Dimers and cluster integrable systems

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci. EN

On the Severi problem in arbitrary characteristic

We show that Severi varieties parametrizing irreducible reduced planar curves of given degree and geometric genus are either empty or irreducible in any characteristic. As a consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of given geometric genus is nodal. As a byproduct, we obtain the first proof of the irreducibility of the moduli space of smooth projective curves of given genus in positive characteristic, that does not involve a reduction to the characteristic zero case.Comment: 34 pages, 9 figures. Comments are welcome