The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points

TL;DR

This paper introduces the Iarrobino scheme, a self-dual analogue of the Hilbert scheme of points, which parametrizes Gorenstein subschemes with additional structure, revealing new geometric properties and applications in deformation and enumerative geometry.

Contribution

It defines the Iarrobino scheme as a self-dual moduli space of Gorenstein subschemes, extending classical Hilbert schemes with new geometric and deformation-theoretic insights.

Findings

Iarrobino schemes are smooth for smooth curves.

They have rich geometric structures.

Applications to deformation theory and enumerative geometry.

Abstract

For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of oriented Gorenstein zero-dimensional subschemes of $X$ together with some additional data (a self-dual filtration) which is vacuous over a big open set but non-trivial over the compactification. Via the link between Hilbert schemes and varieties of commuting matrices, Iarrobino schemes correspond to commuting symmetric matrices. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. A crucial role in the construction is played by the variety of completed quadrics. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve and that they have very rich geometry. We give applications, in particular to deformation theory of (usual) Hilbert schemes of points on threefolds, and to enumerative geometry \`a la June Huh.

Equations

$${\mathrm{Iar}}_d(X)\to {\mathrm{Hilb}}_d(X)$$

$${\tau }_X:{\mathrm{Iar}}_d(X)\to {\mathrm{Hilb}}_d(X),$$

$$X\supseteq Z_0⊋Z_1⊋Z_2⊋\dots$$

$$q_i:{\left(\frac{I_{Z_{i+1}}}{I_{Z_i}}\right)}^{\vee }\to \frac{I_{Z_{i+1}}}{I_{Z_i}},$$

$${\tau }_C:{\mathrm{Iar}}_d(C)\to {\mathrm{Hilb}}_d(C)$$

$$\mathbb{P}({\mathrm{Sym}}^{2}V{)}^{\mathrm{full}\mathrm{rk}}\hookrightarrow \mathbb{P}({\mathrm{Sym}}^{2}V)\times \mathbb{P}\left({\mathrm{Sym}}^2{\Lambda }^{2}V\right)\times \dots \times \mathbb{P}\left({\mathrm{Sym}}^2{\Lambda }^{d-1}V\right)\times \mathbb{P}\left({\mathrm{Sym}}^2{\Lambda }^{d}V\right)$$

$$0\to {\mathcal{O}}_{V_i}\to {\mathcal{O}}_{Z_i}\to {\mathcal{O}}_{Z_{i+1}}\to 0,$$

$$\mathbb{Z}{L}_1\oplus \dots \oplus \mathbb{Z}{L}_{d-1}$$

$$\mathrm{gr}(A)=\underset{i=0}{\overset{\infty }{⨁}}\frac{{\mathfrak{m}}^i}{{\mathfrak{m}}^{i+1}}.$$

$$H_A=\sum _{\delta =0}^{s-2}{\Delta }_{\delta }$$

$$\sum _{d\ge 0}\sum _{k=0}^d\chi (L_{k,d})q^d{r}^k=\left(\sum _{d\ge 0}\frac{1}{{\prod }_{i=1}^d(1-t^i)}{q}^d\right)\cdot \left(\sum _{d\ge 0}\frac{1}{{\prod }_{i=1}^d(1-t^i)}{q}^d{r}^d{t}^{(d-1)d}\right)$$

$$\left\{(X_1,\dots ,X_n,v)\in \mathrm{End}(V{)}^{\oplus n}\times V|\mathbb{C}[X_1,\dots ,X_n]v=V\text{mbox}and{\forall }_{i,j}{X}_i{X}_j=X_j{X}_i\right\}$$

$$\left\{(X_1,\dots ,X_n,v)\in {\mathrm{End}}^{\mathrm{sym}}(V{)}^{\oplus n}\times V|\mathbb{C}[X_1,\dots ,X_n]v=V\text{mbox}and{\forall }_{i,j}{X}_i{X}_j=X_j{X}_i\right\}$$

$$(s\cdot \phi )(m):=\phi (sm),$$

$$\Phi :M\times M\to \mathbb{k},$$

$$\Phi (a_1,a_2)=\phi (a_1)(a_2)=\left(\phi (a_1\cdot 1)\right)(a_2)=\left(a_1\cdot \phi (1)\right)(a_2)=(a_1\cdot \alpha )(a_2)=\alpha (a_1{a}_2),$$

$$\frac{{\mathfrak{m}}^i}{{\mathfrak{m}}^{i+1}}\times \frac{(0:{\mathfrak{m}}^{i+1})}{(0:{\mathfrak{m}}^i)}\to \mathbb{k}$$

$$\frac{{\mathfrak{m}}^i\cap (0:{\mathfrak{m}}^{j+1})}{{\mathfrak{m}}^{i+1}\cap (0:{\mathfrak{m}}^{j+1})+{\mathfrak{m}}^i\cap (0:{\mathfrak{m}}^j)}\times \frac{{\mathfrak{m}}^j\cap (0:{\mathfrak{m}}^{i+1})}{{\mathfrak{m}}^{j+1}\cap (0:{\mathfrak{m}}^{i+1})+{\mathfrak{m}}^j\cap (0:{\mathfrak{m}}^i)}\to \mathbb{k}$$

