Noncommutative conic pencils classified same as 4D Frobenius algebras
A complete classification shows the two sets of objects stand in one-to-one correspondence.
math.AG
Algebraic Geometry
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
sort pith recommended most recent
Tiny direction set at infinity forces algebraicity
A d-dimensional complex analytic set is algebraic exactly when its infinity directions have Hausdorff dimension below 2d.
A finite blowup sequence resolves positive-characteristic singularities
The proof replaces numerical invariants by a well-founded history of certified blowup steps.
Quiver mutations determine plane-curve monodromy
The mutation class of a malleable divide uniquely recovers the Alexander module and the topological type.
· “Monodromy of plane curve singularities and quiver mutation”
Toric mirror curves obey a universal conifold gap at every genus
Each genus≥2 free energy has one polar term, no logarithms, and a holomorphic remainder across the node.
Two mirror constructions coincide for K3 surfaces
A proof shows the intrinsic mirror recovers the classical toric degeneration mirror after resolving the family.
· “Mirrors to toric degenerations via intrinsic mirror symmetry”
Every dualizable additive category is an almost-module category
The characterization links dualizable additive categories to idempotent ideals, nuclear modules, and algebraic K-theory.
Central configurations reduce to a single rank-one matrix identity
A single rank-one factorization unifies the classical five-body and Dziobek systems, even on degenerate strata.
All isogeny graphs of non-CM curves over char 0 now classified
Each p-primary piece is a small explicit tree family; only one shape never occurs, and the rest are all realizable.
· “Isogeny graphs of elliptic curves in characteristic zero”
Integrable observables prove Π-hierarchy equivalences
They replace psi classes while keeping integrability, establish Miura links to Dubrovin-Zhang and ramification hierarchies, and give a short
· “Beyond descendants: integrable observables for cohomological field theories”
All (p,q)-pinwheel embeddings in B_{p,q} are Hamiltonian isotopic
The nearby Lagrangian conjecture holds for these singular Lagrangians because the symplectomorphism group is generated by a single twist.
Tame sheaves arise from étale data plus local tame sections
The construction yields a comparison of tame syntomic cohomology with the Nygaard filtration on the tame de Rham-Witt complex.
· “A construction of tame sheaves and tame de Rham--Witt cohomology”
Quasihole Laughlin bundles yield Chern classes matching Berry phase split
The decomposition into Aharonov-Bohm and fractional statistical terms follows from Grothendieck-Riemann-Roch applied to the symmetric power.
· “Chern classes of Laughlin bundles on the quasihole moduli space”
Drinfeld modules carry crystalline Galois representations
Delta Frobenius on jet spaces gives Hodge-Pink structures, recovering the Tate module for the Carlitz case.
· “Delta theory of Anderson Modules II: Hodge-Pink structure”
Affine space appears only for divisors of neighbouring degrees
Two boundary divisors differing by one in degree, meeting in one reduced fibre — that is the exact criterion.
· “Divisors in projective bundles over the projective line whose complement is affine space”
Numerical Godeaux surfaces cap at six disjoint (-2)-curves
The bound is sharp, and it settles the last open case for involutions on K^2=7 surfaces.
· “Numerical Godeaux Surfaces with many disjoint (-2)-curves and Applications”
Games with constrained equilibria form predictable algebraic loci
Constraining strategies to a fixed algebraic variety cuts the game space in a codimension that is now explicit.
This paper proves a quadratic upper bound
Over F_q(t) with large constant field and characteristic above the degree, a smooth degree-d hypersurface satisfies the circle-method…
· “Birch's theorem over function fields with quadratically many variables”
One cluster variety coordinates local systems for every root type
Jordan split groups unify flag configurations, double Bruhat cells, and positive representations in one cluster structure.
· “Noncommutative Cluster Varieties and Moduli Spaces of Local Systems”
Negative curves catch every unstable line bundle on surfaces
If a line bundle or its shift fails stability, a subobject from a negative self-intersection divisor already shows why.
