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Wednesday, 4 November 2026: 22 new papers, 12 cross-lists and 25 replacements in Algebraic Geometry (math.AG), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Let ${\mathcal L}$ be an arrangement of $d$ lines in the complex projective plane and let $Z={\operatorname{Sing}}({\mathcal L})$. A question of Pokora asks whether the multipoint Seshadri constant of ${\mathcal O}_{\mathbb{P}^2}(1)$ at $Z$ is always computed by a line of the arrangement containing the largest number of points of $Z$. We prove that the answer is affirmative for every arrangement of at most twelve lines. The proof uses a B\'ezout-type covering criterion, elementary restrictions on the intersection lattice, and a detailed analysis of the two extremal combinatorial types that can occur for twelve lines. One type is the Hesse arrangement; it is handled using a net of quartics through its twelve double points. The other type would yield a uniquely determined incidence structure on thirteen lines. We show that this incidence structure is not realizable over any field.
We study min-plus Bruhat cells $B_{trop}^{(n)} \odot p\odot B_{trop}^{(n)}$, where $B_{trop}^{(n)}$ denotes the set of all upper-triangular matrices over $\RR \cup \{\infty\}$, and $p$ is a permutation. It is proved that $\dim (B_{trop}^{(n)} \odot p \odot B_{trop}^{(n)})= \frac{n(n+1)}{2} + l(p)$, where $l(p)$ is the length of $p$ in Coxeter generators. Also we provide an explicit description of the cell $B_{trop}^{(n)}(\RR) \odot p_0\odot B_{trop}^{(n)}(\RR)$ (up to a subset of dimension less than $n^2$) for the highest in Bruhat order permutation $p_0$. Finally, it is shown that for a tropically non-singular matrix $A$ for at most one permutation $p_A$ it can hold $A\in B_{trop}^{(n)}\odot p_A\odot B_{trop}^{(n)}$.
We show that birationally equivalent hyperk\"ahler varieties of generalized Kummer type are derived equivalent, confirming the D-equivalence conjecture in this case. Combined with recent results of other authors, this settles the conjecture for all known deformation types of hyperk\"ahler varieties. The kernel is built from an equivariant Fourier-Mukai kernel on the sum-zero loci of powers of an abelian surface. We deform it along diagonal twistor paths, using the hyperholomorphic endomorphism algebra. Comparing wall hyperplanes gives a common twisted derived category for two adjacent birational models, and a finite sequence of wall crossings then gives the equivalence.
For a connected regular semisimple Hessenberg variety other than the full flag variety, we prove that the identity component of its automorphism group is an algebraic torus. The argument uses equivariant local coordinates and weight-homogeneous coefficient functions of vector fields.
We develop an anticanonical minimal model program for projective $\mathbb{Q}$-factorial klt pairs $(X,\Delta)$ such that $-(K_{X}+\Delta)$ is pseudoeffective. We first prove the cone and contraction theorems for potentially klt triples and establish the existence of flips for such triples. Using these results, we introduce the notion of admissible extremal rays and show that $(K_{X}+\Delta)$-positive admissible extremal rays are precisely the rays that are to be contracted by $-(K_{X}+\Delta)$-MMP. The elementary MMP steps associated to such admissible rays preserve klt singularities. For potentially klt pairs $(X,\Delta)$ with $\mathbb{Q}$-divisors $\Delta$, we prove that every $(K_{X}+\Delta)$-positive extremal ray is admissible and obtain a cone theorem for the $(K_{X}+\Delta)$-positive part of the Mori cone. Assuming termination of the anticanonical minimal model program, we establish a dichotomy: potentially klt pairs admit anticanonical minimal models, whereas non-potentially klt pairs admit non-admissible models.
