De geometrie van contactlocussen en de boog-Floer-conjectuur

The goal of this thesis is to explore the geometry of the \emph{contact loci} associated to a hypersurface singularity. Contact loci are subsets of the arc space of a variety that parametrize arcs meeting a given hypersurface with some prescribed order of contact. They are known to be related to motivic integration, the monodromy conjecture, birational geometry and symplectic geometry. Nevertheless, even basic geometric properties of contact loci, such as the structure of their irreducible components, remain poorly understood. We begin by studying the \emph{embedded Nash problem}, which consists in describing the irreducible components of contact loci in terms of a log resolution of singularities. Our main result in this direction is proving that divisors appearing in minimal models of the pair given by the hypersurface and the ambient space always give distinct irreducible components of the contact loci. We tackle the embedded Nash problem in two particular cases: plane curves and semihomogeneous hypersurfaces. In each case we are able to characterize for a particular log resolution of singularities which exceptional components come from a minimal model, which ones correspond to an irreducible component of the contact locus, and which ones appear in every resolution. We find that in both cases the three sets of divisors might be different, showing that the situation is more subtle than in the classical Nash problem. Once we have understood the irreducible components of contact loci in these two cases, we study their geometry. As a result, we are able to explicitly compute the compactly supported cohomology of the $m$-contact locus for all integers $m$. We use these computations to verify the \emph{arc-Floer conjecture} in some cases. For plane curves, this involves computing the Floer cohomology of iterates of the monodromy map of the Milnor fibration, which we are able to do by ruling out the existence of certain pseudoholomorphic curves via topological arguments. For semihomogeneous singularities, we show that a spectral sequence due to McLean degenerates in many cases. The study of these examples shows that contact loci have a richer geometry than previously thought; for example they might have singular irreducible components and a certain natural decomposition into locally closed subvarieties may not be a stratification. As an application of our results, we give a formula for the motivic zeta function of a plane curve singularity in terms of its Puiseux data which recovers the result that the only divisors contributing poles are those with valency greater or equal to three in the resolution graph.