DeformationTheoryAG · Deformation Theory and Moduli Spaces in Algebraic Geometry

We will look for proving a number of different results in Deformation Theory and Moduli Spaces in algebraic geometry. In our project, we will mainly use the modern language of algebraic geometry, based on DGLAs and L-infinity algebras.As first objective, we seek to give an explicit description of the DGLA controlling infinitesimal deformations of morphisms of schemes.We will also try to extend this construction to the case of morphisms of stacks. In particular, we will be dealing with the key case of morphisms from a singular curve in a smooth variety (like the case of stable maps).Once we have the DGLA for this problem, we will attempt to define a semiregularity map.By semi-regularity map we mean a map whose kernel contains all obstructions to deformations. This map will allow us to handle the problem of obstructions, providing a technical tool to understand the local behavior of the associated moduli space (in particular the one of stable maps).Moreover, we will try to apply these techniques to construct new examples of dg-schemes and dg- manifolds.Finally, we will seek to use all these new tools for computing new Gromov-Witten invariants. Mainly, we will be dealing with the cases in which the classical theory fails, working out new examples of reduced Gromov-Witten invariants.