$$(\mathcal{Q}(\delta ){)}_i:=\frac{{\mathfrak{m}}^i\cap (0:{\mathfrak{m}}^{s-i-\delta +1})}{{\mathfrak{m}}^{i+1}\cap (0:{\mathfrak{m}}^{s-i-\delta +1})+{\mathfrak{m}}^i\cap (0:{\mathfrak{m}}^{s-i-\delta })}$$

$$\mathcal{Q}(A):=\underset{\delta =0}{\overset{s-2}{⨁}}\mathcal{Q}(\delta )$$

$$H_A=\sum _{\delta =0}^{s-2}{\Delta }_{\delta }.$$

$$(\mathcal{Q}(\delta ){)}_i:=\frac{{\mathfrak{m}}^{i}M\cap (0{:}_M{\mathfrak{m}}^{s-i-\delta +1})}{{\mathfrak{m}}^{i+1}M\cap (0{:}_M{\mathfrak{m}}^{s-i-\delta +1})+{\mathfrak{m}}^{i}M\cap (0{:}_M{\mathfrak{m}}^{s-i-\delta })}.$$

$$\mathcal{Q}(\delta )\simeq \mathcal{Q}(\delta {)}^{\vee }[-(s-\delta )]$$

$$\mathrm{CQ}(\mathcal{V})\to B,$$

$$\mathrm{CQ}({\phi }^{\ast }\mathcal{V})$$

$$\displaystyle\operatorname{CQ}(V)(\Bbbk)=\Big{\{}([q_{0}],[q_{1}],[q_{2}],\ldots)\ \Big{|}\$$

$$q_1:\ker q_0\to \mathrm{coker}{q}_0,q_1^{\top }=q_1,$$

$$q_2:\ker q_1\to \mathrm{coker}{q}_1,q_2^{\top }=q_2,$$

$$\displaystyle\phantom{q_{2}\colon\ker q_{1}\to}\ldots\phantom{\operatorname{coker}q_{1}\ q_{2}^{\top}=q_{2}}\Big{\}}$$

$$F_0=V^{\vee }⊋F_1⊋F_2⊋\dots$$

$$0$$

Full text

Supported by National Science Centre grant 2023/50/E/ST1/0033.

1. Introduction

Let $\operatorname{Hilb}_{d}(X)$ be the Hilbert scheme of points on a quasi-projective $\Bbbk$-scheme $X$. The $\Bbbk$-points of $\operatorname{Hilb}_{d}(X)$ are finite degree $d$ closed subschemes $Z\subseteq X$. For $X$ a smooth curve or a smooth surface, the Hilbert scheme $\operatorname{Hilb}_{d}(X)$ is smooth and prominently appears throughout combinatorics [Hai01, Hai02b], enumerative geometry and representation theory [Göt94, KV00, Nak01], topology [FT11, KZ25], and knot theory [GNR21, GN15], here for singular curves.

In this work we introduce the Iarrobino scheme of points $\operatorname{Iar}_{d}(X)$, which serves as analogue of $\operatorname{Hilb}_{d}(X)$ in the self-dual setting. The scheme $\operatorname{Iar}_{d}(X)$ parameterises pairs $(Z\subseteq X,\,[q_{\bullet}])$, where $Z$ is as above and $q_{\bullet}$ is a broken quadric that makes an associated graded of the sheaf $\mathcal{O}_{Z}$ self-dual. An important case is when $q_{\bullet}$ is a single full rank quadric and $\mathcal{O}_{Z}$ itself is self-dual; this is the oriented Gorenstein locus. The Iarrobino scheme compactifies this locus: the forgetful map

[TABLE]

is projective. We discuss $\operatorname{Iar}_{d}(X)$ more precisely in §1.1 below.

The motivation for introducing $\operatorname{Iar}_{d}(X)$ comes from four independent sources:

1. (1)

As an alternative compactification of the locus of oriented Gorenstein subschemes, especially for threefolds §1.2.1. This has already led to smoothness results on the Hilbert scheme itself [JRS24]. 2. (2)

As a convenient moduli space for intersection theory in completed quadrics §1.2.2, as developed by Mateusz Michałek and others [Mic23, MMM*+*24, JPS25], extending the work of June Huh (and others) on matroids [HK12, Huh12, AHK18]. 3. (3)

As an ambient scheme for the Iarrobino’s symmetric decomposition of the Hilbert function [Iar94], which is a central tool for classifying and understanding finite non-graded Gorenstein algebras, see for example [ER12, CJN15, IMM21], 4. (4)

As a moduli space with interesting enumerative properties.

We discuss these more thoroughly in §1.2 below.

A certain beauty in the Iarrobino scheme is that it links distant topics, both in applications and in the construction. For example, two of main ingredients of the construction are:

• •

the variety of completed quadrics, which dates back to classical times [Kle80, Sem48, Tyr56], and is reinterpreted as a wonderful compactification [DCP83].

• •

moduli spaces and stacks: the Hilbert and Quot schemes of points, the stack of finite algebras etc.

