Algebraic Geometry
arXiv:math.AG
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
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1,463 papers
Open $r$-spin theory in genus one, and the Gelfand-Dikii wave function
We construct the $g=1$ sector of the open $r$-spin theory, that is, an open $r$-spin theory on the moduli space of cylinders. This is the second construction of a $g>0$ open intersection theory, which includes descendents (the first is the all genus construction of the intersection theory on moduli of open Riemann surfaces with boundaries [23,30], whose $g=1$ case equals to the $r=2$ case of our construction). Unlike the construction of [30], in order to construct the $r$-spin cylinder theory we had to overcome the foundational problem of dimension jump loci, which in analogous closed theories has been treated using virtual fundamental class techniques, that are currently absent in the open setting. For this reason our construction is much more involved, and relies on the point insertion technique developed in [31,32]. We prove that the open $g=1$ potential equals, after a coordinate change, to the $g=1$ part of the Gelfand-Dikii wave function, thus confirming a conjecture of [7]. We also prove that our $g=1$ intersection numbers satisfy a $g=1$ recursion, also predicted in [7,15]. This recursion is the $g=1$ analogue of Solomon's famous $g=0$ Open WDVV equation [25], with descendents, and is also the universal $g=1$ recursion for $F$-Cohomological field theories [1]. Again, this is first geometric construction which is not the $g=1$ sector of [23,30], proven to satisfy this universal recursion.
2601.04114Jan 2026
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2601.04109Automorphisms of odd dimensional $(2,2)$-complete intersections in characteristic $2$
We compute the automorphism scheme of a generic odd dimensional $(2,2)$-complete intersection in characteristic $2$. This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from quadric hypersurfaces and genus $1$ curves.
2601.04109Jan 2026
View2601.04064AKSZ construction for shifted Poisson structures
We prove the AKSZ theorem for shifted Poisson structures: if $X$ is an $n$-shifted Poisson derived stack, and $Y$ a $d$-oriented derived stack, then the mapping stack \[\underline{\mathrm{Map}}(Y,X)\] is naturally endowed with an $(n-d)$-shifted Poisson structure. For this, we prove that the data of an $n$-shifted Poisson structure on a derived Artin stack is equivalent to the data of an $(n+1)$-shifted Lagrangian thickening of it. We also extend the definition of shifted Poisson structures to derived prestacks having a deformation theory and give two applications, one for mapping stacks with a non-proper source and one in BV formalism.
2601.04064Jan 2026
View2601.04053Thomason condition for regular algebraic stacks
We show that any immersion into a concentrated regular Noetherian algebraic stack with quasi-finite and locally separated diagonal satisfies the Thomason condition. In particular, the derived category of quasi-coherent sheaves on such an algebraic stack is singly compactly generated. This extends results of Hall--Rydh from separated to locally separated diagonal, under additional regularity hypotheses.
2601.04053Jan 2026
View2601.03816Residue Balancing on Singular Curves
This paper investigates residue maps and their spanning properties for singular algebraic curves, with particular emphasis on three interconnected themes: the \emph{scheme--theoretic residue span}, the \emph{residue--balancing principle}, and \emph{residue balancing in the presence of arbitrary singularities}. Starting from the theory of dualizing sheaves on nodal curves, we reinterpret canonical and higher--order differentials as meromorphic objects on the normalization whose local principal parts are constrained by explicit residue conditions. A key result is the scheme--theoretic residue span theorem, which asserts that % for nodal curves of geometric genus $g$ with $δ$ nodes, when $δ\ge g$ the residue functionals at the nodes span $H^0(C,ω_C)^\vee$, so canonical differentials are completely determined by their residue data. This provides a concrete, linear description of $H^0(C,ω_C)$ and yields powerful applications to deformation theory, Severi varieties, and moduli problems. \vspace{0.1cm} We then develop the residue--balancing principle, showing that global residue conditions on each irreducible component of a singular curve are equivalent to local balancing conditions at the singular points. This equivalence clarifies the local--to--global structure of dualizing sheaves and extends naturally to $k$--differentials. Finally, we address the case of arbitrary singularities, where nodes no longer suffice to describe local geometry. Using normalization and the conductor ideal, we formulate a refined balancing principle that replaces simple residue cancellation by higher--order and conductor--level constraints. Together, these results provide a unified framework for understanding how local singular behavior governs global differentials and their deformations.
