Equivariant Chow-Witt groups and moduli stacks of elliptic curves

We introduce equivariant Chow-Witt groups in order to define Chow-Witt groups of quotient stacks. We compute the Chow-Witt ring of the moduli stack of stable (resp. smooth) elliptic curves, providing a geometric interpretation of the new generators. Along the way, we also determine the Chow-Witt ring of the classifying stack of $\mu_{2n}$.Comment: 29 pages, comments welcome

Integral Picard group of the stack of quasi-polarized K3 surfaces of low degree

We compute the integral Picard group of the stack $\mathcal{K}_{2l}$ of quasi-polarized K3 surfaces of degree $2l=4,6,8$. We show that in this range the integral Picard group is torsion-free and that a basis is given by certain elliptic Noether-Lefschetz divisors together with the Hodge line bundle. To achieve this result, we investigate certain stacks of complete intersections and their Picard groups by means of equivariant geometry. In the end we compute an expression of the class of some Noether-Lefschetz divisors, restricted to an open substack of $\mathcal{K}_{2l}$, in terms of the basis mentioned above.Comment: 24 pages, comments are welcome

On quantum and relativistic mechanical analogues in mean field spin models

Conceptual analogies among statistical mechanics and classical (or quantum) mechanics often appeared in the literature. For classical two-body mean field models, an analogy develops into a proper identification between the free energy of Curie-Weiss type magnetic models and the Hamilton-Jacobi action for a one dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrodinger equation in quantum mechanics. We show that this identification can be remarkably extended to include a wide family of magnetic models classified by normal forms of suitable real algebraic dispersion curves. In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton-Jacobi type. We observe that the mechanical analog of these models can be viewed as the relativistic analog of the Curie-Weiss model and this helps to clarify the connection between generalised self-averaging and in statistical thermodynamics and the semi-classical dynamics of viscous conservation laws.Comment: Dedicated to Sandro Graffi in honor of his seventieth birthda

Cohomological invariants of root stacks and admissible double coverings

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the stack of admissible double coverings.Comment: 16 pages, comments welcome

A complete description of the cohomological invariants of even genus hyperelliptic curves

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristicComment: 25 pages, comments are welcome! To appear on Documenta Mathematic

Brauer groups of moduli of hyperelliptic curves via cohomological invariants

We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.Comment: 28 pages, comments are very welcome

Brauer groups of moduli of hyperelliptic curves via cohomological invariants

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves Hg over any field of characteristic 0. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.Peer Reviewe