1.1. Completed quadrics and the Iarrobino

scheme

Throughout, $\Bbbk$ is a field of characteristic different from $2$. A finite $\Bbbk$-scheme $Z$ is Gorenstein if its dualising sheaf $\omega_{Z}=\mathcal{O}_{Z}^{\vee}$, see §2.1, is invertible, or, equivalently, trivial: there is an isomorphism $q\colon\omega_{Z}\to\mathcal{O}_{Z}$. For a variety $X$, the locus of Gorenstein schemes $\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)\hookrightarrow\operatorname{Hilb}_{d}(X)$ is open [CN09a, p.2056], but it is usually far from being projective even if $X$ is projective, see Example 2.4. The Iarrobino scheme remedies this.

Theorem 1.1** (Definition 4.6, Proposition 4.7).**

Let $X$ be a quasi-projective $\Bbbk$-scheme. There is a quasi-projective scheme $\operatorname{Iar}_{d}(X)$ with a projective morphism

[TABLE]

such that the $\Bbbk$-points of $\operatorname{Iar}_{d}(X)$ correspond bijectively to sequences (of varying length):

[TABLE]

of zero-dimensional subschemes of $X$ such that $\deg Z_{0}=d$, together with a choice (up to scalar multiplication) of $\mathcal{O}_{X}$-module isomorphisms

[TABLE]

for $i=0,1,\ldots$, which satisfy $q_{i}^{\top}=q_{i}$. The map $\tau_{X}$ sends such a $\Bbbk$-point to $[Z_{0}]\in\operatorname{Hilb}_{d}(X)$.

Less formally put, the scheme $Z_{0}$ comes with an exhaustive filtration by self-dual quotients.

The “good” locus in Theorem 1.1 corresponds to the case with only $Z_{0}$. Here, the subquotient $I_{Z_{1}}/I_{Z_{0}}$ is isomorphic to $\mathcal{O}_{Z_{0}}$ and $q_{0}\colon\mathcal{O}_{Z_{0}}^{\vee}\to\mathcal{O}_{Z_{0}}$ is the trivialisation of the dualising sheaf, hence $Z_{0}$ is Gorenstein. This locus is open in $\operatorname{Iar}_{d}(X)$, see Definition 4.12. The cases with “residual” quadrics $q_{1}$, $q_{2}$, etc. serve to compactify it.

The Iarrobino scheme, just like the usual Hilbert scheme of points, comes accompanied by the Quot scheme version: for a coherent sheaf $\mathcal{E}$ on $X$, we have the self-dual Quot scheme of points $\operatorname{CQuot}_{d}(\mathcal{E})$, see Definition 4.8. As with Hilb and Quot, we have $\operatorname{Iar}_{d}(X)\simeq\operatorname{CQuot}_{d}(\mathcal{O}_{X}^{\oplus 1})$. We also obtain the stacky counterparts: the Artin stack $\operatorname{CQAlg}_{d}$ which plays the role of the stack $\operatorname{Alg}_{d}$ of finite algebras, and the self-dual Coherent Stack $\operatorname{CQCoh}_{d}X$ on a quasi-projective scheme $X$, which plays the role of the stack of coherent sheaves $\operatorname{Coh}_{d}X$, for both see §4.2.2.

The main emphasis of this paper is on constructing the above schemes. However, as a proof of concept, we provide results in the case of curves.

Theorem 1.3** (Theorem 4.14).**

Let $C$ be a smooth irreducible curve and $\mathcal{E}$ be a locally free sheaf on $C$. Then the scheme $\operatorname{CQuot}_{d}(\mathcal{E})$ is smooth connected of dimension $d\cdot(1+\operatorname{rk}\mathcal{E})-1$.

For the Iarrobino scheme, we have also interesting connections to the Hilbert scheme (which is isomorphic to the symmetric product in this case).

Theorem 1.4** (Theorem 4.18).**

Let $C$ be a smooth connected curve. The scheme $\operatorname{Iar}_{d}(C)$ is smooth irreducible of dimension $2d-1$. The Iarrobino-to-Hilbert morphism

[TABLE]

is flat and projective with integral fibres of dimension $d-1$ which are reduced complete intersections in $\operatorname{Iar}_{d}(C)$. The morphism satisfies $(\tau_{C})_{*}\mathcal{O}_{\operatorname{Iar}_{d}(C)}=\mathcal{O}_{\operatorname{Hilb}_{d}(C)}$.

The Hilbert scheme of points admits a fairly elementary construction [Hai02a, Hui06, GLS07], [MS05, Chapter 18]. The construction of the Iarrobino scheme is in hindsight reasonably elementary as well, but involves one additional element, which is quite involved itself: the variety of completed quadrics.

This variety shows up also as a special case of the construction. In the classical setup, the Quot scheme $\operatorname{Quot}_{d}\left(\mathcal{O}_{\operatorname{Spec}(\Bbbk)}^{\oplus d}\right)=\operatorname{Gr}(d,d)$ is a single point corresponding to the $d$-dimensional vector space $V=\Bbbk^{\oplus d}$. In contrast, the self-dual Quot scheme $\operatorname{CQuot}_{d}\left(\mathcal{O}_{\operatorname{Spec}(\Bbbk)}^{\oplus d}\right)$ is much more involved. It is isomorphic to the variety of completed quadrics $\operatorname{CQ}(V)$. This is a smooth projective variety of dimension $\binom{d+1}{2}-1$, which can be defined as the closure of the embedding

[TABLE]

given by the exterior powers. We discuss this variety thoroughly in §3.

