We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new examples of projective manifolds for which such global Torelli theorem holds. As applications we prove that, in these cases, a canonical completion of the Torelli space is a bounded pseudoconvex domain in complex Euclidean space, and show that generic Torelli implies global Torelli on moduli space with certain level structure.

We develop a new technique to study ranks of multiplication maps for linear series via limit linear series and degenerations to chains of elliptic curves. We prove an elementary criterion and apply it to proving cases of the Maximal Rank Conjecture. We give a new proof of the case of quadrics, and also treat several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.

No abstract uploaded!

In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.