We construct a bubble-tree enhancement of the KSBA moduli space of surfaces of general type using index-one-cover Deligne--Mumford stacks. The construction is inspired by bubbling phenomena for K\"ahler--Einstein metrics. Non-klt semi-log-canonical singularities are replaced by admissible simple-normal-crossing configurations: the cusp and degenerate-cusp components are rational, while the simple-elliptic case also contains elliptic ruled components. A key feature is that the construction avoids lci covers. We construct the moduli stack of polarized bubble-tree index-one-cover Deligne-Mumford stacks. The stack is in general not separated, and we construct its underlying separated algebraic space. We also construct the moduli stack of bubble-tree index-one-cover Deligne-Mumford stacks with labeled sections that record algebraic bubbling centers and compare it with the pointed metric bubble tree. The labeled sections determine a numerical class whose stability proves the separatedness of the moduli stack. We establish a perfect obstruction theory and a virtual fundamental class on the bubble-tree moduli stacks. This provides a novel and geometric pathway to prove the existence of a virtual fundamental class on the original KSBA moduli space, thus offering a new approach to a conjecture of Donaldson. We expect that parts of the construction extend to higher dimensions.
We prove Conjecture B of Akhtar--Coates--Corti--Heuberger--Kasprzyk--Oneto--Petracci--Prince--Tveiten unconditionally for the normalized family of maximally mutable Laurent polynomials with prescribed $T$-binomial edge coefficients associated with the degree-10 orbifold del Pezzo hypersurface $X_{10}\subset\mathbb{P}(1,2,3,5)$. An explicit $S$-extended quantum-period calculation determines the affine mirror map $\lambda=a-346$ and proves that the regularized quantum period equals the classical period as a formal power series throughout the family.
We construct a canonical projectivization of every smooth proper algebraic variety $X$ in characteristic zero. Once a principalization procedure is fixed, the construction gives a finite sequence of blowups in smooth centers with smooth projective endpoint, natural under semilinear isomorphisms and preserving the quasi-projective core, the intersection of all maximal quasi-projective open subsets. At each stage we select finitely many ideals near the generic points of the image in $X$ of the boundary of a maximal quasi-projective open. Principalizing their product strictly lowers the dimension of that image. The selection is defined intrinsically, without a polarization or a chosen projective modification; a Hilbert scheme argument proves its finiteness. Naturality embeds the automorphism group in that of a smooth projective variety of the same dimension. We then extend the construction to smooth proper integral algebraic spaces, obtaining canonical projectivizations of all compact connected Moishezon manifolds.
We prove the algebraicity of period images in arithmetic quotients without using o-minimality. The proof uses infinitesimal neighborhoods and formal algebraization to recover the images and the maps from their algebraic sources. Hodge positivity and an algebraic ampleness criterion yield the Griffiths polarization and quasi-projectivity. For lifted images, we construct Stein neighborhoods and holomorphic matrix realizations of the Hodge filtration. On closed horizontal subspaces, we bound canonical Bergman density below by the Hodge volume form. We compare the Bergman and Griffiths--Schmid metrics near one-parameter boundaries with nontrivial unipotent monodromy and nondegenerate graded period maps, and in all directions near general one- and two-node quintic threefolds. A period surface arising from abelian threefolds shows that the upper metric comparison fails in general.
We study hypersurfaces of degree $d+k$ containing a prescribed zero-dimensional fat point scheme $Z\subset\mathbb{P}^n$ and having multiplicity at least $d$ along a general codimension-two linear subspace. We associate with this interpolation problem a vector bundle on $\mathbb{P}^1$ whose splitting type determines the dimensions of all linear systems of type $(d+k,d)$. For a fat point of multiplicity $m$, we determine the exact number of conditions detected by this bundle and introduce the corresponding effective length. This leads to an excess formula separating the contributions of the splitting type, the part of the fat structure invisible to degree-$k$ fibrewise sections, and the Hilbert-function defect of the fixed scheme. We also prove a truncation theorem showing that multiplicities larger than $k$ contribute only a uniform twist of the bundle and, geometrically, fixed hyperplane components. Finally, we exhibit explicit planar and three-dimensional examples showing that genuinely unexpected phenomena persist for nonreduced fixed base schemes, including geometrically irreducible unexpected hypersurfaces.
Let $M=X_6^\sigma$ be a smooth Debarre-Voisin hyperk\"ahler fourfold and $X=X_3^\sigma$ the associated smooth hyperplane section of $\operatorname{Gr}(3,10)$. Using the incidence correspondence between $X$ and $M$, we construct a family $\mathcal E$ of objects in the Kuznetsov component $\operatorname{Ku}(X)$. We write $v$ for the common Mukai vector of its fibers. We prove that the associated Mukai homomorphism \[ \Theta_{\mathcal E}|_{v^\perp}\colon v^\perp\xrightarrow{\sim}H^2(M,\mathbb Z) \] is an integral Hodge isometry and explicitly determine the inverse image of the Pl\"ucker polarization. For a very general point of the trivector divisor corresponding to discriminant $24$, we also construct an integral Hodge isometry between the full Mukai lattices of $\operatorname{Ku}(X)$ and the associated twisted K3 surface.
