We study the multi-height distribution of rational points of smooth, proper and split toric varieties over Q using the lift of the number of points to universal torsor.

Abstract Let X be either a smooth K3 surface or a smooth Fano variety (i.e., $$-K_X$$ - K X is ample) of dimension n and index $$i_X\ge n-2$$ i X ≥ n - 2 and let $$\mathcal {E}$$ E be an initialized Ulrich bundle on X . In this paper, we show that the syzygy bundle $$S_{\mathcal {E}}$$ S E , defined as the kernel of the evaluation map $$eval:H^{0}(X,\mathcal {E})\otimes \mathcal {O}_{X}\rightarrow \mathcal {E},$$ e v a l : H 0 ( X , E ) ⊗ O X → E , is semistable.

Our main result is a combinatorial characterization of when a horospherical variety has (at worst) quotient singularities. Using this characterization, we show that every quasiprojective horospherical variety with quotient singularities is globally the quotient of a smooth variety by a finite abelian group.

Let $X$ be a complex smooth Fano variety of dimension $n$. Assume that $X$ admits a birational contraction of an extremal ray. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\frac{\dim X}{2}$.

The mirror symmetry predicts that the bounded derived category of a smooth Fano variety is equivalent to the Fukaya–Seidel category of its Landau–Ginzburg model. It is expected that the fibers of Landau–Ginzburg model with ordinary double points correspond to an exceptional collection of a Fano variety. We verify this expectation at a numerical level for Fano complete intersections and Calabi–Yau compactifications of their toric Landau–Ginzburg models of Givental’s type.

We construct a quasi-coherent sheaf of associative algebras which controls a category of finitely generated A V AV -modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in étale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.

When two varieties X, X′ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one point of projection, there are many very different shadows with very different degrees. We give the geometric properties of some of them. These shadows are birational to the variety in which they are a shadow. We compute the minimum degree of all such shadows. For most smooth varieties X⊂Pr, r≥3, it is the integer deg(X)−1.

We present a simple and fast embedded resolution of varieties and principalization of ideals using torus actions on ambient smooth varieties with simple normal crossings (SNC) divisors. The canonical functorial resolution in characteristic zero is achieved via the newly introduced cobordant blow-ups along smooth weighted centers. These centers are defined by a geometric invariant measuring the singularities on smooth schemes with SNC divisors. The output is a smooth variety with a torus action and an SNC exceptional divisor. Its geometric quotient is birational to the resolved variety, has only abelian quotient singularities, and can be desingularized by purely combinatorial methods. The method is rooted in ideas from the joint work with Abramovich and Temkin [ATW24] and is closely related to McQuillan's resolution via stack-theoretic weighted blow-ups [McQ19]. As an application, we establish resolution results for certain classes of singularities in positive and mixed characteristic. This paper is a revised version of an earlier preprint.

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

Abstract The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C , introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$ . In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

Abstract This paper studies rings of integral piecewise‐exponential functions on rational fans. Motivated by lattice‐point counting in polytopes, we introduce a special class of unimodular fans called Ehrhart fans, whose rings of integral piecewise‐exponential functions admit a canonical linear functional that behaves like a lattice‐point count. In particular, we verify that all complete unimodular fans are Ehrhart and that the Ehrhart functional agrees with lattice‐point counting in corresponding polytopes, which can otherwise be interpreted as holomorphic Euler characteristics of vector bundles on smooth toric varieties. We also prove that all Bergman fans of matroids are Ehrhart and that the Ehrhart functional in this case agrees with the Euler characteristic of matroids, introduced recently by Larson, Li, Payne, and Proudfoot. A key property that we prove about the Ehrharticity of fans is that it only depends on the support of the fan, not on the fan structure, thus providing a uniform framework for studying ‐rings and Euler characteristics of complete fans and Bergman fans simultaneously.

Abstract In this paper, we consider two variations on Mann’s $\infty $ -categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty $ -categories, and have recollements. Using Nagata’s compactification theorem, we show that Nagata six-functor formalisms on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of “varieties and spans consisting of an open immersion and a proper map”. Using this characterization, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty $ -categories and adjoint triples. We characterize which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty $ -categories and adjoint triples, extend to local six-functor formalisms.

Using the global Kuranishi charts constructed by Hirschi–Swaminathan, we define gravitational descendants and equivariant Gromov–Witten invariants for general symplectic manifolds. We prove that these invariants satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it to derive an explicit formula for the equivariant Gromov–Witten invariants of Hamiltonian GKM manifolds. In particular, the symplectic Gromov–Witten invariants of smooth toric varieties agree with their algebro-geometric counterpart. In the semipositive case, the invariants studied here recover those of Ruan and Tian.

