Torus embeddings and deformations of simple singularities of space curves

Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

“…In [18] Slodowy pointed out that the Chevalley-type theorem of [7] is a consequence of the results of Looijenga and Wirthmüller [13,14,20], and in fact it holds true for any choice of the base element α k , or equivalently, for any fixed vertex of the Dynkin diagram. Hence: 18,20,13,14]). The ring A is isomorphic to the ring of polyno-…”

The discriminants of a series of unimodular singularities of space curves

“…The theorem has consequences for the deformation theory of the curves Tpq2 [* ] of the kind described in [13]. More precisely we will prove the following result.…”

“…They are labelled Pp, q in Wall's classification [11], and are unimodular unless either 2~{p,q} or 3~{p,q} C {3, 4, 5}. For these we have a singularity that is simple in the sense of Arnold, and in [13] we have described the discriminant A of its semi-universal deformation. The description exhibits A as the branch locus of a certain finite algebraic Galois cover l which is constructed from combinatorial data represented by a Dynkin type In the final paragraph of [13] it was explained how the construction of a Galois cover ~/W2 generalizes to a much wider class of diagrams including the Tpq2[* ] for arbitrary values of p and q.…”

“…For these we have a singularity that is simple in the sense of Arnold, and in [13] we have described the discriminant A of its semi-universal deformation. The description exhibits A as the branch locus of a certain finite algebraic Galois cover l which is constructed from combinatorial data represented by a Dynkin type In the final paragraph of [13] it was explained how the construction of a Galois cover ~/W2 generalizes to a much wider class of diagrams including the Tpq2[* ] for arbitrary values of p and q. In general the number of sheets will be infinite and ~r will not be a variety but a collection of such; nevertheless the quotient X/W2 still has the structure of a Stein manifold with ~*-action.…”

“…In view of [13] we will throughout restrict attention to those pairs (p, q) that yield unimodular singularities (1.1). For these, some of the rational surfaces with anti-canonical cycle studied by will play a role similar to that of del Pezzo surfaces in [13].…”