· “Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces”
Cremona isometries give polyhedral nef cones on generic CY surfaces
The conjecture holds for very general klt Calabi–Yau generalised pairs, with the decisive Sakai-surface case delegated to a sequel.
One Hamiltonian reduction builds all nilpotent-cone resolutions
Reducing the cotangent bundle of base affine space at varying parameters reproduces all crepant partial resolutions.
· “Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction”
Cyclic covers split even over singular bases
The branch divisor's log canonical threshold decides when the cover has Du Bois or rational singularities.
Quasimap spaces are critical loci of one explicit function
An explicit finite-dimensional equation describes every stable quasimap to a Nakajima quiver variety.
Rank n−1: odd-quadric uniform bundles are all homogeneous
New orthogonal kernel bundles of rank n are uniform yet non-homogeneous, settling the threshold.
Weighted Veronese rings disprove Koszul conjecture
With weights (1,4,7,9), every 9k+12-th Veronese subring fails nonstandard Koszulness.
· “Eventual Nonstandard Koszulness Fails for Veronese Subrings of Weighted Polynomial Rings”
Multiplicity of curve singularities is unbounded on abelian surfaces
Beyond genus 2, primitive polarizations on Picard-rank-1 abelian surfaces contain arbitrarily singular curves.
· “An approach to curves in abelian surfaces using Fourier--Mukai and quadratic forms”
Fermat cubic's 72 sixers split into two symmetry classes
Plane symmetries of order 36 vs 12, plus a determinant-based arithmetic signature, tell the two families apart.
Pipe-dream RSK variant reduces to ordinary dual RSK
For any binary matrix, the second output of the pipe-dream algorithm is the complement of the usual recording tableau.
Calabi-Yau wall-crossing conjecture now a theorem
The Gross-Joyce-Tanaka formula is proved for all CY4 categories with framing functors.
Spin-parity volumes of minimal strata now have a closed formula
The even-minus-odd Masur-Veech volume reduces to Hodge integrals via a recursion on spin-parity square cones.
This paper determines the full invariant theory of degree-four rational maps on the…
The invariant ring for degree-four rational maps on P^1 is minimally generated by fifty explicit joint transvectants, and the moduli space…
· “The invariant ring of degree-four rational maps on the projective line”
Small points on quasi-projective varieties equidistribute over function fields
Small-height points spread by the Chern measure at every place, settling the function-field case of the conjecture.
· “Equidistribution for quasi-projective varieties over function fields”
Special intersections on a modular product converge to one point
For smooth curves over Cp, these intersections concentrate at the unique canonical point, except in a supersingular exceptional case.
· “p-adic Equidistribution of Special Loci in a Product of Modular Curves”
An affine complex surface whose rational mixed Hodge structure equals that of a positive…
A smooth affine complex surface with the rational mixed Hodge structure of a smooth projective curve of positive genus is the total space…
· “On smooth affine surfaces with the cohomology of a smooth projective curve”
This paper proves that a local invariant called the delta invariant always has a…
For any log Fano cone singularity, the infimum defining the local delta invariant is attained by a torus-invariant quasi-monomial valuation.
· “On the existence of minimizer on a log Fano cone singularity”
This paper computes the Hilbert series of quotient rings by powers of general linear forms
The Iarrobino-Fröberg conjecture fails for all sufficiently large n, with explicit Hilbert series for n+2 powers of general linear forms…
· “On the Hilbert series of ideals generated by general linear forms”
This number theory paper proves that
For Zariski-generic coefficient vectors, hyperelliptic families y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t are generically ordinary at every large…
· “Generically Ordinary One-parameter Hyperelliptic Families are Dense”
A new theorem in singularity theory shows that adding the Hessian relation to the moduli…
For isolated hypersurface singularities, the derivation algebra of the generalized moduli algebra has the same dimension as the usual one…
· “Derivations of Generalized Moduli Algebras of Isolated Hypersurface Singularities”
This paper proves an effective Second Main Theorem for entire curves in the complement of…
Every algebraically nondegenerate entire curve in the complement of a general ordered pair of smooth transverse plane curves of admissible…
· “Invariant two-jets and effective hyperbolicity for complements of two plane curves”
This math paper proves that if a complex algebraic variety carries a variation of Hodge…
For varieties supporting a variation of Hodge structure with an immersive period map, all exterior powers of the logarithmic cotangent…
· “Intermediate hyperbolicity of varieties supporting a variation of Hodge structure”
Derived pushforward is a localization for threefold resolutions
The paper confirms a 2002 conjecture about resolutions of threefolds, over any field, not just characteristic zero.
