arXiv subjects

Michele Graffeo

Publications and source records attributed to Michele Graffeo.

17 recordsLinked to original sources

arXiv2026

Higgs Grassmannians

We review and study the notion of Higgs Grassmannians, which are schemes parametrizing the Higgs subbundles of a given Higgs bundle over a smooth variety. We write their equations as closed subschemes of the usual Grassmann bundles and investigate their geometry. Often the Higgs Grassmannians generically have 0-dimensional fibers over the base variety, thus implying that Higgs subbundles are "scarce." We characterize the structure of the Higgs Grassmannians by analyzing the local Jordan type of the Higgs field. A refined analysis of the rank 2 case is also provided in terms of the discriminant of the characteristic polynomial. We apply our characterizations to the Simpson system of a smooth variety to provide a streamlined proof of its semistability, and we establish a structural relationship between the rank 1 Higgs Grassmannian and the spectral cover of the Higgs bundle. Finally, we introduce the schemes of flags of Higgs subbundles of a given Higgs bundle, and the Quot schemes parametrizing Higgs quotients; we conclude with some examples.

math.AG↗

arXiv2026

5d Trinions and Tetraons

Recently, an atomic classification scheme of 5d SCFTs has been proposed, relying on the identification of indecomposable building blocks that can be fused together to produce large classes of 5d SCFTs. These novel SCFTs are known as bifundamental 5d conformal matter theories, and their fusion produces linear generalized quivers. We generalize such picture employing M-theory geometric engineering to construct trinions and tetraon 5d SCFTs with flavor symmetry of type D. They correspond to non-linear generalized quivers displaying novel patterns of instantonic symmetry enhancement, and their fusion produces singular geometries that often are non-toric non-complete intersections. Finally, within our setup, we rule out trinions and tetraons of type E.

hep-th↗

arXiv2026

Motivic and cohomological stabilisation of the Quot scheme of points

We prove that the motive of the punctual Quot scheme $\mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})_0$ stabilises, when $n \to \infty$, to $[\mathrm{Gr}(d-1,\infty)]\cdot \sum_{i=0}^{r-1}\mathbb L^{di}$. We similarly show that the Poincaré polynomial of the Quot scheme $ \mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})$ stabilises and we compute the limit in terms of the infinite Grassmannian. Finally, we prove that the motive of the nested Hilbert scheme stabilises to the motive of the infinite flag variety and we compute the cohomology ring in the limit. These results provide affirmative evidence to a question of Pandharipande concerning the cohomology of Quot schemes on $\mathbb A^\infty$.

math.AG↗

arXiv2026

Components of the nested Hilbert scheme of few points

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth $n$-fold, for $n\geqslant 4$. Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of $\mathbb A^3$. Explicitly, we show that the Hilbert scheme of points on a hypersurface of $\mathbb{A}^3$ having a singularity of multiplicity at least 5 admits elementary components.

math.AG↗

arXiv2025

Classical Algebraic Geometry and Discrete Integrable Systems

The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry side, which is applied to differential and discrete Painlevé equations on the integrable systems side. Along the way we also discuss the theory of resolution of indeterminacies, which is applied to the cohomological computation of algebraic entropy of birational transformations of projective spaces, which is closely related to the integrability of the discrete systems they define.

math.AG↗

arXiv2025

New components of Hilbert schemes of points and 2-step ideals

This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite colength ideals that are introduced in this paper and referred to as 2-step ideals. The most notable reducibility results pertain to the 4-nested Hilbert scheme of points on a smooth surface, the reducibility of $\text{Hilb}^{3,7}\mathbb{A}^4$, and a method to detect a large number of generically reduced elementary components. To demonstrate the feasibility of this approach, we provide an explicit description of 215 new generically reduced elementary components in dimensions 4, 5 and 6.

