Groupoides symplectiques by Coste A., Dazord P. et Weinstein A.

By Coste A., Dazord P. et Weinstein A.

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Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

Generally considered as a vintage of contemporary arithmetic, this accelerated model of Felix Klein's celebrated 1894 lectures makes use of modern suggestions to envision 3 recognized difficulties of antiquity: doubling the quantity of a dice, trisecting an attitude, and squaring a circle. latest scholars will locate this quantity of specific curiosity in its solutions to such questions as: below what conditions is a geometrical building attainable?

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There is a natural involution ι I(X) −→ I(X) (ξ, σ) → (ξ, σ −1 ) which gives rise to an involution, denoted by the same symbol, ι : I(X) → I(X). On the level of gerbes, it sends G → X to itself, but for an object y of G changes the isomorphism µr Aut(Y ) by composing with the homomorphism µr → µr sending ζr → ζr−1 . We define a new twisted evaluation map eˇi = ι ◦ ei : Kg,n (X, β) −→ I(X). 32 D. 4 The Boundary of Moduli We can finally answer the tow questions asked before: Given a nodal curve C = C1 Σ C2 , we have Hom(C, X) = Hom(C1 , X) × Hom(C2 , X).

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1. Define a category I(X) = Ir (X), r where each component has objects ⎧ ⎫ φ ⎨ G −→ X⎬ Ir (X)(T ) = ⎩↓ ⎭ T where • • G → T is a gerbe banded by µr . φ : G → X is representable. A priori this is again a 2-category, but again and for a different reason, it is equivalent to a category. 1. I(X) is a Deligne–Mumford stack. There is a close relationship between I(X) and the inertia stack I(X). In fact it is not too difficult to see that there is a diagram I(X) GX  I(X) making I(X) the universal gerbe over I(X)!

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