· “The Bondal-Orlov Localization Conjecture Holds for Threefolds”
Motivic Fourier transform is involutive on derived bundles
For every derived vector bundle over a derived Artin stack, the motive-based transform inverts itself up to a shift.
Counterexample found to curve semistability conjecture
A rank-four Higgs bundle is semistable along every curve yet has nonzero discriminant.
· “A counterexample to the curve semistability conjecture for Higgs bundles”
First smooth unirational degree-one del Pezzo threefolds
A 28-parameter sextic family is unirational, and its automorphism orbit sweeps 50 dimensions.
Exact formula counts cluster-variety points over finite fields
The F2 count also reveals how many tori are needed to cover the cluster manifold.
· “Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields”
Every product of two elliptic curves maps to the plane in degree 3
An explicit coordinate map gives a degree-3 cover, and degree 2 is impossible.
· “Degree of irrationality of a product of two elliptic curves”
Orbifold threefolds get integer Gopakumar–Vafa counts
For local A_N Calabi–Yau orbifolds the new invariants are finite, and on singular K3 surfaces they reproduce Yau–Zaslow.
· “Gopakumar-Vafa Invariants for Local Calabi-Yau Orbifolds of A_(N)-type”
Projective compactified universal Jacobians are classical
Classification settles an open question: non-classical compactified universal Jacobian spaces are never projective.
Algebraically trivial tropical cycles vanish after self-products
For any such Z, some k!·Z^k is rationally trivial in Y^k — a tropical analogue of a classical nilpotence theorem.
· “Algebraically trivial tropical cycles are smash-nilpotent”
Morse splitting: the quadratic part never changes Lê numbers
All generic Lê numbers come from the remainder g; the squared part contributes nothing.
One elliptic quotient map integrates the exceptional foliations
Leaves are parametrized; symmetry groups, first integrals, and the positive-characteristic dichotomy are made explicit.
Fourteen K3-fibred Calabi-Yau families classified
Orbifold scrolls add four families to the ten known; all Hodge numbers are computed.
· “A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls”
One Ext class decides Frobenius liftability mod p²
For flat schemes, vanishing of the new class is equivalent to existence of a Frobenius lift on the mod p² reduction.
· “Arithmetic Kodaira--Spencer Class and Frobenius Liftings via Frobenius--Witt Cotangent Complex”
Matroid Schubert variety is Gorenstein iff every cocircuit sums to -2
A purely combinatorial test: assign rational weights to elements so each cocircuit totals -2.
· “When is the matroid Schubert variety mathbb{Q}-Gorenstein?”
Tropical Prym–Torelli map is continuous
A new basis of anti-symmetric cycles survives edge contractions, making the Prym construction extend to the boundary.
Every pair on or above the curve m=2^(e-1) is realized
Plus (1,1), these are exactly the (embedding dimension, multiplicity) pairs that complete intersections realize.
Generalised Kummer surfaces classified in every characteristic
The last open cases, where the characteristic divides the group order, are now settled.
· “The classification of generalised Kummer surfaces in positive characteristic”
Every hyperelliptic Brill-Noether locus is locally catalecticant
Locally each germ is cut out by minors of one catalecticant matrix: log resolutions, multiplicities, Betti numbers.
No mysterious points in surface or finite-type cluster algebras
Every point outside all cluster tori is fixed by a non-trivial cluster automorphism.
