We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible, and non-degenerate curve of degree d and genus g in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus g and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

We answer an open problem posed by Iarrobino, Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert', in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276] $\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$ with dimension less than $(n-1)(d-1)$ ? For each $n\geq 4$ , we construct an infinite class of elementary components in $\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$ producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev, Power sums, Gorenstein algebras, and determinantal loci (Springer-Verlag, Berlin, 1999), 221–226] of the form with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$ .

We determine all the extremal Gromov-Witten invariants of the Hilbert scheme of $3$ points on a smooth projective complex surface. Our result for the genus- $1$ case verifies a conjecture that we propose for the genus- $1$ extremal Gromov-Witten invariant of the Hilbert scheme of n points with n being arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

We propose a variation of the classical Hilbert scheme of points, the double nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss $K$-theoretic refinements à la Nekrasov–Okounkov.

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on ${\mathbb {C}}^2$ . For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty $ . For t-hooks, we prove distributions that are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products

$$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$

Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb {C}^{2})^{[n]}$ on $n$ points, as $n\rightarrow +\infty ,$ is a Gumbel distribution. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_{k}(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$ , then we obtain the asymptotic

$$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$

Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d’une variété lisse—c’est-à-dire l’espace de modules paramétrant les drapeaux de quotients de dimension $0$ d’un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah dans le cadre des schémas de Hilbert ponctuels emboîtés.

We prove an equivalence between the Bryan-Steinberg theory of $\pi $-stable pairs on $Y = \mathcal {A}_{m-1} \times \mathbb {C}$ and the theory of quasimaps to $X = \text{Hilb}(\mathcal {A}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on Y arising from 3D mirror symmetry for quasimaps to X, including the Donaldson-Thomas crepant resolution conjecture.

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$-fold ${{\mathbb {A}}}^3$. We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$, that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$. Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$, where F is an equivariant exceptional locally free sheaf on a projective toric $3$-fold X.

As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

We investigate families of minimal rational curves on Schubert varieties, their Bott–Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families on small resolutions of minuscule Schubert varieties.

We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$. We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface $S$.

We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.