Host plant resistance (HPR) has shown potential for suppressing sweetpotato whitefly, Bemisia tabaci Gennadius (Hemiptera: Aleyrodidae), in smooth (glabrous) crop varieties lacking leaf trichomes. The objective of this study was to investigate the interaction between HPR and insecticidal control, aiming to enhance their collective efficacy in whitefly management. Field trials were conducted in cotton and cantaloupe planted as strip crops at 2 locations in southern Georgia, United States: Tifton and Camilla. Treatments comprised 2 insecticides, based on the active ingredients pyriproxyfen and cyantraniliprole, with 2 different trichome conditions: pubescent (hairy), or smooth varieties. During the crop growing season, B. tabaci adult, egg, and nymph populations were monitored, and whitefly preferences were evaluated. Results indicate a preference of whiteflies for cotton and cantaloupe pubescent varieties, largely attributed to the presence of leaf trichomes. Pyriproxyfen predominantly reduced nymph populations, while cyantraniliprole was effective against both immatures and adults. Significant interactions among crop type, trichome presence, and insecticide application in determining B. tabaci abundance were measured. The glabrous cotton variety demonstrated greater whitefly suppression compared to glabrous melon, and cyantraniliprole exhibited a heightened initial mortality in pubescent cultivars. The study underscores the importance of selecting smooth leaf crop varieties in integrated B. tabaci management strategies. The results illuminate the need for developing real-world testing models with compatible strategies of integrated pest management (IPM) programs for B. tabaci and provide a wide-ranging insight into the interactive effects and dependency of multiple components involved in whitefly control in multicropping systems.

Abstract The modified Macdonald functions are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the ‐fixed points of the Hilbert schemes and the functions realizing a derived equivalence between ‐equivariant coherent sheaves on and ‐equivariant coherent sheaves on . Carlsson–Gorsky–Mellit introduced a larger family of smooth varieties called the parabolic flag Hilbert schemes. They showed that an algebra , directly related to the double Dyck path algebra employed in Carlsson–Mellit's proof of the Shuffle Theorem, acts naturally on the ‐equivariant K‐theory of these spaces, and moreover, there is a ‐isomorphism where is the polynomial representation. The isomorphism is known to extend Haiman's correspondence. In this paper, we explicitly compute the images of the normalized ‐fixed point classes of the spaces and show that they agree with the modified partially symmetric Macdonald polynomials introduced by Goodberry–Orr, confirming their prior conjecture. We use this result to give an explicit formula for the action of the involution on .

Given a vector bundle \mathcal{E} on a smooth projective variety B , the flag bundle \mathcal{F}l(1,2,\mathcal{E}) admits two projective bundle structures over the Grassmann bundles \mathcal{G}r(1,\mathcal{E}) and \mathcal{G}r (2,\mathcal{E}) . The data of a general section of a suitably defined line bundle on \mathcal{F}l(1,2,\mathcal{E}) defines two varieties: a cover X_{1} of B , and a fibration X_{2} on B with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of X_{2} which consists of a list of exceptional objects and a subcategory equivalent to the derived category of X_{1} . As a by-product, we obtain a new full exceptional collection for the Fano fourfold of degree 12 and genus 7 . Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor \mathcal{F}l(1, 2, \mathcal{E}) is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal–Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.

Abstract We study two classes of morphisms in infinite type: tamely presented morphisms and morphisms with coherent pullback. These are generalizations of finitely presented morphisms and morphisms of finite Tor-dimension, respectively. The class of tamely presented schemes and stacks is restricted enough to retain the key features of finite-type schemes from the point of view of coherent sheaf theory, but wide enough to encompass many infinite-type examples of interest in geometric representation theory. The condition that a diagonal has coherent pullback is a natural generalization of smoothness to the tamely presented setting, and we show such objects retain many good cohomological properties of smooth varieties. Our results are motivated by the study of convolution products in the double affine Hecke category and related categories in the theory of Coulomb branches.

We prove that every 0-shifted Poisson structure on a derived Artin n-stack admits a curved A 1 deformation quantisation whenever the stack has perfect cotangent complex; in particular, this applies to LCI schemes, where it gives a DQ algebroid quantisation.Whereas the Kontsevich-Tamarkin approach to quantisation for smooth varieties hinges on invariance of the Hochschild complex under affine transformations, we instead exploit the observation that the Hochschild complex carries an anti-involution, and that such anti-involutive deformations of the complex of polyvectors are essentially unique.We also establish analogous statements for deformation quantisations in C 1 and analytic settings.14A30, 53D55 Introduction 3717 1. Involutively filtered deformations of Poisson algebras 3720 2. Quantisations on derived Deligne-Mumford stacks 3729 3. Quantisations on derived Artin stacks 3746

On formal non-commutative deformations of smooth varieties

Abstract To a smooth variety X with simple normal crossings divisor D, we associate a sheaf of vertex algebras on X, denoted $$\Omega ^{ch}_{X}(\operatorname {log}D)$$ Ω X ch ( log D ) , whose conformal weight 0 subspace is the algebra $$\Omega _{X}(\operatorname {log}D)$$ Ω X ( log D ) of forms with log poles along D. We prove various basic structural results about $$\Omega ^{ch}_{X}(\operatorname {log}D)$$ Ω X ch ( log D ) . In particular, if $$X^{*}=X\setminus D$$ X ∗ = X \ D has a volume form then we show that $$\Omega ^{ch}_{X}(\operatorname {log}D)$$ Ω X ch ( log D ) admits a topological structure of rank $$d=\operatorname {dim}(X)$$ d = dim ( X ) , which is enhanced to an extended topological structure if $$D\sim -K_{X}$$ D ∼ - K X is in fact anticanonical. In this latter case, we also show that the resulting (q, y) character $$\operatorname {Ell}(X,D)(q,y)$$ Ell ( X , D ) ( q , y ) is a section of the line bundle $$\Theta ^{\otimes d}$$ Θ ⊗ d on the elliptic curve $$E=\textbf{C}^{*}/q^{\textbf{Z}}$$ E = C ∗ / q Z . We further show how $$\Omega ^{ch}_{X}(\operatorname {log}D)$$ Ω X ch ( log D ) can be understood in terms of a simple birational modification of the space of jets into X.