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 that are not necessarily uniform, and give a presentation of the Picard group of the substacks corresponding to smooth triple cyclic covers.

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of F -manifolds due to Manin [22], we consider a special class of F -manifolds, called bi-flat F -manifolds.A bi-flat F -manifold is given by the following data (M, ∇ 1 , ∇ 2 , •, * , e, E), where (M, •) is an F -manifold, e is the identity of the product •, ∇ 1 is a flat connection compatible with • and satisfying ∇ 1 e = 0, while E is an eventual identity giving rise to the dual product * , and ∇ 2 is a flat connection compatible with * and satisfying ∇ 2 E = 0. Moreover, the two connections ∇ 1 and ∇ 2 are required to be hydrodynamically almost equivalent in the sense specified in [2].First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat F -manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric.Although any Frobenius manifold possesses automatically the structure of a bi-flat F -manifold, we show that the latter is a strictly larger class.In particular we study in some detail bi-flat F -manifolds in dimensions n = 2, 3. For instance, we show that in dimension three bi-flat F -manifolds are parametrized by solutions of a two parameters Painlevé VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat F -manifolds.

We show that bi-flat F -manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's ∨-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank 2 and 3. On the Veselov's ∨-systems side, we provide a generalization of the notion of ∨-systems that gives rise to a dual connection which coincides with a Dunkl-Kohno-type connection associated with such groups. In particular, this allows us to treat on the same ground several different examples including Coxeter and Shephard groups. Remarkably, as a byproduct of our results, we prove that in some examples basic flat invariants are not uniquely defined. As far as we know, such a phenomenon has never been pointed out before.

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