### A character approach to Looijenga's invariant theory for generalized root systems

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We consider a generic complex polynomial in two variables and a basis in the first homology group of a nonsingular level curve. We take an arbitrary tuple of homogeneous polynomial 1-forms of appropriate degrees so that their integrals over the basic cycles form a square matrix (of multivalued analytic functions of the level value). We give an explicit formula for the determinant of this matrix.

We introduce two entire functions ${f}_{{\mathrm{A}}_{\frac{1}{2}\infty}}$ and ${f}_{{\mathrm{D}}_{\frac{1}{2}\infty}}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps ${\mathbf{C}}^{2}\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}\mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\phantom{\rule{-0.166667em}{0ex}}\setminus \phantom{\rule{-0.166667em}{0ex}}\{0,1\}$. All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type ${\mathrm{A}}_{\frac{1}{2}\infty}$ and ${\mathrm{D}}_{\frac{1}{2}\infty}$, respectively. Coxeter elements of type ${\mathrm{A}}_{\frac{1}{2}\infty}$ and...

Applications of singularity theory give rise to many questions concerning deformations of singularities. Unfortunately, satisfactory answers are known only for simple singularities and partially for unimodal ones. The aim of this paper is to give some insight into decompositions of multi-modal singularities with unimodal leading part. We investigate the ${J}_{k,0}$ singularities which have modality k - 1 but the quasihomogeneous part of their normal form only depends on one modulus.

We study deformations of hypersurfaces with one-dimensional singular loci by two different methods. The first method is by using the Le numbers of a hypersurfaces singularity — this falls under the general heading of a “polar” method. The second method is by studying the number of certain special types of singularities which occur in generic deformations of the original hypersurface. We compare and contrast these two methods, and provide a large number of examples.