Max Lieblich
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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: Locally free twisted sheaves of infinite rank
Locally free twisted sheaves of infinite rank Open
We study twisted vector bundles of infinite rank on gerbes, giving a new point of view on Grothendieck’s famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question ha…
View article: Brauertsch fields
Brauertsch fields Open
We prove a local-to-global principle for Brauer classes: for any finite collection of non-trivial Brauer classes on a variety over a field of transcendence degree at least 3, there are infinitely many specializations where each class stays…
View article: Deformation theory of perfect complexes and traces
Deformation theory of perfect complexes and traces Open
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where …
View article: Derived equivalences over base schemes and support of complexes
Derived equivalences over base schemes and support of complexes Open
Let $X$ and $Y$ be smooth projective varieties over a field $k$ admitting morphisms $f:X \to T$ and $g:Y \to T$ to a third variety $T$. We formulate conditions on a derived equivalence $Φ:D(X) \to D(Y)$ ensuring that $Φ$ is induced by a co…
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: Transcendental splitting fields of division algebras
Transcendental splitting fields of division algebras Open
We examine when division algebras can share common splitting fields of certain types. In particular, we show that one can find fields for which one has infinitely many Brauer classes of the same index and period at least 3, all nonisomorph…
View article: Splitting Brauer classes using the universal Albanese
Splitting Brauer classes using the universal Albanese Open
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the …
View article: Locally free twisted sheaves of infinite rank
Locally free twisted sheaves of infinite rank Open
We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affi…
View article: Fourier–Mukai partners of Enriques and bielliptic surfaces in positive characteristic
Fourier–Mukai partners of Enriques and bielliptic surfaces in positive characteristic Open
We prove that a twisted Enriques (respectively, untwisted bielliptic) surface over an algebraically closed field of positive characteristic at least 3 (respectively, at least 5) has no non-trivial Fourier-Mukai partners.
View article: A Tannakian approach to patching
A Tannakian approach to patching Open
We use Tannakian methods to show that patching for coherent sheaves implies patching for objects in any Noetherian algebraic stack with affine stabilizers. Among other things, this gives a straightforward way to prove patching for torsors …
View article: Topological reconstruction theorems for varieties
Topological reconstruction theorems for varieties Open
We study Torelli-type theorems in the Zariski topology for varieties of dimension at least 2, over arbitrary fields. In place of the Hodge structure, we use the linear equivalence relation on Weil divisors. Using this setup, we prove a uni…
View article: Derived categories and birationality
Derived categories and birationality Open
We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categor…
View article: Two Hilbert Schemes in Computer Vision
Two Hilbert Schemes in Computer Vision Open
We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli sp…
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: A case-crossover study of heat exposure and injury risk among outdoor construction workers in Washington State
A case-crossover study of heat exposure and injury risk among outdoor construction workers in Washington State Open
Objectives The primary objective of this study was to assess the relationship between heat exposure and occupational traumatic injuries among construction workers. Methods We assessed the relationship between humidex, a measure of apparent…
View article: A reconstruction theorem for varieties
A reconstruction theorem for varieties Open
We show that varieties of dimension at least 2 over infinite fields are determined as abstract schemes by their Zariski topological spaces together with the rational equivalence relation on the set of effective divisors. This gives a unive…
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 …
View article: A note on the cone conjecture for K3 surfaces in positive characteristic
A note on the cone conjecture for K3 surfaces in positive characteristic Open
We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative i…
View article: Moduli of sheaves: a modern primer
Moduli of sheaves: a modern primer Open
We give a modern introduction to the moduli of sheaves. After reviewing the classical theory, we give a catalogue of results from the last decade. We then consider a more "symmetric" formulation of the theory by working with gerbes from th…
View article: A Case-Crossover Study of Heat Exposure and Injury Risk in Outdoor Agricultural Workers
A Case-Crossover Study of Heat Exposure and Injury Risk in Outdoor Agricultural Workers Open
Agricultural workers laboring in warm conditions are at risk for heat-related traumatic injuries. Combined heat-related illness and injury prevention efforts should be considered in high-risk populations exposed to warm ambient conditions …