. The aim of this short note is to study the relative canonical resolution of C on a rational normal scroll swept out by a g 1 k = |L| with L ∈ W 1 k (C) general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if ρ > 0 and (k − ρ − 7 2 ) 2 − 2k + 23 4 > 0.

The spherical Couette system, consisting of a viscous fluid between two differentially rotating concentric spheres, is studied using numerical simulations and compared with experiments performed at BTU Cottbus-Senftenberg, Germany. We concentrate on the case where the outer boundary rotates fast enough for the Coriolis force to play an important role in the force balance, and the inner boundary rotates slower or in the opposite direction as compared to the outer boundary. As the magnitude of differential rotation is increased, the system is found to transition through three distinct hydrodynamic regimes. The first regime consists of the emergence of the first non-axisymmetric instability. Thereafter one finds the onset of ‘fast’ equatorially antisymmetric inertial modes, with pairs of inertial modes forming triadic resonances with the first instability. A further increase in the magnitude of differential rotation leads to the flow transitioning to turbulence. Using an artificial excitation, we study how the background flow modifies the inertial mode frequency and structure, thereby causing departures from the eigenmodes of a full sphere and a spherical shell. We investigate triadic resonances of pairs of inertial modes with the fundamental instability. We explore possible onset mechanisms through numerical experiments.

We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analogous of the Kuznetsov's Conjecture for cubic fourfolds: a Gushel-Mukai fourfold is rational if and only if it admits an associated K3 surface.

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that $$h^0 (C, V \otimes E) \ge k$$ h 0 ( C , V ⊗ E ) ≥ k . We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and “nontwisted” Brill–Noether loci. We describe the tangent cones to the twisted Brill–Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill–Noether loci with negative expected dimension.