We give a proof that an isolated algebraic symplectic singularity of dimension $2n\ge 4$ admitting a projective symplectic resolution is analytically isomorphic to the minimal nilpotent orbit closure in $\mathfrak{sl}_{n+1}$ at the origin.
Logarithmic Fitting ideals encode the complexity of the corresponding morphisms in a natural and intrinsic way, and their principalisation offers a new perspective on problems such as the resolution of cotangent sheaves and the monomialisation of morphisms. In this paper, we give a local resolution theorem for log-rank adapted morphisms: after finite sequences of smooth local blow-ups, the first two log-Fitting ideals \mathcal{F}\_{n--1}(\sigma) and \mathcal{F}\_{n--2}(\sigma) become principal monomial ideals.
Let $f$ be a morphism from a log canonical pair $(X,\Delta)$ to an abelian variety $A$ and $D\sim_{\mathbb Q}m(K_X+\Delta)$ a Cartier divisor on $X$ with $m>1$ a rational number. We prove that $f_*\mathcal{O}_X(D)$ is globally generated after pullback by an isogeny. Our methods involve birational geometry, mixed Hodge modules and singular Hermitian metrics with the minimal extension property. Moreover, we provide simple counterexamples where the above result fails for plt pairs when $m=1$ and where the existence of Chen--Jiang decompositions fails for plt pairs for any $m\ge1$.
We compare genus zero degree two Gromov--Witten invariants with the number of embedded conics on general projective complete intersections whose conic space has expected dimension zero. Their difference comes from double covers of lines. We compute this contribution as an integral of Chern classes over the Fano scheme of lines, using an explicit obstruction bundle on the compactified space of double covers.
Inspired by the quadratic involutions of Corti-Pukhlikov-Reid and Gavin Brown's description of hypersurface flips via toric GIT, we apply similar symmetries to local covering models of terminal threefold flops. We recover the derived equivalence of their canonical stacks by pullback and pushforward along the fibre product over the canonical stack of the contraction target, revisiting the local picture in Kawamata's work.
We study the least pole order of a rational function whose nonzero evaluations scale a prescribed code to a Euclidean LCD code. A residue-one differential yields an obstruction for odd-dimensional one-point AG codes in characteristic two: for $C=C_L(D,aQ)$ with $0\le a<n$ and odd dimension, the threshold is at least $n+2g-a-1$. The obstruction uses alternating Gram matrices and does not require the scaled code to be self-orthogonal. For the dimension-three codes $C_L(D,(r+1)Q)$ on Hermitian curves over $\F_{r^2}$, $r=2^e\ge4$, we obtain \[ n+2g-r-2\le\tau_D(C;Q)\le n+2g-3. \] Consequently the leading genus contribution is asymptotically sharp even in dimension three. On the elliptic curve $y^2+y=x^3$ over $\F_4$, we prove the exact dimension-three threshold $6$ by an explicit multiplier. For repetition codes on the Hermitian family, the threshold equals the general interpolation bound $n+2g-1$. The upper bounds use information sets and hold for $q>3k$, and for $q>2k$ in characteristic two.
We show that every smooth complex double cover of $(\mathbf P^1)^3$ branched along a smooth divisor of tridegree $(2,2,2)$ is algebraically Gromov elliptic. These irrational Fano threefolds form Mori--Mukai family 3-1 and have anticanonical genus seven and Picard rank three. Their three coordinate projections define degree-four del Pezzo fibrations. Each threefold also admits a dominant rational map of degree two from $\mathbf P^3$.