1.2. Applications

1.2.1. Compactifications in codimension

three

In this subsection $X$ is a smooth threefold. In this case, the Gorenstein locus $\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$ is smooth by Buchsbaum-Eisenbud structure theorem [BE77, KMR98]. The whole Hilbert scheme of points is rather complicated: it is reducible and much is unknown, see for example [Jel24]. The Gorenstein locus is far from being complete, even for projective scheme. It is natural to ask:

Question 1.6**.**

Is there a smooth modular compactification of $\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$?

This is wide open. Here, we fix orientations $[q]$, hence rather than with $\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$, we work with a scheme $f\colon\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(X)\to\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$ over it, see §4.3; the map $f$ is smooth, so $\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(X)$ is smooth as well.

The scheme $\operatorname{Iar}_{d}(X)$ is projective for a projective $X$. It is natural to ask whether it is smooth. The answer is negative: already the scheme $\operatorname{Iar}_{d}(\mathbb{A}^{2})$ is singular [JRS25]. So the construction of $\operatorname{Iar}_{d}(X)$ seems not to yield the desired result. Surprisingly, this is not the end of the story, because the intuition behind $\operatorname{Iar}_{d}(X)$ yields very interesting results in this direction.

Namely, consider a point of $\operatorname{Iar}_{d}(X)$, as in Theorem 1.1, and assume additionally that every $I_{Z_{i+1}}/I_{Z_{i}}$ is principal, so that we obtain sequences

[TABLE]

where $V_{i}\subseteq X$ is a Gorenstein subscheme corresponding to the principal $\mathcal{O}_{X}$-module ${I_{Z_{i+1}}}/{I_{Z_{i}}}$. The following result was directly inspired by the contents of this work.

Theorem 1.7** **(broken Gorenstein without flips are smooth in

codimention three, [JRS24, Theorem 1.5]).

Let $X$ be a smooth threefold and let $Z\subseteq X$ be a finite degree $d$ subscheme that admits a sequence as in Theorem 1.1 with every $I_{Z_{i+1}}/I_{Z_{i}}$ principal. Then $[Z]\in\operatorname{Hilb}_{d}(X)$ is a smooth point.

Conjecturally, together with the flipped cases, these are all smooth points of $\operatorname{Hilb}_{d}(X)$ on its main component, see [JRS24, Conjecture 1.8].

Passing to the limit of Gorenstein algebras in the Iarrobino scheme yields a structure similar to a broken Gorenstein algebra without flips, yet the intermediate ideals need not be principal, see Example 4.30. Thus, it seems that the condition for smoothness is that the intermediate ideals are principal. This suggests that the desired construction of smooth compactification may be obtained using a blowup, similarly as one obtains the Hilbert scheme from the symmetric product [Hai02a, §2], [Hui06, ES14].

1.2.2. Characteristic numbers of algebras

June Huh and others obtained breakthrough results (for example [HK12, Huh12, AHK18, Huh18] and subsequent developments) in the theory of matroids by realising the coefficients of the reduced chromatic polynomial of a matroid as certain intersections numbers on the permutohedral toric variety.

The permutohedral toric variety can be realised as the closure of a torus inside the variety of completed quadrics $\operatorname{CQ}(V)$, see also Example 4.4. Mateusz Michałek and others used intersections numbers on the whole variety of completed quadrics $\operatorname{CQ}(V)$ to resolve several central conjectures in algebraic statistics and beyond, see, for example [MMM*+*24, MMW21, DMS21, Mic23]. A natural generalisation, proposed by Michałek, is to study characteristic numbers of algebras. In this setup, $V$ is an algebra and the multiplication map $\operatorname{Sym}^{2}V\to V$ yields a subspace $i\colon V^{\vee}\hookrightarrow\operatorname{Sym}^{2}V^{\vee}$. If $V$ is Gorenstein, the image of $i$ contains full rank quadrics. One takes the closure of $(\operatorname{im}i)\cap\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ inside $\operatorname{CQ}(V^{\vee})\simeq\operatorname{CQ}(V)$ and obtains a subvariety $X_{V}\subseteq\operatorname{CQ}(V)$ of dimension $\dim_{\Bbbk}V-1=d-1$.

The variety $\operatorname{CQ}(V)$ has Picard group

[TABLE]

for certain distinguished line bundles $L_{1},\ldots,L_{d-1}$, which are pullbacks of $\mathcal{O}(1)$-bundles from respective factors of (1.5). A characteristic number of the algebra $V$ is any intersection number $L_{1}^{b_{1}}\ldots L_{d-1}^{b_{d-1}}|_{X_{V}}$ where $b_{1},\ldots,b_{d-1}\in\mathbb{Z}_{\geq 0}$ sum up to $d-1$. Characteristic numbers are lower semicontinuous.

