Jet Schemes, Newton Polygons and Continued Fractions

Abstract: We study jet schemes of Newton non-degenerate plane curve singularities. We identify a subgraph of the graph of jet components and show that it can be constructed from walks on the lattice points in the first quadrant of the Cartesian plane. In particular, we determine all the irreducible components of the jet schemes. Furthermore, we prove that this subgraph encodes the embedded topological type of the curve singularity in the plane. Finally, we introduce a generating series defined in terms of the irreducible components of the jet schemes and their (co-)dimensions, and we prove that this series is rational and explicitly determine its poles.