RELATIONS BETWEEN MULTIPLICITY AND DIVISOR CLASS GROUP FOR RATIONAL SURFACE SINGULARITIES

In this paper we prove that a rational surface singularity with divisor class group ℤ/(2) is a rational double point. This generalizes a result by Brieskorn: if the divisor class group of a rational singularity is trivial then it is the E8 singularity [3]. We also prove several inequalities involving the integers e, δ, mi, [Formula: see text], where [Formula: see text] is the fundamental cycle. The proof of this result uses ideas from Minkowski's theory of reduction of positive-definite quadratic forms. We also give some interesting counterexamples to some of the related questions in this context.

CONNECTING RANDOM MATRIX AND TOPOLOGICAL LG MODELS VIA INTEGRABILITY

We prove that a bi-Hamiltonian system (e.g., a KdV system) is integrable if and only if its Krichever–Moser–Jacobi matrix L has a double eigenvalue, i.e., if (the characteristic polynomial of) L has rational double point singularities. Therefore, L (precisely, its characteristic polynomial) can be identified with the potential of a Landau–Ginzburg (LG) topological model with the same (rational double point) singularity. A rational curve is naturally defined in the corresponding Kummer variety and this explains the appearance (or non-appearance) of the doubling of the string equation as well as a phenomenon observed by Eguchi et al. Finally, we point out a parallelism between rational two-dimensional theories and rational singularities.