In this article we provide a modular interpretation of $X_{V}$ as the closure of the unbroken part of the fiber over $[\operatorname{Spec}(V)]$ of the Iarrobino-to-Hilbert map. This part is not necessarily the full fibre, which explains the lower-semicontinuity.

For a smooth curve, the Iarrobino-to-Hilbert map $\tau$ is flat with irreducible fibers, see Theorem 1.4 above, so the varieties $X_{V}$ coincide with the fibers and we obtain immediately the following result, first proven by Jakub Jagiełła, Paweł Pielasa, and Anatoli Shatsila.

Theorem 1.8** **([JPS25, Theorem 1.3, Corollary 1.4],

see Proposition 4.21).

For every finite degree $d$ algebra of the form $\Bbbk[x]/(f)$, the characteristic numbers are the same.

There is ongoing work by Mateusz Michałek and others to study characteristic numbers in different generalities; the Iarrobino scheme is the natural moduli space for such computations.

1.2.3. Classification of zero-dimension Gorenstein

algebras and origin of the name

The classification of finite degree $d$ algebras is a classical subject. The theory reduces immediately to the case of local $\Bbbk$-algebras $(A,\mathfrak{m},\Bbbk)$. A key numerical invariant of such an algebra $A$ is its Hilbert function $H_{A}(i):=\dim_{\Bbbk}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$ obtained from its associated graded algebra

[TABLE]

Since $A$ is Artinian, the sum above is actually finite, ending at the maximal number $s$ such that $\mathfrak{m}^{s}\neq 0$. For Gorenstein algebras, their classification and numerical invariants are also very actively researched, see, for example [CN09b, CN11, ER12, IMS22, CJN15, Jel17, JMR23, IMM21] also thanks to their applications to secant varieties [BB14]. For numerical invariants, a foundational discovery of Anthony Iarrobino [Iar94] was the introduction of symmetric decomposition

[TABLE]

into $s-2$ functions $\Delta_{0},\ldots,\Delta_{s-2}$ such that $\Delta_{\delta}(i)=\Delta_{\delta}(s-i-\delta)$ for every $0\leq i\leq s-\delta$ and $\Delta_{\delta}(i)=0$ otherwise. The symmetric decomposition refines $H_{A}$ and restricts possible Hilbert functions of Gorenstein algebras, for example $H_{A}=(1,n,n+1,1,0,0,\ldots)$ is not possible for any $n$. Still, possible $H_{A}$ and $\Delta_{\bullet}$ are far from being classified.

The symmetric decomposition is induced from a filtration on $\operatorname{gr}(A)$ obtained from the Loewy filtration $(0:\mathfrak{m}^{s+1-i})_{i\geq 0}$ on $A$. The associated graded of this filtration is a self-dual $\operatorname{gr}(A)$-module $\bigoplus_{\delta=0}^{s-2}\mathcal{Q}(\delta)$, see §2.2 for details.

The passage from $A$ to the associated graded $\operatorname{gr}(A)$ can be realised by embedding $\operatorname{Spec}(A)\subseteq\mathbb{A}^{n}$ with support at the origin and as taking the torus limit at $t\to\infty$ for the usual action $t\cdot(x_{1},\ldots,x_{n})=(tx_{1},\ldots,tx_{n})$, see §4.5.2. This simple observation has lead, via the Białynicki-Birula decomposition, to surprisingly significant advances in investigation of finite algebras and their smoothability, see [Jel19].

In this work we realise $\bigoplus_{\delta=0}^{s-2}\mathcal{Q}(\delta)$ as a limit of a Gorenstein $\operatorname{Spec}(A)\subseteq\mathbb{A}^{n}$ in the Iarrobino scheme, see Theorem 4.27 and the picture below.

\lim_{t\to\infty}t\cdot(A,[q_{0}])=(\mathcal{Q}(\bullet),[q_{\bullet}])$$\lim_{t\to\infty}t\cdot[A]=[\operatorname{gr}(A)]$$\operatorname{Iar}_{d}(X)$$\operatorname{Hilb}_{d}(X)$$\tau_{X}$$\mathbb{G}_{\mathrm{m}}-orbit closures

We hope that new results can be obtained using Białynicki-Birula decompositions on $\operatorname{Iar}_{d}(\mathbb{A}^{n})$.

1.2.4. Enumerative geometry of the Iarrobino scheme

Enumerative geometry of Hilbert schemes in low dimensions is a vast and incredibly beautiful subject. For the case of surfaces, the references [Nak99, Göt94, HL10, Nak16, OP10] are classical. Here, thanks to the smoothness of $\operatorname{Hilb}_{d}(X)$, no virtual structure is necessary. In the case of threefolds, the Hilbert scheme ceases to be smooth and the virtual structure coming from a symmetric perfect obstruction theory is essential [Beh09, BF08], as is, at least for $X$ toric, virtual localisation. A celebrated result is the GW-DT correspondence [MNOP06a, MNOP06b]. A gentle introduction to the subject, with many references, is [Ric22]. Finally, for 4-folds, there is the Oh-Thomas theory [OT23], and Nekrasov’s “magnificent four” [Nek20], recently proven by Kool-Rennemo [KR25]. The Quot schemes of points, especially for smooth curves, also received much attention, see, for example, [FFFR11, Ric20, BFP20, MN23].

