arXiv subjects
Publications and source records attributed to Daniel Bragg.
We show that if $X$ is a supersingular K3 surface then there exists a curve $D$ on $X$ such that the logarithmic Hodge-de Rham spectral sequence for $(X,D)$ is nondegenerate.
We explain how to define an embedding of a tame stack over a noetherian ring into a certain generalization of a weighted projective stack using a notion of ample vector bundle on the stack. As applications we construct algebraic moduli stacks of tame stacks equipped with an ample vector bundle and algebraic stacks of tame orbicurves.
We show that a Fourier--Mukai equivalence between smooth projective varieties of characteristic $p$ which commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor product with invertible sheaves whose $(p-1)$th tensor power is trivial.
We show that various natural algebro-geometric moduli stacks, including the stack of curves, have the property that every Deligne-Mumford gerbe over a field appears as the residual gerbe of one of their points. These gerbes are universal obstructions for objects of the stack to be defined over their fields of moduli, and for the corresponding coarse moduli space to be fine. Thus, our results show that many natural moduli stacks hold objects that are obstructed from being defined over their fields of moduli in every possible way, and have coarse spaces which fail to be fine moduli spaces in every possible way. A basic insight enabling our arguments is that many classical constructions in equivariant projective geometry generalize to the setting of relative geometry over an arbitrary Deligne-Mumford gerbe over a field.
We show that if $K$ is an arbitrary field and $G$ is a finite group then there exists a curve over $K$ with automorphism group $G$. We also give a positive solution to the weak inverse Galois problem for function fields over an arbitrary field. Finally, we reduce the inverse Galois problem for function fields over an arbitrary field $K$ to the case of $K(T)$.
We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus $g\geq 2$. 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 introduce the theory of unipotent morphisms of algebraic stacks and prove a surprising local to global principle for a class of vector bundles. Two sample applications of our methods are the following: (1) a unipotent analogue of Gabber's Theorem for torsion $\mathbf{G}_m$-gerbes and (2) smooth Deligne-Mumford stacks with quasi-projective coarse spaces satisfy the resolution property in positive characteristic. Our main tool is a descent result for flags, which we prove using results of Sch\"appi.
We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if $f\colon X\rightarrow \mathrm{Spec}(k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^nf_*G$ is representable for all $n$. More generally, we study the derived pushforwards $R^nf_*G$ for $f\colon X\rightarrow S$ a projective morphism and $G$ a finite flat abelian group scheme over $X$. We also define compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height $1$, and prove higher categorical versions of our main representability results.
We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf's result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic.
We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface $X$ in terms of certain integral sublattices of the second rational $\ell$-adic and crystalline cohomology groups of $X$. This is a positive characteristic analog of a result of Huybrechts, and extends results of the second author. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier--Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson.
Deligne showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over $\mathrm{Spec}\mathbf{Z}$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving.
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 describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.
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 isomorphisms between their associated K3 crystals. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov, and its extension to twisted K3 surfaces by Huybrechts and Stellari. We give applications to various questions concerning Fourier-Mukai partners, extending results of C\u{a}ld\u{a}raru and Huybrechts and Stellari. We also give an exact formula for the number of twisted Fourier-Mukai partners of a twisted supersingular K3 surface.
We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to the Brauer groups of supersingular K3 surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted supersingular K3 surfaces, analogous to the analytic moduli space of marked K3 surfaces. We describe a theory of crystals for twisted supersingular K3 surfaces and a twisted period morphism from the moduli space of twisted supersingular K3 surfaces to this space of crystals. As applications of this theory, we give a new proof of the Ogus-Torelli theorem modeled on Verbitsky's proof in the complex analytic setting and a new proof of the result of Rudakov-Shafarevich that supersingular K3 surfaces have potentially good reduction. These proofs work in characteristic 3, filling in the last remaining gaps in the theory. As a further application, we show that each component of the supersingular locus in each moduli space of polarized K3 surfaces is unirational.
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.