Let $k$ be an algebraically closed field of odd characteristic $p$, and let $\varphi_d$ be the monic Chebyshev polynomial defined by $\varphi_d(t+t^{-1})=t^d+t^{-d}$. For $d>4$ with $p\nmid d$, let $H_d\le\text{PGL}_2(k)$ be the stabilizer of the zero set of $\varphi_d$. We show that \[ H_d\cong \begin{cases} D_d,&\text{if }2d=p^r+1\;\text{ for some integer }\;r\geq 1,\\ S_3,&\text{if }4d=p^r+1\;\text{ for some integer }\; r\geq 1,\\ C_2,&\text{otherwise}. \end{cases} \]
In this note we prove a simple generalization of cohomology and base-change which seems to be missing from standard sources. We illustrate its usefulness by providing an alternative proof of a result of Ishimura on descent of vector bundles along blowups at a smooth center.
Using adelic intersection theory and following the method envisioned by Parshin and Moret-Bailly, we prove a conjecture of Vojta for curves of genus two and a weak form of the $abc$ conjecture
We construct numerical Bridgeland stability conditions on the category of graded matrix factorizations of every smooth complex projective hypersurface of degree $d\ge2$. These conditions are of Gepner type $(\tau,2/d)$, where $\tau$ is the grading shift autoequivalence. By Orlov's theorem, this gives stability conditions of Gepner type on all Calabi--Yau hypersurfaces and on Kuznetsov components of all Fano hypersurfaces. These conditions satisfy a mass-Hom bound and the support property on the full numerical Grothendieck group. Semistable objects of fixed numerical class and phase admit proper good moduli spaces.
We determine the positive scalar curvature K\"ahler cone of every minimal compact K\"ahler surface of Kodaira dimension $-\infty$. By the Enriques--Kodaira classification, every such surface is either $\mathbb P^2$ or a geometrically ruled surface $\mathbb P(E)\to\Sigma_g$, where $E$ is a rank-two holomorphic vector bundle over a compact Riemann surface of genus $g$. On $\mathbb P^2$, every K\"ahler class admits a K\"ahler metric of positive scalar curvature, whereas on $\mathbb P(E)$, every K\"ahler class of positive total scalar curvature admits a metric of positive scalar curvature if and only if $g\le1$ or $E$ is slope-semistable. We further prove that if a smooth compact K\"ahler surface admits a birational morphism onto a ruled surface $\mathbb P(E)\to\Sigma_g$, where $g\ge2$ and $E$ is slope-unstable, then it contains a K\"ahler class of positive total scalar curvature admitting no positive scalar curvature K\"ahler metric. Finally, for every smooth compact K\"ahler surface $X$ birational to $Y=\mathbb P^2$ or $Y=\mathbb P(E)$, we show that K\"ahler classes admitting positive scalar curvature metrics on $Y$ are exactly the pushforwards of classes admitting such metrics on $X$.
We study four-dimensional binomial Artin-Schelter regular algebras and their connection with twisted monoid algebras. Starting with $23$ examples up to permutation of generators, we investigate the associated monoids of I-type and the structure of their groups of fractions. We determine the corresponding finite permutation groups and compute the second group cohomology $H^2(G(M),k^\times)$. These computations allow us to construct explicit families of four-dimensional Koszul Artin-Schelter regular algebras through cocycle twists. We illustrate this construction for some of these algebras.
We establish an analogue of the Hilbert irreducibility theorem for all K3 surfaces over number fields that admit an automorphism of positive entropy. Our method draws ingredients from both algebraic geometry and complex dynamics.
For two third-order tensors over $\mathbb{R}$ or $\mathbb{C}$, we define the Strassen CP rank gap as the sum of their CP ranks minus the CP rank of their direct sum. For arbitrary third-order tensors, we derive explicit upper bounds on this gap in terms of CP rank excess and dimension. For symmetric third-order tensors, we similarly define the Strassen symmetric rank gap using symmetric rank and establish explicit upper bounds in terms of symmetric rank excess, CP rank excess, and dimension. These bounds also yield sufficient conditions for the respective gaps to vanish, ensuring additivity of CP rank or symmetric rank under direct sums.
