Arithmetic dynamics and Generalized Fermat's conjecture
This paper proposes a generalized Fermat's conjecture within the framework of arithmetic dynamics, provides supporting evidence for it, and introduces a multi-indexed version of the conjecture.
🔢 Category
math.AG
170 papers
Three formulas for CSM classes of open quiver loci
This paper presents one geometric and two combinatorial formulas for computing the equivariant Chern-Schwartz-MacPherson classes of open quiver loci in type quiver representations, introducing "chained generic pipe dreams" and providing streamlined versions of known formulas for the associated quiver polynomials.
A note on quasi-perfect morphisms
This paper establishes a new characterization of regular Noetherian algebraic spaces via quasi-perfect blowups and demonstrates that the quasi-perfectness of proper morphisms is an étale-local property, thereby proving that the locus of such points is Zariski open.
Desingularization of double covers of regular surfaces
This paper provides explicit equations describing Lipman's desingularization for double covers of regular surfaces, thereby establishing a concrete algorithm to resolve their singularities.
Witt Group of Nondyadic Curves
This paper computes the derived Witt groups of smooth proper curves over nondyadic local fields of characteristic not equal to 2 by utilizing reduction techniques and conducting a general study of the existence of Theta characteristics.
Gromov--Witten theory beyond maximal contacts
This paper generalizes the local-relative correspondence beyond maximal contacts by identifying genus zero relative Gromov–Witten invariants of a smooth projective variety with a smooth nef divisor against specific orbifold invariants of multi-root stacks over a -bundle, thereby enabling the computation of relative invariants via absolute invariants of toric bundles.
Visible Lagrangians for Hitchin Systems and Pillowcase Covers
This paper establishes a general framework for visible Lagrangians in Hitchin systems that factor through proper subvarieties of the Hitchin base, computes their fiber-wise Fourier-Mukai transforms to construct mirror dual branes, and provides a detailed new example arising from pillowcase covers that connects to Hausel's toy model.
Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles
This paper constructs a large family of pairwise non-isomorphic hyperelliptic curves mapping birationally into abelian surfaces isogenous to products of elliptic curves to generate rational equivalences in the Chow group of zero-cycles, thereby providing progress toward Beilinson's conjecture on the vanishing of the kernel of the Albanese map.
Crystalline prisms: Reflections and diffractions, present and past
This paper establishes an equivalence between crystals on the prismatic site and modules with integrable quasi-nilpotent -connections, demonstrating that their cohomology is computed by a -de Rham complex and providing a geometric construction of the prismatic Sen operator that yields an explicit description of the action of on de Rham cohomology.
Functionality for isomorphism classes of curves and hypersurfaces
This paper presents algorithms grounded in invariant theory to address geometric problems concerning curves and hypersurfaces, with a primary focus on those of genus 2, 3, and 4, while also incorporating new theoretical results derived from the first author's PhD thesis.
Linear Code Equivalence via Plücker Coordinates
This paper presents a theoretical algebraic framework for the Linear Code Equivalence problem using Plücker coordinates and invariant rational functions to construct polynomials with the underlying permutation matrix as a root, demonstrating the potential of algebraic geometry in cryptanalysis despite the resulting polynomials being computationally infeasible for practical attacks.
Picard groups of completed period images and the Deng-Robles problem
This paper resolves the Deng-Robles problem for polarized variations of Hodge structure over smooth quasi-projective surfaces with one-dimensional pure period images by demonstrating that the obstruction to an intrinsic algebraic description of the completed period image is divisor-theoretic and proving the necessary Picard-generation statement.
Ulrich bundles on smooth toric threefolds with Picard number $2$
This paper investigates Ulrich bundles on smooth toric threefolds with Picard number 2 by constructing resolutions and monads for bundles of arbitrary rank, classifying those pulled back from , and proving that these varieties are Ulrich wild.
Semi-rigid stable sheaves: a criterion and examples
This paper introduces the concept of semi-rigidity for stable sheaves on smooth polarized varieties to characterize the existence of stable deformations of direct sums, providing a criterion based on the Yoneda pairing and applying it to line bundles on projective varieties and Lagrangian subvarieties of hyper-Kähler manifolds.
Stable Degenerations of log Fano Fibration Germs
This paper proves the stable degeneration conjecture for log Fano fibration germs by introducing the -invariant to identify a unique minimizing valuation that induces a special degeneration to a K-semistable germ, which further admits a unique K-polystable special degeneration.
Frobenius structure on rigid connections and arithmetic applications
This paper constructs natural Frobenius structures on two families of rigid irregular -connections, utilizing them to analyze local monodromy, verify Reeder--Yu's predictions on epipelagic Langlands parameters, and confirm the cohomological and physical rigidity conjectures of Heinloth--Ngô--Yun.
Specialized Simpson's main estimates for cyclic harmonic -bundles
This paper generalizes Simpson's main estimates for cyclic harmonic -bundles induced by split automorphisms and applies these results to classify Toda type -harmonic bundles.
On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties
This paper refutes the conjecture that isolated periodic points of automorphisms on affine spaces have bounded height by providing a counterexample, while simultaneously proving that cohomologically hyperbolic dominant rational self-maps on projective varieties possess a Zariski open subset with height-bounded periodic points and offering evidence that such boundedness may fail for preperiodic points.
The ABCT Variety is a Positive Geometry
This paper proves Lam's conjecture that the ABCT variety is a positive geometry by analyzing its combinatorial and algebraic structure, interpreting its boundary subvarieties as point configurations on , and constructing a top-degree meromorphic form.
Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space
This paper extends the authors' formalism of elliptic virtual structure constants to encompass hypersurfaces and complete intersections within weighted projective spaces that possess a single Kähler class.