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We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings$k[X]$_{$U$}of the one-variable polynomial ring over a commutative ring$k$. Our description is based on the computation of the resultant of polynomials in$k[X]$. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.

We prove the existence of normal forms for some local real-analytic Levi-flat hypersurfaces with an isolated line singularity. We also give sufficient conditions for a Levi-flat hypersurface with a complex line as singularity to be a pullback of a real-analytic curve in $\mathbb{C}$ via a holomorphic function.

In our previous work, we studied positive representations of split real quantum groups U q q~ (g R ) restricted to their Borel part and showed that they are closed under taking tensor products. But the tensor product decomposition was only constructed abstractly using the GNS representation of a C*-algebraic version of the Drinfeld–Jimbo quantum groups. Here, using the recently discovered cluster realization of quantum groups, we write the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.

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