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Abstract An embedding of the complete bipartite graph $$K_{3,3}$$ in $$\mathbb{P}^{2}$$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an important example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $$K_{3,3}$$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. We develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that the example above suggests. In characteristic zero we show that there is a one-to-one correspondence between weak perspective representations of a hyperplane arrangement and polynomials of minimal degree in certain saturations of the Jacobian ideal of the arrangement, providing a connection to algebra. In this setting we can use duality theorems to explain how rigidity theory is reflected in the graded Betti numbers of the module of logarithmic derivations of a line arrangement.
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The six functors for Zariski-constructible sheaves in rigid geometry
We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.
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- PAR ID:
- 10339580
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 158
- Issue:
- 2
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 437 to 482
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X22007291
