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Max Lieblich

Publications and source records attributed to Max Lieblich.

At least 19 recordsLinked to original sources
arXiv2026

Higher-weight Jacobians

We define and study Jacobians of Hodge structures with weight greater than 1. A complex variety has a Jacobian of weight 2 precisely when it has maximal Picard number; these weight 2 Jacobians naturally arise in the context of the Brauer group and the Tate conjecture. The case of surfaces of maximal Picard number has been previously studied by Beauville; the higher-dimensional case is also related to the work of Totaro on Hodge structures with no middle pieces. Higher-weight Jacobians are complex tori, and it is generally quite difficult to tell if they are algebraic. However, for abelian varieties of maximal Picard number, we are able to explicitly calculate their higher-weight Jacobians using algebraic number theory, and prove that they are not just abelian varieties but in fact also have maximal Picard number. We also compute higher-weight Jacobians for Kummer varieties and singular K3 surfaces. Via class field theory, we study the field of definition of abelian surfaces and singular K3 surfaces using their weight 2 Jacobians.

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arXiv2025

The Clifford algebra of a finite morphism

We develop a general theory of Clifford algebras for finite morphisms of schemes and describe applications to the theory of Ulrich bundles and connections to period-index problems for curves of genus 1.

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arXiv2024

Murphy's Law for Algebraic Stacks

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.

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arXiv2023

Transcendental splitting fields of division algebras

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 nonisomorphic and having the same set of finite splitting fields as well as the same splitting fields of transcendence degree $1$ and genus at most $1$. On the other hand, we show that one fixes any division algebra over a field, then any division algebras sharing the same splitting fields of transcendence degree at most 3 must generate the same cyclic subgroup of the Brauer group. In particular, there are only a finite number of such division algebras. We also show that a similar finiteness statement holds for splitting fields of transcendence degree at most $2$.

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arXiv2023

Brauertsch fields

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 non-trivial. This is deduced from a Grothendieck--Lefschetz-type theorem for Brauer groups of certain smooth stacks. This also leads to the notion of a Brauertsch field.

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arXiv2022

An Atlas for the Pinhole Camera

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.

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arXiv2022

Deformation theory of perfect complexes and traces

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 one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.

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arXiv2022

Derived categories and birationality

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 categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.

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arXiv2022

Derived equivalences over base schemes and support of complexes

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 complex $P \in D(X \times_T Y )$, defining derived equivalences between the fibers of $f$ and $g$. We apply our results to the canonical fibration and albanese fibration.

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arXiv2021

Perfect points on genus one curves and consequences for supersingular K3 surfaces

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.

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arXiv2021

Locally free twisted sheaves of infinite rank

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 affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of "very positive" infinite rank vector bundles on projective varieties, and show that an infinite rank vector bundle on a curve in a surface can be lifted to the surface away from finitely many points.

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arXiv2021

Topological reconstruction theorems for varieties

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 universal Torelli theorem in the sense of Bogomolov and Tschinkel. The proofs rely heavily on new variants of the classical Fundamental Theorem of Projective Geometry of Veblen and Young. For proper normal varieties over uncountable algebraically closed fields of characteristic 0, we show that the Zariski topological space can be used to recover the linear equivalence relation on divisors. As a consequence, we show that the underlying scheme of any such variety is uniquely determined by its Zariski topological space. We use this to prove a topological version of Gabriel's theorem, stating that a proper normal variety over an uncountable algebraically closed field of characteristic 0 is determined by its category of constructible abelian étale sheaves. We also discuss a conjecture in arbitrary characteristic, relating the Zariski topological space to the perfection of a proper normal variety.

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arXiv2020

A reconstruction theorem for varieties

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 universal Torelli theorem in the sense of Bogomolov and Tschinkel. The proof relies heavily on a rational version of the classical Fundamental Theorem of Projective Geometry.

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arXiv2020

A Tannakian approach to patching

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 under linear algebraic groups, as well as patching for sheaves and torsors on proper algebraic spaces.

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arXiv2020

Two Hilbert schemes in computer vision

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 spaces always admit open immersions into Hilbert schemes for more than two views, extending and refining work of Aholt-Sturmfels-Thomas. We use these moduli spaces to study and extend the classical twisted pair covering of the essential variety.

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arXiv2019

A note on the cone conjecture for K3 surfaces in positive characteristic

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 integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group.

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arXiv2019

Universal limit linear series and descent of moduli spaces

We introduce a formalism of descent of moduli spaces, and use it to produce limit linear series moduli spaces for families of curves in which the components of fibers may have monodromy. We then construct a universal stack of limit linear series over the stack of semistable curves of compact type, and produce new results on existence of real curves with few real linear series.

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