Needless to say, the above list is miserably incomplete. We believe that some or even many results above have (non-trivial) analogues for the Iarrobino scheme. This aspect will be hopefully investigated in subsequent work [JRS25], so here we only sketch the emerging picture.

In the case of a curve $X$, the Iarrobino-to-Hilbert map $\tau=\tau_{\operatorname{Iar}_{d}(X)}\colon\operatorname{Iar}_{d}(X)\to\operatorname{Hilb}_{d}(X)$ is $\mathcal{O}$-connected (see Theorem 4.18) and actually $R\tau_{*}\mathcal{O}_{\operatorname{Iar}_{d}(X)}=\mathcal{O}_{\operatorname{Hilb}_{d}(X)}$, see [JRS25]. In this case, the counts on the Iarrobino scheme are more interesting than on the Hilbert scheme primarily because we have a much richer variety with more possible insertions. For example for $X=\mathbb{A}^{1}$, in [JRS25] we obtain

[TABLE]

where $t$ is the equivariant parameter for the usual $\mathbb{G}_{\mathrm{m}}$-action on $\mathbb{A}^{1}$, while for $i=1,\ldots,d-1$ the bundle $L_{i,d}$ on $\operatorname{Iar}_{d}(\mathbb{A}^{1})$ is the pullback of $L_{i}$ discussed in relation to characteristic numbers in §1.2.2 above, and $L_{0,d}=\mathcal{O}_{\operatorname{Iar}_{d}(\mathbb{A}^{1})},L_{d,d}$ are defined likewise.

For surfaces, at least for $\mathbb{A}^{2}$, the Iarrobino scheme admits a perfect obstruction theory different from the one on the Hilbert scheme and the results for curves extend to this case, but with the structure sheaf replaced by the virtual structure sheaf [JRS25].

1.2.5. Additional motivation from commuting matrices

In the case $X=\mathbb{A}^{n}$, the Hilbert scheme of points $d$ points admits a well-known description known as the ADHM construction [Nak99, §2]. Fix a $d$-dimensional vector space $V$. Then $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ is the free quotient of

[TABLE]

by the natural $\operatorname{GL}(V)$-action. This description is quite useful in enumerative geometry, yielding the global Kuranishi chart for $n=2$, and in moduli spaces, as it tightly links the Hilbert and Quot schemes to the variety of commuting matrices [JŠ22].

It is natural to ask what happens when we replace $\operatorname{End}(V)$ by some classical subalgebra. Here we discuss the orthogonal group. More precisely, let $V$ come with a fixed full rank quadric $Q$ and let $\widetilde{O}(V,Q)\subseteq\operatorname{GL}(V)$ be the stabiliser of $[Q]\in\mathbb{P}\operatorname{Sym}^{2}(V)$, so that $\widetilde{O}(V,Q)=O(V,Q)\cdot\mathbb{G}_{\mathrm{m}}$, where $O(V,Q)$ is the usual orthogonal group and $\mathbb{G}_{\mathrm{m}}\subseteq\operatorname{GL}(V)$ are the scalar matrices. The quadric $Q$ identifies $V$ with $V^{\vee}$ and in particular allows us to speak about symmetric matrices $\operatorname{End}^{\mathrm{sym}}(V)\subseteq\operatorname{End}(V)$. We have the following analogue:

Proposition 1.9**.**

The oriented Gorenstein locus $\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(\mathbb{A}^{n})$ is a free quotient of

[TABLE]

by the natural $\widetilde{O}(V,Q)$-action.

As we will prove in [JRS25], the $\operatorname{Iar}_{d}(\mathbb{A}^{n})$ admits a similar description, where the commuting matrices are no longer arbitrary, but taken from the compatible bundle on $\operatorname{CQ}(V)$, which we introduce in §3.4. Correspondingly, the commutation condition is also restricted: the commutator lies in the anticompatible bundle.

Acknowledgements

The author is very grateful to Michel Brion, Maciej Dołęga, Martijn Kool, Andrea Ricolfi, Reinier Schmiermann, and Jarosław Wiśniewski for enlightening discussions. I additionally thank Andrea and Reinier for their insights and questions shared during our discussions. I thank Michel Brion also for directing me to [DCS99]. Thanks to Alessio Sammartano and Richard Thomas for helpful comments. I thank Tony Iarrobino for permission to baptise $\operatorname{Iar}_{d}(X)$ using his name. Finally, thanks to the organisers of Hilbert scheme of points day in Berlin, Geometry and Commutative algebra workshop in Milano, Enumerative geometry of the Hilbert scheme of points in Les Diablerets, and the MFO workshop Recent Trends in Algebraic Geometry for giving me the possibility to present this work.

2. Preliminaries

We work over a base field $\Bbbk$. We assume that $2\in\Bbbk$ is invertible.

