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Andrea T. Ricolfi

Publications and source records attributed to Andrea T. Ricolfi.

At least 19 recordsLinked to original sources
arXiv2026

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.

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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 $\mathbb A^\infty$.

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arXiv2026

Derived hyperquot schemes

We define a derived enhancement of the hyperquot scheme (also known as nested Quot scheme), which classically parametrises flags of quotients of a perfect coherent sheaf on a projective scheme. We prove it is representable by a derived scheme, and we compute its global tangent complex. As an application, we provide a natural obstruction theory on the classical hyperquot scheme. The latter recovers the virtual fundamental class recently constructed by the first and third author in the context of the enumerative geometry of hyperquot schemes on smooth projective curves.

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arXiv2025

Moduli spaces of semiorthogonal decompositions in families

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the étale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\operatorname{Perf}\mathcal{X}_V$. We use Artin's criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover étale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace. Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an étale neighbourhood of the point.

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arXiv2025

Indecomposability of derived categories in families

Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a natural conjecture on the structure of the moduli space, we give both general results, and discuss interesting explicit examples of the behaviour of indecomposability in families, by relating it to the behaviour of the canonical base locus in families. These examples are symmetric powers of curves, certain regular surfaces of general type with large canonical base locus, and Hilbert schemes of points on surfaces. Indecomposability for symmetric powers of curves has been settled via other means, the other cases remain open and we expect that our analysis of the base locus will prove instrumental in finding unconditional proofs.

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arXiv2025

On the stack of 0-dimensional coherent sheaves: motivic aspects

Let $X$ be a variety. In this survey, we study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the construction of the support map $\mathscr{C}oh^n(X) \to \mathrm{Sym}^n(X)$ to the symmetric product and we prove that, for any closed point $p \in X$, the motive of the punctual stack $\mathscr{C}oh^n(X)_p$ parametrising sheaves supported at $p$ only depends on a formal neighbourhood of $p$. We perform the same analysis for the Quot-to-Chow morphism $\mathrm{Quot}_X(\mathcal E,n) \to \mathrm{Sym}^n(X)$, for a fixed sheaf $\mathcal E \in \mathrm{Coh}(X)$.

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arXiv2025

Hyperquot schemes on curves: virtual class and motivic invariants

Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed Hilbert polynomial, and offer alternative compactifications to the spaces of maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in this paper we construct a perfect obstruction theory (and hence a virtual class and a virtual structure sheaf) on these moduli spaces, which we use to provide criteria for smoothness and unobstructedness. Under these assumptions, we determine their motivic partition function in the Grothendieck ring of varieties, in terms of the motivic zeta function of $C$.

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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.

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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.

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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.

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arXiv2024

On the stack of 0-dimensional coherent sheaves: structural aspects

Let $X$ be a quasiprojective scheme. In this expository note we collect a series of useful structural results on the stack $\mathscr{C}oh^n(X)$ parametrising $0$-dimensional coherent sheaves of length $n$ over $X$. For instance, we discuss its functoriality (in particular its behaviour along étale maps), the support morphism to $\mathrm{Sym}^n(X)$, and its relationship with the Quot scheme of points $\mathrm{Quot}_X(\mathcal E,n)$ for fixed $\mathcal E\in \mathrm{Coh}(X)$.

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arXiv2023

The d-critical structure on the Quot scheme of points of a Calabi-Yau 3-fold

The Artin stack $\mathcal M_n$ of $0$-dimensional sheaves of length $n$ on $\mathbb A^3$ carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack $[\textrm{crit}(f_n)/\textrm{GL}_n]$, another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points $\textrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n) = \textrm{crit}(f_{r,n})$, which is a global critical locus for every $r>0$, and also carries a derived-in-flavour d-critical structure besides the one induced by the potential $f_{r,n}$. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on $\textrm{Quot}_X(F,n)$, where $F$ is a locally free sheaf of rank $r$ on a projective Calabi-Yau $3$-fold $X$. Finally, we prove that the perfect obstruction theory on $\textrm{Hilb}^n\mathbb A^3=\textrm{crit}(f_{1,n})$ induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential $f_{1,n}$.

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arXiv2023

A sign that used to annoy me, and still does

We provide a proof of the following fact: if a complex scheme $Y$ has Behrend function constantly equal to a sign $σ\in \{\pm 1\}$, then all of its components $Z \subset Y$ are generically reduced and satisfy $(-1)^{\mathrm{dim}_{\mathbb C} T_pY} = σ= (-1)^{\mathrm{dim}Z}$ for $p \in Z$ a general point. Given the recent counterexamples to the parity conjecture for the Hilbert scheme of points $\mathrm{Hilb}^n(\mathbb A^3)$, our argument suggests a possible path to disprove the constancy of the Behrend function of $\mathrm{Hilb}^n(\mathbb A^3)$.

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arXiv2022

On the motive of the nested Quot scheme of points on a curve

Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the nested Quot scheme of points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1$ of fixed length $n_i = χ(T_i)$. The resulting series, obtained by exploiting the Bialynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of $C$. In particular, it is a rational function.

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arXiv2022

Hilbert squares of degeneracy loci

Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.

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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.

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