Search results for "Coherent sheaf"
showing 4 items of 4 documents
Derived categories of irreducible projective curves of arithmetic genus one
2006
We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all $t$ -structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder–Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.
Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves
2006
In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.
Stability conditions and related filtrations for $(G,h)$-constellations
2017
Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raise…
p −1-Linear Maps in Algebra and Geometry
2012
At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.