math.AG↗

arXiv2025

Invariants of nested Hilbert and Quot schemes on surfaces

Let $(S,p)$ be a smooth pointed surface. In the first part of this paper we study motivic invariants of punctual nested Hilbert schemes attached to $(S,p)$ using the Hilbert-Samuel stratification. We compute two infinite families of motivic classes of punctual nested Hilbert schemes, corresponding to nestings of the form $(2,n)$ and $(3,n)$. As a consequence, we are able to give a lower bound for the number of irreducible components of $S_p^{[2,n]}$ and $S_p^{[3,n]}$. In the second part of this paper we characterise completely the generating series of Euler characteristics of all nested Hilbert and Quot schemes. This is achieved via a novel technique, involving differential operators modelled on the enumerative problem, which we introduce. From this analysis, we deduce that in the Hilbert scheme case the generating series is the product of a rational function by the celebrated Euler's product formula counting integer partitions. In higher rank, we derive functional equations relating the nested Quot scheme generating series to the rank one series, corresponding to nested Hilbert schemes.

math.AG↗

arXiv2025

Enumeration of partitions via socle reduction

We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional partitions, satisfying suitable constraints on their embedding dimension and socle type. We provide exact formulas for the generating functions of several infinite families of such partitions, and design a procedure enumerating them in the general case. As a proof of concept, we determine the number of partitions of size up to 30 in any dimension.

math.CO↗

arXiv2024

The Painlevé equivalence problem for a constrained 3D system

In this paper we propose a geometric approach to study Painlevé equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlevé property from a geometric perspective. Utilising the Painlevé identification algorithm we also relate this system to the Painlevé VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlevé property without the need to restrict it to the invariant hypersurface.

nlin.SI↗

arXiv2024

The geometry of double nested Hilbert schemes of points on curves

Let $C$ be a smooth curve. In this paper we investigate the geometric properties of the double nested Hilbert scheme of points on $C$, a moduli space introduced by the third author in the context of BPS invariants of local curves and sheaf counting on Calabi-Yau 3-folds. We prove this moduli space is connected, reduced and of pure dimension; we list its components via an explicit combinatorial characterisation and we show they can be resolved, when singular, by products of symmetric products of $C$. We achieve this via a purely algebraic analysis of the factorisation properties of the monoid of reverse plane partitions. We discuss the (virtual) fundamental class of the moduli space, we describe the local equations cutting it inside a smooth ambient space, and finally we provide a closed formula for its motivic class in the Grothendieck ring of varieties.

math.AG↗

arXiv2024

Unexpected but recurrent phenomena for Quot and Hilbert schemes of points

We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of the elementary components of the nested Hilbert schemes of points on smooth quasi-projective varieties of dimension at least 4 are generically non-reduced. We also deduce that nested Hilbert schemes of points on smooth surfaces have generically non-reduced components. Finally, we give an infinite family of elementary components of the classical Hilbert schemes of points.

math.AG↗

arXiv2024

The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an expression for $\mathsf Z_d(t)$ as the product of the zeta function of $\mathbb P^{d-1}$ and a polynomial $\mathsf P_d(t)$, which in particular implies that $\mathsf Z_d(t)$ is a rational function. Moreover, we reduce the complexity of $\mathsf P_d(t)$ to the computation of $d-8$ initial data, and therefore give explicit formulas for $\mathsf Z_d(t)$ in the cases $d \leq 8$, which in turn yields a formula for $[\mathrm{Hilb}^{\leq 8}(X)]$ for any smooth variety $X$. We perform a similar analysis for the Quot scheme of points, obtaining explicit formulas for the full generating function (summing over all ranks and dimensions) for $d \leq 4$. In the limit $n \to \infty$, we prove that the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0]$ stabilise to the class of the infinite Grassmannian $\mathrm{Gr}(d-1,\infty)$. Finally, exploiting our geometric methods, we conjecture (and partially confirm) a structural result on the 'error' measuring the discrepancy between the count of higher dimensional partitions and MacMahon's famous guess.

