Open research questions in Algebraic Geometry and Number Theory
99 unresolved questions extracted from the limitations and future-work sections of 427 Algebraic Geometry and Number Theory papers in our library. Each links back to the study that raised it.
What the literature leaves open
The paper identifies a gap in the understanding of the local Langlands correspondence. The paper aims to fill this gap by introducing a new object and proving new theorems.
The paper notes that the asymptotics of Mertens-like functions are subtle. It also acknowledges that the analysis of the Möbius contribution is approximate. The paper identifies the need for further research to understand the structure of murmurations in various contexts.
The authors face the challenge of completing the classification of these surfaces. The authors must overcome the gap in the understanding of singular rational surfaces with quotient singularities. The authors must develop new techniques for analyzing the properties of these surfaces and their Galois covers.
The complexity of the algebraic geometry and number theory involved. The lack of a general method for calculating the number of principal polarizations.
To study the distribution of algebraic tori over ℚ using the results of the paper. To generalize the results of the paper to higher-dimensional tori over ℚ.
The paper identifies a gap in the understanding of the Hochschild-Serre spectral sequence for semiabelian varieties over non-closed fields. The paper seeks to fill this gap by providing a new proof of the formula for the first potentially non-zero differential of the sequence.
The lack of understanding of the Hasse principle for algebraic families of del Pezzo surfaces and hyperelliptic curves. The need for a constructive method to produce algebraic families that violate the Hasse principle.
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4 · 2026 · DOIThe paper suggests that future research should focus on the case where C is not reduced. The paper proposes that future research should investigate the relationship between the group scheme Σ D and the non-commutative Witt Hopf-algebras. The paper recommends that future research should explore the applications of the results to the study of principal bundles over projective varieties.
The theory of the fundamental group scheme initiated by M. V. Nori is difficult to compute explicitly. There is a lack of examples for which the fundamental group scheme is certainly local and non-trivial. The paper identifies a gap in the understanding of the fundamental group scheme of non-normal varieties.
Further study of the geometry of elliptic surfaces. Extension of the results to more general fibrations M → B. Application of the results to other areas of algebraic geometry and number theory.
The paper identifies a gap in the understanding of Hilbert schemes of elliptic surfaces. The paper addresses the lack of a general construction for commutative group schemes that embed as open subschemes of Hilbert schemes.
There is a gap in the literature regarding the homological stability of hypersurfaces. The paper fills this gap by proving a homological stability result for hypersurfaces of increasing degree.
The bounded negativity conjecture is still an open problem in the field. The paper identifies a gap in the existing literature by proving a weaker version of the conjecture.
Very little is known in general for varieties of dimension greater than or equal to 2. The problem of determining the degree of irrationality of a variety is a natural invariant measuring how far the variety is from being rational, but it has only recently received considerable attention.
Further study of the Brauer-Manin obstruction and its behavior with respect to primes of good reduction is needed. Investigation of the geometry of algebraic varieties and the implications of the results for cryptography and coding theory may be fruitful.
The paper identifies a gap in the understanding of log Calabi-Yau pairs of index one and birational complexity zero. The paper aims to fill this gap by proposing a conjecture and analyzing the geometry of such pairs.
The paper identifies a gap in the understanding of nearly Gorenstein singularities. The paper aims to fill this gap by providing a criterion for rational surface singularities to be nearly Gorenstein.
Further study of the conjecture proposed in the paper. Analysis of the geometry of log Calabi-Yau pairs in higher dimensions.
The problem of counting algebraic tori over ℚ by Artin conductor is not well understood. There is a lack of results on the asymptotics of the number of tori over ℚ.
However, though the Euler restriction has a longer history than the Ziegler restriction, the cokernel and its dimension of the Euler restriction have not been studied at all.
The behavior of the Brauer-Manin obstruction with respect to primes of good reduction is not well understood. There is a need for new conditions and examples that shed light on this problem.
The paper establishes quadratic degree growth for Ϸ via the Jordan block structure of matrix M, but does not provide a comparison of this degree growth with the degree growth rate of the original maps ϕ ◦ ψ on general fibers or explain how degeneration of the family affects the transition from polynomial to quadratic growth.
The map Ϸ is stated to behave 'almost the same as Ϸ' with similar indeterminacy points, but the specific differences in behavior, the explicit defining polynomials for Ϸ, and how these differences affect the degree growth and dynamical degree are not detailed in the provided excerpt.
We place the construction in the context of prior work (Ulam spiral, Sacks spiral, phyllotaxis models, polygonal spiral sequences), identify the dual role of each prime as both spiral index and polygon vertex count as the novel element, and list seven open problems connecting the PSGG to spectral graph theory, algebraic topology, and the distribution of prime…
The paper identifies a gap in the study of additive actions on projective surfaces - The authors note that the case of singular del Pezzo surfaces over Q was previously studied, but the general case was not fully understood
Most-cited papers in Algebraic Geometry and Number Theory
- Generalized de Jonqui eres divisors on generic curves · Studia Universitatis Babes-Bolyai Matematica · 2023 · 3 citations
- A Prime-Indexed Logarithmic Spiral Geometric Graph via the Golden Angle · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 2 citations
- Meromorphic vector bundles on the Fargues–Fontaine curve · Journal of the European Mathematical Society · 2026 · 2 citations
- The Manin–Peyre Conjecture for Three Del Pezzo Surfaces · The Quarterly Journal of Mathematics · 2026 · 1 citations
- On the Hodge and V-filtrations of mixed Hodge modules · Selecta Mathematica · 2026 · 1 citations
- A note on certain Tannakian group schemes · Archivum Mathematicum · 2020 · 1 citations
- On global invertibility of semi-algebraic local diffeomorphisms · Topological Methods in Nonlinear Analysis · 2021 · 1 citations
- ON CONSTANT COEFFICIENT PDE SYSTEMS AND INTERSECTION MULTIPLICITIES · Proceedings of the YSU A Physical and Mathematical Sciences · 2020 · 1 citations
- Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles · Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica · 2021 · 1 citations
- ARBOREAL GALOIS GROUPS OF POSTCRITICALLY FINITE QUADRATIC POLYNOMIALS: THE STRICTLY PREPERIODIC CASE · Canadian Journal of Mathematics · 2026 · 0 citations
Most recent work
- A Prime-Indexed Logarithmic Spiral Geometric Graph via the Golden Angle · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Meromorphic vector bundles on the Fargues–Fontaine curve · Journal of the European Mathematical Society · 2026
- The Manin–Peyre Conjecture for Three Del Pezzo Surfaces · The Quarterly Journal of Mathematics · 2026
- On the Hodge and V-filtrations of mixed Hodge modules · Selecta Mathematica · 2026
- ARBOREAL GALOIS GROUPS OF POSTCRITICALLY FINITE QUADRATIC POLYNOMIALS: THE STRICTLY PREPERIODIC CASE · Canadian Journal of Mathematics · 2026
- The degeneration of a family of rational surface automorphisms · European Journal of Mathematics · 2026
- Remarks on Higgs bundles twisted by a vector bundle · International Journal of Mathematics · 2026
- Symplectic automorphisms of a surface with genus two fibration and their action on CH0 · Pacific Journal of Mathematics · 2026
- Confirm Goldfeld Conjectured for Infinitely Many Elliptic Curves · Indian Journal of Advanced Mathematics · 2026
- Torsion points of small order on cyclic covers of ℚ <sup>1</sup> . II · International Journal of Algebra and Computation · 2026
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