Daniel Bragg
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View article: On the logarithmic Hodge-de Rham spectral sequence for curves on K3 surfaces
On the logarithmic Hodge-de Rham spectral sequence for curves on K3 surfaces Open
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.
View article: Fourier–Mukai transforms commuting with Frobenius
Fourier–Mukai transforms commuting with Frobenius Open
We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible …
View article: Ample vector bundles and moduli of tame stacks
Ample vector bundles and moduli of tame stacks Open
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 sta…
View article: Fourier--Mukai transforms commuting with Frobenius
Fourier--Mukai transforms commuting with Frobenius Open
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 invert…
View article: Murphy's Law for Algebraic Stacks
Murphy's Law for Algebraic Stacks Open
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 o…
View article: Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic
Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic Open
Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic
View article: Automorphism groups of curves over arbitrary fields
Automorphism groups of curves over arbitrary fields Open
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 f…
View article: Projectivity of the moduli space of vector bundles on a curve
Projectivity of the moduli space of vector bundles on a curve Open
We discuss the projectivity of the moduli space of semistable vector bundles\non a curve of genus $g\\geq 2$. This is a classical result from the 1960s,\nobtained using geometric invariant theory. We outline a modern approach that\ncombine…
View article: Projectivity of the moduli space of vector bundles on a curve
Projectivity of the moduli space of vector bundles on a curve Open
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 th…
View article: Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces
Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces Open
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to sho…
View article: Derived invariants from topological Hochschild homology
Derived invariants from topological Hochschild homology Open
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 behavio…
View article: Lifts of Twisted K3 Surfaces to Characteristic 0
Lifts of Twisted K3 Surfaces to Characteristic 0 Open
Deligne [9] 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…
View article: Unipotent morphisms
Unipotent morphisms Open
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…
View article: Representability of cohomology of finite flat abelian group schemes
Representability of cohomology of finite flat abelian group schemes Open
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 finit…
View article: Hodge numbers are not derived invariants in positive characteristic
Hodge numbers are not derived invariants in positive characteristic Open
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 …
View article: Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic
Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic Open
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 cry…
View article: Derived invariants from topological Hochschild homology
Derived invariants from topological Hochschild homology Open
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 behavi…
View article: Perfect points on genus one curves and consequences for supersingular K3 surfaces
Perfect points on genus one curves and consequences for supersingular K3 surfaces Open
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 s…
View article: Perfect points on genus one curves and consequences for supersingular K3\n surfaces
Perfect points on genus one curves and consequences for supersingular K3\n surfaces Open
We describe a method to show that certain elliptic surfaces do not admit\npurely inseparable multisections (equivalently, that genus one curves over\nfunction fields admit no points over the perfect closure of the base field) and\nuse it t…
View article: Twistor spaces for supersingular K3 surfaces
Twistor spaces for supersingular K3 surfaces Open
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 …