The projectivity of the moduli space of stable curves, III: The line bundles on M..., and a proof of the projectivity of M... in characteristic 0.
We review a construction of Ellingsrud-Strømme relating instantons of charge n on the ordinary projective space and theta-characteristics on a plane curve of degree n with some extra-structure.
Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension , time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...
We prove: For a local analytic family of analytic space germs there is a largest subspace in such that the family is trivial over . Moreover the reduction of equals the germ of those points in for which is isomorphic to the special fibre .
We classify all complex - and -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex -dimensional dual mock-Lie algebras.
One has two notions of vanishing cycles: the Deligne's general notion and a concrete one used recently in the study of polynomial functions. We compare these two notions which gives us in particular a relative connectivity result. We finish with an example of vanishing cycle calculation which shows the difficulty of a good choice of compactification.
This work contains an extended version of a course given in Arrangements in Pyrénées. School on hyperplane arrangements and related topics held at Pau (France) in June 2012. In the first part, we recall the computation of the fundamental group of the complement of a line arrangement. In the second part, we deal with characteristic varieties of line arrangements focusing on two aspects: the relationship with the position of the singular points (relative to projective curves of some prescribed degrees)...
We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.