Abstract
We study a relation between coupling introduced by Ebeling (Kodai Math J 29:319-336, 2006) and the polytope duality among families of K3 surfaces.
Highlights
A notion of coupling is introduced by Ebeling (2006) as a tone-down of the duality of weight systems by Kobayashi (2008)
It is proved that the duality is “polar dual”, in the sense that certain rational polytopes associated to weight systems are dual
In Ebeling (2006), there is given a list of coupling pairs for 95 weight systems of simple K 3 hypersurface singularities classified by Yonemura (1990), and it is proved that the duality induces Saito’s duality, which is a relation between the zeta functions of the Milnor fibre of the singularities
Summary
A notion of coupling is introduced by Ebeling (2006) as a tone-down of the duality of weight systems by Kobayashi (2008). It is proved that the duality is “polar dual”, in the sense that certain rational polytopes associated to weight systems are dual. The polytope duality is focusing on more details in geometry of K 3 surfaces such as resolution of singularities, as a compactification of some singularities in three dimensional space which should affect the geometry of the surfaces while the polar duality in Ebeling (2006), Kobayashi (2008) is determined only by the weight systems. Determine whether or not there exist reflexive polytopes and , and projectivisations F and F of f and f in the weighted projective spaces P(a) and P(b), respectively, such that they are polytope dual in the sense that they satisfy the following conditions:.
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