Open research questions in Algebraic Geometry and Number Theory

What the literature leaves open

• Future research should continue to study the action of symplectic automorphisms on CH0 of algebraic surfaces. The results of this paper should be generalized to other classes of algebraic surfaces. The study of algebraic cycles and the Bloch-Beilinson conjecture should continue to be a focus of research.

  Symplectic automorphisms of a surface with genus two fibration and their action on CH0 · 2026 · DOI

• The study of symplectic automorphisms of algebraic surfaces is an active area of research, but there are still many open questions. The action of automorphisms on cohomology groups and Chow groups is not well understood. The paper aims to address this gap by studying the action of symplectic automorphisms on CH0 of surfaces with genus two fibration.

  Symplectic automorphisms of a surface with genus two fibration and their action on CH0 · 2026 · DOI

• The Kodaira classification of M g,n is an open problem for certain cases. The paper identifies a gap in the literature for the Kodaira classification of M g,n for genus g = 3.

  The Kodaira classification of the moduli space of pointed curves in genus 3 · 2026 · DOI

• Further study of dormant differential modules and their properties may lead to new insights into algebraic geometry and number theory. The authors suggest examining the relationships between dormant differential modules and other objects in algebraic geometry.

  Differential modules and dormant opers of higher level · 2026 · DOI

• The study of differential modules and D-modules in characteristic p > 0 is less well understood than the case of characteristic 0. The Grothendieck-Katz p-curvature conjecture is an important open problem in the field.

  Differential modules and dormant opers of higher level · 2026 · DOI

• The lack of a construction of p-adic Artin L-functions for noncommutative Artin representations defined over CM fields. The lack of a proof of the Iwasawa main conjecture for Artin representations on CM fields.

  On the $p$-adic $L$-function and Iwasawa main conjecture for an Artin motive over a CM field · 2026 · DOI

• The authors identify a need for further research on the study of algebraic cycles and the arithmetic of algebraic varieties.

  The minimum value of the first dynamical degrees of automorphisms of complex simple abelian varieties with fixing dimensions · 2026 · DOI

• The gap in prior work is the lack of an explicit description of the hypergeometric motives. The paper identifies a need for a more detailed analysis of the point counts and periods of Delsarte K3 pencils.

  Hypergeometric decomposition of Delsarte K3 pencils · 2026 · DOI

• The lack of understanding of the logarithmic Hilbert scheme of points on a smooth pointed curve. The need to calculate the integral Chow rings in terms of those of the symmetric products.

  Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings · 2026

• The authors note that classical examples arising from modular and Drinfeld modular curves exhibit asymptotically good behavior, but a new construction is needed to study the asymptotic behavior of dormant modular curves.

  Genus formulas for dormant modular curves and asymptotic behavior of their function fields · 2026

• Further study of the properties of isoPrymians is needed. The paper's approach can be applied to other problems in algebraic geometry and complex analysis.

  2-dimensional finite-gap Schrödinger operator whose spectrum admits two involutions · 2026

• The problem of identifying isoPrymians of smooth double coverings of curves is addressed. The paper fills a gap in the study of 2-dimensional finite-gap Schrödinger operators.

  2-dimensional finite-gap Schrödinger operator whose spectrum admits two involutions · 2026

• The paper identifies a gap in the study of abelian surfaces, namely the lack of a characterization of derived isogenies. It also notes the need to extend the filtered Torelli theorem to twisted abelian surfaces.

  Derived isogenies and isogenies for abelian surfaces · 2026 · DOI

• There is a gap in the literature regarding the computation of the V-filtration of Kashiwara and Malgrange for equivariant D-modules along semi-invariant functions. The paper aims to fill this gap using representation-theoretic techniques.

  Filtrations of D-modules along semi-invariant functions · 2026 · DOI

• The classification of finite left-simple skew braces is not complete. The relationship between skew braces and Hopf-Galois structures is not fully understood.

  Skew braces with no proper left ideals · 2026 · DOI

• 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.

  Variations on Murmurations · 2026 · DOI

• 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 degeneration of a family of rational surface automorphisms · 2026 · DOI

• The paper only considers complex simple abelian varieties, and does not address the case of non-simple varieties. The authors assume that the dimension of the variety is a prime number, and do not consider the case of non-prime dimensions. The paper uses computational methods to enumerate totally real algebraic integers, but does not provide a general formula for the minimum value of the first dynamical degrees.

  The minimum value of the first dynamical degrees of automorphisms of complex simple abelian varieties with fixing dimensions · 2026 · DOI

• The authors note that the theory of mixed Hodge modules is not well-known to singularity theorists.

  On the Hodge and V-filtrations of mixed Hodge modules · 2026 · DOI

• Further study of the dynamics of rational surface automorphisms. Application of the techniques developed to other areas of mathematics. Investigation of the implications of the results for other fields of study.

  The degeneration of a family of rational surface automorphisms · 2026 · DOI

• 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…

  A Prime-Indexed Logarithmic Spiral Geometric Graph via the Golden Angle · 2026 · DOI

• The paper identifies a gap in the understanding of the structure of the Z2-orbifolds of lattice vertex operator algebras. The authors aim to fill this gap by computing the 1-point correlation functions of all states for these orbifolds.

  1-Point Functions for $\mathbb{Z}_2$-Orbifolds of Lattice VOAs · 2026 · DOI

• 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…

  Additive actions on projective surfaces with a finite number of orbits · 2026 · DOI

• 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.

  Meromorphic vector bundles on the Fargues–Fontaine curve · 2026 · DOI

• To further develop the theory of walls and chambers to study moduli spaces on higher-dimensional varieties. To apply this theory to other areas of algebraic geometry.

  Moduli spaces of stable bundles on ruled 3-folds and Brill-Noether problems · 2026 · DOI

Most-cited papers in Algebraic Geometry and Number Theory

• NAKED SINGULARITIES · Annals of the New York Academy of Sciences · 1973 · 423 citations
• The QT Interval and Torsade de Pointes · Drug Safety · 1999 · 114 citations
• Singularities of linear systems and boundedness of Fano varieties · Annals of Mathematics · 2021 · 114 citations
• A test of the Heckscher-Ohlin-Vanek theorem: The Leontief commonplace · Journal of International Economics · 1985 · 74 citations
• Intersecting generalized Lorenz curves and the Gini index · Social Choice and Welfare · 1999 · 62 citations
• A Note On Projective Identification · Journal of the American Psychoanalytic Association · 1980 · 41 citations
• The Jordan-Schonflies Theorem and the Classification of Surface · American Mathematical Monthly · 1992 · 40 citations
• Graded modalities, II (canonical models) · Studia Logica · 1988 · 32 citations
• Some investigations of varieties ofN-lattices-lattices · Studia Logica · 1984 · 24 citations
• Multiplicity for symmetric indefinite functionals: applications to Hamiltonian and elliptic systems · Topological Methods in Nonlinear Analysis · 1998 · 19 citations

Most recent work

• Dedekind's problem in the hypergrid · Advances in Mathematics · 2026
• Irreducible symplectic varieties via relative Prym varieties · Advances in Mathematics · 2026
• 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
• Enumerative geometry of quantum periods · Advances in Mathematics · 2026
• Counting rational points on Hirzebruch–Kleinschmidt varieties over number fields · Mathematische Annalen · 2026
• Linear Diophantine equations and conjugator length in 2‐step nilpotent groups · Bulletin of the London 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
• Almost toric fibrations on K3 surfaces via degenerations · Advances in Mathematics · 2026

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