For any power q of the positive ground field characteristic, a smooth q-bic threefold—the Fermat threefold of degree $q+1$, for example—has a smooth surface S of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of S. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute the cohomology of the structure sheaf of S when q is prime.

We describe the behavior of a free reduced plane projective curve with respect to the addition, respectively, deletion, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.

This paper considers $A_\infty$-algebras satisfying an analytic bound with respect to a fixed norm. We define a notion of right Calabi–Yau (CY) structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich and Soibelman [Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv: 0811.2435], and yields a flexible method for obtaining the analytic potentials of Hua and Keller [Quivers with analytic potentials, Preprint (2019), arXiv: 1909.13517]. We apply these results to the endomorphism differential graded algebra (DGA) of polystable sheaves considered by Toda [Moduli stacks of semistable sheaves and representations of Ext-quivers, Geom. Topol. 22 (2018), 3083–3144], for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms. As a result, we find a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This yields an extension of Toda’s result [Geom. Topol. 22 (2018), 3083–3144] to the quasi-projective setting, and a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi–Yau varieties.

In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogs of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

We study the relationship between the enumerative geometry of rational curves in local geometries and various versions of maximal contact logarithmic curve counts. Our approach is via quasimap theory, and we show versions of the [vGGR19] local/logarithmic correspondence for quasimaps, and in particular for normal crossings settings, where the Gromov-Witten theoretic formulation of the correspondence fails. The results suggest a link between different formulations of relative Gromov-Witten theory for simple normal crossings divisors via the mirror map. The main results follow from a rank reduction strategy, together with a new degeneration formula for quasimaps.

We construct semiorthogonal decompositions of Donaldson–Thomas (DT) categories for reduced curve classes on local surfaces into products of quasi-BPS categories and Pandharipande–Thomas (PT) categories, giving a categorical analogue of the numerical DT/PT correspondence for Calabi–Yau 3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT quivers (which appear as Ext quivers in the DT/PT wall-crossing) proved in our previous paper. We also study quasi-BPS categories of points on local surfaces and propose conjectural computations of their K-theory analogous to formulas already known for the three-dimensional affine space.

We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a surface with two irreducible components glued along a smooth divisor, we prove a degeneration formula that relates the moduli space associated to the surface with the relative spaces associated to the two components. For a smooth surface and no markings, our notion of stability agrees with slope stability with respect to a suitable choice of polarization. We apply our results to compute elliptic genera of moduli spaces of stable sheaves on some elliptic surfaces.

A few years ago, G. Oberdieck conjectured a multiple cover formula that determines the number of curves of fixed genus and degree passing through a configuration of points in an abelian surface. This formula was proved by the author using tropical techniques and Nishinou’s correspondence theorem. Using the same techniques, we give a much shorter proof of the multiple cover formula for point insertions, relying on the same geometrical idea, but avoiding any kind of tropical enumeration.

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

We prove a motivic integral identity relating the motivic Behrend function of a $(-1)$-shifted symplectic stack to that of its stack of graded points. This generalizes analogous identities for moduli stacks of objects in $3$-Calabi–Yau abelian categories obtained by Kontsevich and Soibelman, and Joyce and Song, which are crucial in proving wall-crossing formulae for Donaldson–Thomas invariants. We expect our identity to be useful in extending motivic Donaldson–Thomas theory to general $(-1)$-shifted symplectic stacks.

Let X be a zero-dimensional reduced subscheme of a multiprojective space $\mathbb {V} $. Let $s_i$ be the length of the projection of X onto the ith component of $\mathbb {V}$. A result of Van Tuyl states that the Hilbert function of X is completely determined by its restriction to the product of the intervals $[0, s_i - 1]$. We extend this result to arbitrary zero-dimensional subschemes of $\mathbb {V}$.

Segre and Verlinde series have been studied in many cases, including virtual geometries of Quot schemes on surfaces and Calabi–Yau 4-folds. Our work is the first to address the equivariant setting for both ${\mathbb{C}}^2$ and ${\mathbb{C}}^4$ by examining higher degree contributions which have no compact analogue.

1. (i) For ${\mathbb{C}}^2$, we work mostly with virtual geometries of Quot schemes. After connecting the equivariant series in degree zero to the existing results of the first author for compact surfaces, we extend the Segre–Verlinde correspondence to all degrees and to the reduced virtual classes. Additionally, we conjecture that there is an equivariant symmetry of Segre series, which was also observed in the compact setting.

2. (ii) For ${\mathbb{C}}^4$, we give further motivation for the definition of the Verlinde series. Based on empirical data andtorsiopn additional structural results, we conjecture that there is an equivariant Segre–Verlinde correspondence and Segre symmetry analogous to the one for ${\mathbb{C}}^2$.

We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi–Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking–Keel–Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov–Witten theory, and mirror symmetry.

Schubert Vanishing is a problem of deciding whether Schubert coefficients are zero. Until this work it was open whether this problem is in the polynomial hierarchy ${{\mathsf {PH}}}$. We prove this problem is in ${{\mathsf {AM}}} \cap {{\mathsf {coAM}}}$ assuming the Generalized Riemann Hypothesis ($\mathrm{GRH}$), that is, relatively low in ${{\mathsf {PH}}}$. Our approach uses Purbhoo’s criterion [57] to construct explicit polynomial systems for the problem. The result follows from a reduction to Parametric Hilbert’s Nullstellensatz, recently analyzed in [2]. We extend our results to all classical types.

We prove a ‘Whitney’ presentation, and a ‘Coulomb branch’ presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm {Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $\lambda _y$ classes of the tautological bundles. In physics, the $\lambda _y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on $\mathrm {Gr}(k;n)$, using the ‘quantum=classical’ statement.

In the present notes, we study a generalization of the Peterson subalgebra to an oriented (generalized) cohomology theory which we call the formal Peterson subalgebra. Observe that by recent results of Zhong the dual of the formal Peterson algebra provides an algebraic model for the oriented cohomology of the affine Grassmannian.

Our first result shows that the centre of the formal affine Demazure algebra (FADA) generates the formal Peterson subalgebra. Our second observation is motivated by the Peterson conjecture. We show that a certain localization of the formal Peterson subalgebra for the extended Dynkin diagram of type $\hat A_1$ provides an algebraic model for “quantum” oriented cohomology of the projective line. Our last result can be viewed as an extension of the previous results on Hopf algebroids of structure algebras of moment graphs to the case of affine root systems. We prove that the dual of the formal Peterson subalgebra (an oriented cohomology of the affine Grassmannian) is the zeroth Hochschild homology of the FADA.

We study certain categories associated to symmetric quivers with potential, called quasi-Bogomol’nyi–Prasad–Sommerfield (BPS) categories. We construct semiorthogonal decompositions of the categories of matrix factorizations for moduli stacks of representations of (framed or unframed) symmetric quivers with potential, where the summands are categorical Hall products of quasi-BPS categories. These results generalize our previous results about the three-loop quiver. We prove several properties of quasi-BPS categories: wall-crossing equivalence, strong generation, and a categorical support lemma in the case of tripled quivers with potential. We also introduce reduced quasi-BPS categories for preprojective algebras, which have trivial relative Serre functor and are indecomposable when the weight is coprime with the total dimension. In this case, we regard the reduced quasi-BPS categories as noncommutative local hyperkähler varieties and as (twisted) categorical versions of crepant resolutions of singularities of good moduli spaces of representations of preprojective algebras. The studied categories include the local models of quasi-BPS categories of K3 surfaces. In a follow-up paper, we establish analogous properties for quasi-BPS categories of K3 surfaces.