2.1. Duality for zero-dimensional modules

In this section we collect elementary, but useful, facts about duality. We use the word finite to mean finite rank over $\Bbbk$. For a $\Bbbk$-vector space $V$, we denote by $V^{\vee}$ the dual space. If $S$ is a $\Bbbk$-algebra and $M$ is an finite $S$-module, then the dual vector space $M^{\vee}$ is also an $S$-module via contraction:

[TABLE]

for all $s\in S$, $\varphi\in M^{\vee}$, $m\in M$. For a finite algebra $A$, the $A$-module $A^{\vee}$ is the dualising module, see for example [Eis95, Proposition 21.4], and we write it as $\omega_{A}$.

We say that $M$ is self-dual if there is an isomorphism $\varphi\colon M\to M^{\vee}$ of $S$-modules. If this is the case, then $\varphi^{\top}\colon M^{\vee}\to(M^{\vee})^{\vee}\simeq M$ is another isomorphism of $S$-modules. We say that $\varphi$ is symmetric if $\varphi=\varphi^{\top}$. An isomorphism $\varphi$ comes with an associated nondegenerate $\Bbbk$-bilinear map

[TABLE]

that satisfies the condition $\Phi(sm_{1},m_{2})=\Phi(m_{1},sm_{2})$ for all $s\in S$, $m_{1},m_{2}\in M$. We refer to this condition as $S$-linearity. The map is obtained by $\Phi(m_{1},m_{2})=\varphi(m_{1})(m_{2})$. Moreover, the map $\Phi$ is symmetric if and only if $\varphi$ is symmetric.

Example 2.2** (quadrics from functional).**

Let $A$ be a zero-dimensional $\Bbbk$-algebra. The $A$-module $A$ is self-dual if and only if $A$ is Gorenstein. Moreover, every isomorphism $\varphi\colon A\to\omega_{A}$ is symmetric. Indeed, let $\alpha:=\varphi(1)\in A^{\vee}$, then

[TABLE]

which is symmetric since $A$ is commutative. We see that $\varphi$ is completely determined by $\alpha$. Conversely, a functional $\alpha$ yields an isomorphism $\varphi$ if and only if $\alpha$ is nonzero on a socle element of every local algebra of $A$. The isomorphism $\varphi^{-1}$ gives rise to a quadric on $A$.

2.2. Iarrobino’s symmetric decomposition

Iarrobino’s symmetric decomposition of the Hilbert function of a Gorenstein algebra appears in particular in [Iar94]. It plays the role of (refined) Hilbert function in the Gorenstein setup and it is widely used [ER12, BR13, RS00, ER21, IMS22, CJN15, IMM21]. We recall it below.

Let $(A,\mathfrak{m},\Bbbk)$ be a local Gorenstein $\Bbbk$-algebra and fix a symmetric nondegenerate $A$-linear pairing $\Phi\colon A\times A\to\Bbbk$. For every $i$, $j$, the pairing descends to a nondegenerate pairings

[TABLE]

Let $s$ be the largest integer such that $\mathfrak{m}^{s}\neq 0$, it is called the socle degree of $A$. Following [Iar94, §1], we denote for every $\delta=0,1,\ldots,s-2$ the subquotient

[TABLE]

and $\mathcal{Q}(\delta):=\bigoplus_{i}(\mathcal{Q}(\delta))_{i}$. For every $\delta$, the linear space $\mathcal{Q}(\delta)$ is actually a graded $\operatorname{gr}(A)$-module and (2.3) yields a canonical isomorphism $\mathcal{Q}(\delta)\to\mathcal{Q}(\delta)^{\vee}[-(s-\delta)]$ of graded $\operatorname{gr}(A)$-modules, where $[-(s-\delta)]$ denotes the shift in degree by $-(s-\delta)$. Therefore, the graded module

[TABLE]

is self-dual (but not graded self-dual).

Let $\Delta_{\delta}$ be the Hilbert function of $\mathcal{Q}(\delta)$. Applying Hilbert function to the duality above, we obtain $\Delta_{\delta}(i)=\Delta_{\delta}(s-i-\delta)$ for every $0\leq i\leq s-\delta$ and $\Delta_{\delta}(i)=0$ for $i$ outside this range. Moreover, we have

[TABLE]

This is Iarrobino’s symmetric decomposition of the Hilbert function.

Example 2.4**.**

Let $A=\Bbbk[x,y]/(xy,x^{2}-y^{3})$. The associated graded is $\operatorname{gr}(A)=\Bbbk[x,y]/(xy,x^{2},y^{4})$ and so $H_{A}=(1,2,1,1)$ and $s=3$. The algebra $\operatorname{gr}(A)$ is not Gorenstein.

The element $x\in A$ lies in $\mathfrak{m}\cap(0:\mathfrak{m}^{2})$ and yields a nonzero class in $(\mathcal{Q}(1))_{1}$, in fact $\mathcal{Q}(1)=\left\langle x\right\rangle$ is spanned by this class. The space $\mathcal{Q}(0)$ is $\left\langle 1,y,y^{2},y^{3}\right\rangle$, so that as $\operatorname{gr}(A)$-modules, we have $\mathcal{Q}(0)\simeq\frac{\operatorname{gr}(A)}{(x)}$ and $\mathcal{Q}(1)\simeq\frac{\operatorname{gr}(A)}{(x,y)}$.

