Stacks of cyclic covers of projective spaces

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer-Severi schemes of rank 1, which are not necessarily uniform, and give a presentation of the Picard group for substacks corresponding to smooth triple cyclic covers.Comment: 23 pages; some minor changes; to appear in Compositio Mathematic

On near-optimal time samplings for initial data best approximation

Leveraging on the work of De Vore and Zuazua, we further explore their methodology and deal with two open questions presented in their paper. We show that for a class of linear evolutionary PDEs the admissible choice of relevant parameters used to construct the near-optimal sampling sequence is not influenced by the spectrum of of the operator controlling the spatial part of the PDE, but only by its order. Furthermore, we show that it is possible to extend their algorithm to a simple version of a non-autonomous heat equation in which the heat diffusivity coefficient depends explicitly on time

Integrable viscous conservation laws

We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven. We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe

Collision trajectories and regularisation of two-body problem on S2S^2

In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position vectors of the two bodies. We describe the $\omega$-limit set of the variables in the symplectically reduced system corresponding to initial data that lead to collisions. Furthermore we provide a geometric description of the dynamics. Lastly, we regularise the system and investigate its behaviour on near collision orbits. This involves the study of completely degenerate equilibria and the use of high-dimensional non-homogenous blow-ups