2601.03816Jan 2026
View2601.03691Surfaces of general type and sl_2-triples
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a general theory for actions of the corresponding height-one group scheme G=SL_2[F]. Sending a point to the Lie algebra of its stabilizer defines rational maps to various Grassmann varieties. For surfaces of general type, this yields fibrations in curves of genus g at least 2 over the projective line. Using properties of the corresponding moduli stack M_g, we prove that there are no smooth surfaces of general type with an sl_2-triple. On the other hand, employing Lefschetz pencils and Frobenius pullbacks we show that canonical surfaces of general type with such triples exist in abundance. In this connection, we classify the rational double points where the tangent sheaf is free or the evaluation pairing with Kähler differentials is surjetive, including characteristic two.
2601.03691Jan 2026
View2601.03406On the Ulrichness of twisted syzygies and dual syzygies bundles
Given a projective variety $X$ and a very ample line bundle $\mathcal{L}$ on $X$, we classify for which $X$ and $\mathcal{L}$ the twisted syzygies and twisted dual syzygies bundles are Ulrich with respect to the polarizations $\mathcal{L}^a$. We obtain some partial results when considering an arbitrary polarization $H$.
2601.03406Jan 2026
View2601.03179Deformations of the connected sum of Gorenstein algebras
We prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n>9$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we get a non-reducedness result for the cactus scheme. We generalise the Białynicki-Birula decomposition to abstract deformation functors, providing a new method of studying deformation theory. Our construction gives us fractal structures on the nested Hilbert scheme.
2601.03179Jan 2026
View2601.03068Hamiltonian reductions as affine closures of cotangent bundles
For an irreducible non-singular affine $G$-variety $Y$ whose action is $2$-large, we prove that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ for $Z:=Y/\!/G$. As applications, we provide a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G\rightarrow \mathcal{D}(Z)$, and we establish the surjectivity of the symbol map on $Z$.
2601.03068Jan 2026
View2601.03036On the Hilbert-Chow crepant resolution conjecture
We prove the Hilbert-Chow crepant resolution conjecture in the exceptional curve classes for all projective surfaces and all genera. In particular, this confirms Ruan's cohomological Hilbert-Chow crepant resolution conjecture. The proof exploits Fulton-MacPherson compactifications, reducing the conjecture to the case of the affine plane. As an application, using previous results of the author, we also deduce the families DT/GW correspondence for threefolds $S \times C$ in classes that are zero on the first factor, yielding a wall-crossing proof of the correspondence in this case. Finally, we speculate on the relationship between Hilbert schemes and Fulton-MacPherson compactifications beyond the topics considered in this work.
2601.03036Jan 2026
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K-stability of Fano weighted hypersurfaces via plt flags and convex geometry
We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible.
2601.02974Jan 2026
View2601.02741Stacks of p-adic shtukas and spatial kimberlites
The main purpose of this article is to show that the special Newton polygon map from the stack of p-adic shtukas to the stack of G-bundles on the Fargues--Fontaine curve is representable in diamonds and sufficiently nice for cohomological considerations (i.e. fdcs). The second purpose is to show that the $\bar{\mathbb{F}}_p$-fibers of the special Newton polygon map behave like formal schemes, and in particular, satisfy henselianity properties with respect to their reduced locus. These two goals achieved in this article are two of the crucial ingredients used in our collaboration with Hamman, Ivanov, Lourenço and Zou to construct the equivalence that compares the schematic and analytic local Langlands categories of Zhu and of Fargues--Scholze. To achieve these goals, we introduce and study spatial kimberlites, which is a better behaved variant of the theory previously developed by the author.
2601.02741Jan 2026
View2601.02719Symmetric quiver varieties and critical stable envelopes
Symmetric quiver varieties with potentials are natural generalizations of Nakajima quiver varieties, and their equivariant critical cohomologies provide more flexible settings for geometric representation theory and enumerative geometry. In this paper, we study their geometric properties and show that they behave like universally deformed Nakajima quiver varieties. Based on this, we provide a new proof of the existence of critical stable envelopes on them. Following an idea of Nakajima, we give a sheaf theoretic interpretation of critical stable envelopes by the hyperbolic restriction in the affinization of symmetric quiver varieties. The associativity of hyperbolic restrictions implies the triangle lemma of critical stable envelopes.