We study Jack Lee's conjecture that, for a compact strictly pseudoconvex CR manifold $M$, the condition $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{R})$ implies the existence of a pseudo-Einstein structure. For links of isolated singularities, we construct counterexamples in every odd dimension at least three, with $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{Z})$ and $b_1(M)=0$. For isolated algebraic singularities of log canonical type and complex dimension at least three, we prove that the real Chern class condition is equivalent to $\mathbb{Q}$-Cartierness of the canonical divisor, log canonicity, and the existence of a pseudo-Einstein structure. Finally, we construct a Du Bois threefold singularity whose link satisfies the real Chern class condition but admits no pseudo-Einstein structure. The initial versions of the proofs were generated with AI assistance and were subsequently revised and verified by the authors.
We show that the Dirichlet-to-Neumann map of the elastic wave equation determines the density and the anisotropic stiffness tensor on the boundary for generic density and stiffness tensor fields. The proof builds on an interplay between microlocal analysis and algebraic geometry. Microlocal analysis yields a construction of special solutions, valid for many real principal type systems, that concentrate near a boundary point. Generic conditions ensure that this construction holds and allow for extraction of information about stiffness tensor and density at the boundary. We also describe the generic sets through algebraic conditions.
We prove Douglas' conjecture for $m$-Kronecker quivers for each $m \geq 3$ on the normalized logarithms of Euler characteristics of moduli spaces. Also, we state a log-concavity conjecture on Euler characteristics near slope one for the 3-Kronecker quiver.
An important ingredient in fault-tolerant quantum computation is asymptotically good codes with fully addressable transversal non-Clifford gates. Existing constructions, however, provide only small explicit guarantees for rate and relative distance. Here, we substantially improve these guarantees for fully addressable transversal phase gates at every fixed level $h\ge3$ of the Clifford hierarchy, using a cyclic encoding beyond the binary CSS framework. The encoding separates logical phase control from the joint design of rate and distance, enabling two constructions with different computational costs. A deterministic polynomial-time algebraic-geometric construction gives a common asymptotic lower bound of $52/(675h)$ on both quantities. A counting argument gives an exponential-time construction with exact verification, improving this bound by a factor of approximately $2.86$. We further establish a universal upper bound for codes supporting fully addressable transversal single-qubit phase gates. Within our cyclic encoding framework, we obtain a stronger upper bound that is approximately $1.28$ times the lower bound achieved by our exponential-time construction. As concrete examples, we construct codes with $48$ and $90$ physical qubits supporting fully addressable transversal $T$ gates. These results narrow the gap between achievable code parameters and addressability bounds, providing quantitative targets for further improvements in efficient code constructions.
We establish homotopy invariance for connective continuous K-theory of affinoid Tate adic spaces of good reduction with respect to the algebraic affine line, distinct from the adic affine line as analytic stacks. We approach this through an investigation of boundedness of t-structures on nuclear module categories, to which end we also study ultrasolid modules, a variant of solid modules, and their properties.
Let X be a closed oriented surface of genus at least 2 and P a finite subset of X. We bound the action of the mapping class group of (X,P) on the rational cohomology groups of the ordered configuration spaces F_n(X-P) of X-P. In particular, we show that, when #P < 2, the diagonal action of the mapping class group on the direct sum of the H^n(F_n(X-P)) is faithful, and determine its kernel when #P > 1.
We give an example of a tiling of the $m=4$ amplituhedron that has fewer cells than that of the magic number conjecture. The tiling uses an injective degree two cell.
Motivated by examples such as the local rings of projective integral schemes and the Zariski-Riemann space of a field, we develop a geometry for continuously indexed families of subrings of a commutative ring $R$. The indexing spaces are Stone spaces, and continuity is taken with respect to the Zariski topology on the space of subrings. To each such family we associate a subsheaf of the constant sheaf with value $R$ whose stalks are the indexed subrings. This gives a patch ringed space. Its global sections are locally constant $R$-valued functions whose value at each point belongs to the corresponding subring. The ring of global sections, with this representation, is the associated patch algebra. Our main result establishes an algebra-geometry correspondence between these continuously indexed families of subrings and patch ringed spaces that can be expressed as a category equivalence, with each of the two categories dually equivalent to the category of patch algebras. Thus the indexing space, its sheaf, and the indexed subrings can be recovered functorially from the associated patch algebra, which serves as a coordinate ring for this geometry. We describe kernels and factorizations of patch algebra morphisms in geometric terms and characterize their injectivity and surjectivity. We include applications to Pierce duality and Bhatt-Scholze $w$-localizations of integral separated quasi-compact schemes.