· “Locally Acyclic Surface Type and Finite Type Cluster Algebras Have No Mysterious Points”
Every prime Fano threefold is a Mukai section
New cases g=7,9,10,12 complete Mukai's models in positive characteristic, matching characteristic zero.
Unique Mukai bundles exist on all prime Fano threefolds
In positive characteristic, each factorization g=rs gives a stable vector bundle with first Chern class K_X and no cohomology.
· “Mukai bundles and Brill-Noether generality for prime Fano threefolds in positive characteristic”
Motzkin path in, Richardson shape out
A new shape algorithm reads a Motzkin path and returns the shape of its Richardson tableau, skipping the Robinson–Schensted step.
Relative hyperbolic curves are anabelian in families
Dominant maps into such a family are classified by π1-open maps of profinite étale homotopy types over the base.
DPS conjecture proven for compact Kähler spaces
Nef anti-canonical class on total space forces pseudo-effective anti-canonical class on base, with no projectivity.
· “On a conjecture of Demailly-Peternell-Schneider: the Kahler case”
Exact point count hits both Hasse–Weil bounds for one curve family
The sign flips with the parity of r, so the same family is maximal or minimal over F_{q^6}.
· “Maximal and minimal curves of the form y³=x^((q²+1)/2)+x”
The paper proves that two basic operations in the minimal model program
Flips and divisorial contractions preserve the Kähler condition for generalized klt pairs with big B+β_X, and a new Kählerness criterion…
· “Fujiki Class mathcal C Varieties and a K\"ahler Criterion”
Jet test verifies hyperbolicity for many plane-curve complements
Bigraded-module computation finds exact exceptional loci for families of degree-14-and-up plane curves.
A stability number detects positive scalar curvature Kähler metrics
In Kähler geometry, the threshold σ>0 marks exactly the classes carrying a PSC metric; σ>s marks a unique cscK metric.
· “Positive scalar curvature K\"ahler metrics and shifted K-stability”
Norms survive all field extensions under three exact conditions
Tells when a multiplicative norm stays multiplicative over every larger valued field.
All genera of plane-curve counts share one exponential rate
Complete expansions in half-integer powers, with an explicit leading coefficient, now hold for every fixed genus.
· “All-Genus Large-Degree Asymptotics for Gromov--Witten Invariants of the Projective Plane”
Toric aCM bundles on P^2 are wild for all d ≥ 3
A Klyachko-filtration defect criterion shows d=1,2 stay finite while d≥3 embed all matrix pairs.
Henselian field K-theory splits via its value group
The empty summand is the generic restriction map, yielding Gersten injectivity for henselian regular local rings.
· “Finite-coefficient K-theory of henselian valued fields and Gersten injectivity”
Every sextic double plane has a weighted Galois point
Counting these points forces the defining polynomial into special forms and ties them to automorphisms.
· “Weighted Galois points for K3 surfaces in mathbb{P}(1,1,1, 3)”
Quadratic Cremona maps of P4 split into nine families
Every quadratic birational self-map of P4 is accounted for, with all possible degrees and base schemes.
Twisted Witt groups in low dimensions reduce to real-locus topology
Algebraic cycles enter only through a subgroup generated by curves, making Witt groups computable from real points.
p-adic solubility odds are rational functions of p
Three gcds with p-1 decide the local formula; pairwise-coprime exponents yield positive densities.
Log-Voronoi cell boundaries can be transcendental
Two rational one-dimensional models in the probability simplex yield non-algebraic cell boundaries.
· “Boundaries of logarithmic Voronoi cells can be transcendental”
Conical Kähler–Einstein limits have unique tangent cones
A log version of Donaldson–Sun theory recovers the boundary divisor from the metric and pins down the tangent cone at every point.
Intrinsic perturbation provably loses hypergeometric coefficients
The obstruction module measures the gap; in rank 2 its dimension reaches q² for every q.
· “Obstructions to intrinsic perturbation for A-hypergeometric series”