2601.02719Jan 2026
View2601.02592The fiber product of the Torelli map with any product $\mathcal{A}_{g_1}\times \dots \times \mathcal{A}_{g_k}\to\mathcal{A}_g$ is reduced
We prove that the fiber product of the Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ with any product $\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k} \to \mathcal{A}_g$ for $g=g_1+\dots+g_k$ has a reduced scheme structure. As a consequence, letting $d=\text{codim}(t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}])$, we find that the class $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}]\in \mathsf{CH}^{d}(\mathcal{M}^{ct}_g)$ is tautological. In particular, we obtain $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}] = 0$ for $d > 2g-3.$
2601.02592Jan 2026
View2601.02235Hilbert scheme of smooth projective curves of unexpected dimension \& existence of a component with less than the expected number of moduli
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Denoting by $\mathcal{M}_g$ the moduli space of smooth curves of genus $g$, let $μ: \mathcal{H}_{d,g,r}\dasharrow \mathcal{M}_g$ be the natural map sending $X\in\mathcal{H}_{d,g,r}$ to its isomorphism class $μ(X)=[X]\in\mathcal{M}_g$. It has been conjectured that a component $\mathcal{H}\subset\mathcal{H}_{d,g,r}$ has the minimal possible dimension $$\Xx(d,g,r):=3g-3+ρ(d,g,r)+\dim\operatorname{Aut}(\mathbb{P}^r)$$ \noindent if $\codim_{\mathcal{M}_g}μ(\mathcal{H})\le g-5$ provided $\Xx(d,g,r)\ge 0$, where $ρ(d,g,r):=g-(r+1)(g-d+r)$ is the Brill-Noether number. In this article, we exhibit examples against the conjecture discuss further for the study of the functorial map $μ: \\mathcal{H}{d,g,r}\dasharrow\mathcal{M}_g$ along this line. A component $\mathcal{H}\subset \mathcal{H}_{d,g,r}$ is said to have the {\it expected number of moduli} if $$\dimμ({\Hh})=\min\{3g-3, 3g-3+ρ(d,g,r)\},$$provided $3g-3+ρ(d,g,r)\ge 0$. The existence of a component with strictly less than the expected number of moduli has not been known. In this paper, we show the existence of components with less than the expected number of moduli.
2601.02235Jan 2026
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Poles of real motivic zeta functions for curves
To a given real polynomial function f $\in$ R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z $\pm$ top,0 (f\,; s) $\in$ Q(s), analogous to the topological zeta function of Denef and Loeser in the complex case. These functions are specializations of the real motivic zeta functions studied in [Fic05a] and [Cam17]. Therefore, these functions and their sets of poles are invariants of the blow-Nash equivalence. Using the approach of [Vey95], we study the poles of these real topological zeta functions, as well as real motivic zeta functions, when f is a real polynomial in two variables.
2601.02180Jan 2026
View2601.02178Local Monodromy of Constructible Sheaves
Given a morphism $f: X \rightarrow S$ of complex algebraic varieties and a constructible sheaf $\mathcal{G}$ on $X$, we compute the local monodromy of $Rf_*(\mathcal{G})$ and $Rf_!(\mathcal{G})$ in terms of the local monodromy of $\mathcal{G}$. Our results generalize previous results by Brieskorn, Borel, Clemens, Deligne, Landsman, Griffiths, Grothendieck, and Kashiwara in the setting of quasi-unipotent sheaves. In the following, we consider the general setting of sheaves of $R$-modules for a commutative noetherian ring $R$, and give applications to computing local monodromy of abelian covers in a uniform manner. We also obtain applications in the context of `generalized Alexander modules' and intersection cohomology with torsion coefficients.
2601.02178Jan 2026
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Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries
We show that the Hilbert polynomial of a Calabi-Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a convex polytope $Δ$ is given by a lattice point count in the polytope boundary $\partial Δ,$ just as the Hilbert polynomial of $M$ is known to be given by a lattice point count in the convex polytope $Δ.$ Our main tool is a computation of the Euler class in $K$-theory of the normal line bundle to the hypersurface $Z,$ in terms of the Euler classes of the divisors corresponding to the facets of the moment polytope. We observe a remarkable parallel between our expression for the Euler class and the inclusion-exclusion principle in combinatorics. To obtain our result we combine these facts with the known relation between lattice point counts in the facets of $Δ$ and the Hilbert polynomials of the smooth toric varieties corresponding to these facets.
2601.02176Jan 2026
View2601.02004The motivic Satake equivalence using perverse Nori motives
In this article, we develop the theory of stratified perverse Nori motives to prove a refinement of the geometric Satake equivalence of Mirković-Vilonen, for which we call the Nori motivic Satake equivalence, in constrast to the "Tate motivic" Satake equivalence of Richarz-Scholbach.
2601.02004Jan 2026
View2601.01824On type three complex plane curves
The type of a complex projective plane curve has been recently introduced by T. Abe, P. Pokora and the first author. In the same paper they have studied the type two curves. In this paper we study plane curves of type three, with special attention to low degree curves and line arrangements.
2601.01824Jan 2026
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