math.AG↗

arXiv2024

Moduli spaces of $\mathbb{Z}/k\mathbb{Z}$-constellations over $\mathbb{A}^2$

Let $ρ:\mathbb{Z}/k \mathbb{Z}\rightarrow \text{SL}(2,\mathbb{C})$ be a representation of a finite abelian group and let $Θ^{\text{gen}}\subset \text{Hom}_\mathbb{Z}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subsetΘ^{\text{gen}}$ and prove that there are $k!$ of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric $G$-constellations, it is enough to build all simple chambers. We also prove that there are $k\cdot 2^{k-2} $ simple chambers. Finally, we provide an explicit formula for the tautological bundles $\mathscr{R}_C$ over the moduli spaces $\mathscr{M} _C$ for all chambers $C\subset Θ^{\text{gen}}$ which only depends upon the chamber stair which is a combinatorial object attached to the chamber $C$.

math.AG↗

arXiv2024

A counterexample to the parity conjecture

Let $[Z]\in\text{Hilb}^d \mathbb A^3$ be a zero-dimensional subscheme of the affine three-dimensional complex space of length $d>0$. Okounkov and Pandharipande have conjectured that the dimension of the tangent space of $\text{Hilb}^d \mathbb A^3$ at $[Z]$ and $d$ have the same parity. The conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for points $[Z]$ defined by monomial ideals and very recently by Ramkumar and Sammartano for homogeneous ideals. In this paper we exhibit a family of zero-dimensional schemes in $\text{Hilb}^{12} \mathbb A^3$, which disproves the conjecture in the general non-homogeneous case.

math.AG↗

arXiv2023

5d Conformal Matter

Six-dimensional superconformal field theories (SCFTs) have an atomic classification in terms of elementary building blocks, conformal systems that generalize matter and can be fused together to form all known 6d SCFTs in terms of generalized 6d quivers. It is therefore natural to ask whether 5d SCFTs can be organized in a similar manner, as the outcome of fusions of certain elementary building blocks, which we call 5d conformal matter theories. In this project we begin exploring this idea and we give a systematic construction of 5d generalized ``bifundamental'' SCFTs, building from geometric engineering techniques in M-theory. In particular, we find several examples of $(\mathfrak {e}_6,\mathfrak {e}_6)$, $(\mathfrak {e}_7,\mathfrak {e}_7)$ and $(\mathfrak {e}_8,\mathfrak {e}_8)$ 5d bifundamental SCFTs beyond the ones arising from (elementary) KK reductions of the 6d conformal matter theories. We show that these can be fused together giving rise to 5d SCFTs captured by 5d generalized linear quivers with exceptional gauge groups as nodes, and links given by 5d conformal matter. As a first application of these models we uncover a large class of novel 5d dualites, that generalize the well-known fiber/base dualities outside the toric realm.

hep-th↗

arXiv2023

Growth and integrability of some birational maps in dimension three

Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $\mathrm{c}_3\in\mathrm{Bir}(\mathbb{P}^3)$ with projectivities that permute the fixed points of $\mathrm{c}_3$ and the points over which $\mathrm{c}_3$ performs a divisorial contraction. More specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant.

math.AG↗

arXiv2022

On the Behrend function and the blowup of some fat points

The Behrend function of a $\mathbb C$-scheme $X$ is a constructible function $ν_X\colon X(\mathbb C) \to \mathbb Z$ introduced by Behrend, intrinsic to the scheme structure of $X$. It is a (subtle) invariant of singularities of $X$, playing a prominent role in enumerative geometry. To date, only a handful of general properties of the Behrend function are known. In this paper, we compute it for a large class of fat points (schemes supported at a single point). We first observe that, if $X \hookrightarrow \mathbb A^N$ is a fat point, $ν_X$ is the sum of the multiplicities of the irreducible components of the exceptional divisor $E_{X}\mathbb A^N$ in the blowup $\textrm{Bl}_{X}\mathbb A^N$. Moreover, we prove that $ν_X$ can be computed explicitly through the normalisation of $\textrm{Bl}_{X}\mathbb A^N$. The proofs of our explicit formulas for the Behrend function of a fat point in $\mathbb A^2$ rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of $E_{X}\mathbb A^2$, where $X \hookrightarrow \mathbb A^2$ is a fat point such that $\textrm{Bl}_{X}\mathbb A^2$ is normal.

math.AG↗