We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus g ≥ 2. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the recent machinery of good moduli spaces with determinantal line bundle techniques. The crucial step producing an ample line bundle follows an argument by Faltings with improvements by Esteves-Popa. We hope to promote this approach as a blueprint for other projectivity arguments.
We consider derived invariants of varieties in positive characteristic arising from topological Hochschild homology. Using theory developed by Ekedahl and Illusie-Raynaud in their study of the slope spectral sequence, we examine the behavior under derived equivalences of various p-adic quantities related to Hodge-Witt and crystalline cohomology groups, including slope numbers, domino numbers, and Hodge-Witt numbers. As a consequence, we obtain restrictions on the Hodge numbers of derived equivalent varieties, partially extending results of Popa-Schell to positive characteristic.
We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of the twisted K3 crystals introduced in [1]. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov [2] and its extension to twisted K3 surfaces by Huybrechts and Stellari [3,4]. We give applications to various questions concerning Fourier-Mukai partners, extending results of Căldăraru [5] and Huybrechts and Stellari [3]. We also give an exact formula for the number of twisted Fourier-Mukai partners of a twisted supersingular K3 surface. Contents 1. Introduction 1.1. Notation 1.2. Acknowledgements 2. Supersingular K3 surfaces 2.1. The Brauer group 2.2. K3 crystals and Ogus's crystalline Torelli theorem 3. Twisted K3 crystals and derived equivalences 3.1. Crystalline B-fields 3.2. The twisted Mukai crystal 3.3. Twisted Chern characters and the twisted Néron-Severi group 3.4. Action on cohomology 3.5. Crystalline Torelli theorems for twisted supersingular K3 surfaces 4. Applications to Fourier-Mukai equivalences 4.1. Characteristic subspaces 4.2. Fourier-Mukai partners of twisted supersingular K3 surfaces 4.3. Interlude: the orthogonal group of a strictly characteristic subspace 4.4. Counting twisted Fourier-Mukai partners Appendix A. Twisted Chern characters Appendix B. Deformations of kernels of Fourier-Mukai transforms References
The maximal subgroup of unipotent upper-triangular matrices of the finite general linear groups are a fundamental family of p-groups. Their representation theory is well-known to be wild, but there is a standard supercharacter theory, replacing irreducible representations by super-representations, that gives us some control over its representation theory. While this theory has a beautiful underlying combinatorics built on set partitions, the structure constants of restricted super-representations remain mysterious. This paper proposes a new approach to solving the restriction problem by constructing natural intermediate modules that help "factor" the computation of the structure constants. We illustrate the technique by solving the problem completely in the case of rainbow supercharacters (and some generalizations). Along the way we introduce a new q-analogue of the binomial coefficients that depend on an underlying poset. MSC 2010: 20C33, 05E10be polynomial in the size of the underlying field q). As a preliminary step, [16] uses matchings in bipartite graphs to give a combinatorial characterization of when such a coefficient is nonzero; however, only a small set of examples have a direct computation of the coefficients.The supercharacters of UT N are indexed by set partitions of the set N . In this subject, it seems preferable to view set partitions as a set of pairs, as follows. Given a set partition blpλq of N , we can store the block information as a set of pairs λ "
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