ScalingStacks

Example 4.92 . [02S6]

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Example 4.92.

We particularize (4.90) to the case of one-dimensional vertical orbits. Let Ξ›\Lambda be a (nβˆ’1)(n-1)-dimensional polyhedron. Hence V⁑(Ξ›)V(\Lambda) is a vertical curve. Let Ξ›1\Lambda_{1} and Ξ›2\Lambda_{2} be the two nn-dimensional polyhedron that have Ξ›\Lambda as a common face. Let v∈Nβ„šv\in N_{\mathbb{Q}} such that the class [(v,0)][(v,0)] is a generator of the lattice N~​(Ξ›){\widetilde{N}}(\Lambda) and the affine space (v,0)+ℝ​c⁑(Ξ›)(v,0)+\mathbb{R}\operatorname{c}(\Lambda) meets c⁑(Ξ›1)\operatorname{c}(\Lambda_{1}). This second condition fixes one of the two generators of N~​(Ξ›){\widetilde{N}}(\Lambda). Then, by equation (4.25)

(4.93) degDψ⁑(V⁑(Ξ›))=deg⁑([Dψ|V⁑(Ξ›)])=mΞ›2​(v)βˆ’mΞ›1​(v).\deg_{D_{\psi}}(V(\Lambda))=\deg([D_{\psi}|_{V(\Lambda)}])=m_{\Lambda_{2}}(v)-m_{\Lambda_{1}}(v).

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