For quotients of $\Bbbk[x,y]$, it is true in general that $\mathcal{Q}(\bullet)$ are principal $\operatorname{gr}(A)$-modules [Iar94, §2]. This becomes false in more variables.

While the example above perhaps looks innocent, there are many novel phenomena that occur for $\Delta_{\bullet}$, see [IMM21] for some of them. Compared to the usual theory of Hilbert functions, much less is known. For example, the Hilbert functions of Gorenstein quotients of $\Bbbk[x,y,z]$ are not classified and neither are their symmetric decompositions. See [JMR23] for a very partial result.

In Section 4.5 we will give a geometric interpretation of the passage from $A$ to $\mathcal{Q}(\bullet)$ as a $\mathbb{G}_{\mathrm{m}}$-limit on the Iarrobino scheme; this is the motivation for the name.

2.2.1. The case of modules

Self-dual modules received much less attention than Gorenstein algebras, yet there are interesting results [Kun11]. Iarrobino’s symmetric decomposition has been generalised to self-dual modules [Woj24]. The argument is smooth once one chooses the correct generalisation of $(0:\mathfrak{m}^{i})$.

Let $M$ be a zero-dimensional self-dual module over a local $\Bbbk$-algebra $(A,\mathfrak{m},\Bbbk)$. For an ideal $I\subseteq A$ we define $(0:_{\scriptscriptstyle M}I)$ to be the set of elements of $M$ annihilated by $I$. Let $s$ be the largest integer such that $\mathfrak{m}^{s}M\neq 0$. We define

[TABLE]

Then, as proved in [Woj24, Theorem 1.1], the graded $\operatorname{gr}(A)$-module $\mathcal{Q}(\delta)$ is self-dual:

[TABLE]

and all other general properties from the algebra case generalise to this case. Again, we will see in §4.5 that the passage to $\mathcal{Q}(\bullet)$ is realised by a $\mathbb{G}_{\mathrm{m}}$-limit on the Quot analogue of the Iarrobino scheme.

3. Completed quadrics

In this chapter we discuss the variety of completed quadrics, which plays for the Iarrobino scheme a role similar to that played by the Grassmannian for the Quot scheme. We work in the generality of vector bundles, rather than vector spaces, so officially our source of reference is [TK88] and, for §3.3, also [Tyr56]. Still, some of the observations below do not appear explicitly in these sources (or, more generally, in the literature) and we need to do a comprehensive presentation.

Let $\mathcal{V}$ be a locally free sheaf of rank $d$ over a scheme $B$. The variety of completed quadrics on $\mathcal{V}$ is a smooth projective morphism

[TABLE]

which we recall more precisely below. The variety $\operatorname{CQ}(\mathcal{V})$ commutes with base change (by construction, see §3.2 below, see also [TK88]): for any morphism $\varphi\colon B^{\prime}\to B$, we have a cartesian diagram

[TABLE]

Therefore, below, all interesting features are present already in the case $B=\operatorname{Spec}(\Bbbk)$. In this case, the variety of completed quadrics is classical and can be found for example in [Tyr56, Lak87, DCP83, TK88, DCS99, Tha99]. It has plenty of applications to classical enumerative geometry, see for example the derivation of the number 3264 in [EH16, §8.2]. It is also a prominent example of a wonderful compactification.

The theory is very rich and beautifully connects various aspects such as enumerative geometry, moduli spaces and representation theory. It is also rather involved, hence below we will include ample explanations. We discuss $\operatorname{CQ}(\mathcal{V})$ from three angles.

1. (1)

description of $\Bbbk$-points, 2. (2)

construction and an explicit open affine atlas, 3. (3)

representation theoretic aspects.

In parallel with recalling the theory of $\operatorname{CQ}(\mathcal{V})$, we introduce two vector bundles on this variety, which we name the compatible and anticompatible bundle. They will be essential for the construction of the Iarrobino scheme in Section 4. Frequently, $\operatorname{CQ}(V)$ is called the variety of complete quadrics. We prefer the name completed to avoid confusion with the notion of a complete variety.

3.1. Points of $\operatorname{CQ}(\mathcal{V})$

In this section we describe the $\Bbbk$-points of $\operatorname{CQ}(\mathcal{V})$ to give the reader a feel of its geometry. For a nonzero quadric $q\in\operatorname{Sym}^{2}(V)$ on some vector space $V$, we denote by $[q]\in\mathbb{P}\operatorname{Sym}^{2}(V)$ its class up to rescaling. Since $2$ is invertible, we also identify such a $q$ with a $\Bbbk$-linear map $q\colon V^{\vee}\to V$ that satisfies $q^{\top}=q$ and denote by $\operatorname{coker}(q)$ the space $V/\operatorname{im}(q)$.

A $\Bbbk$-point of $\operatorname{CQ}(\mathcal{V})$ is just a $\Bbbk$-point of $\operatorname{CQ}(\mathcal{V}|_{b})$ for some $b\in B(\Bbbk)$, hence we assume $B=\operatorname{Spec}(\Bbbk)$ and write $V$ instead of $\mathcal{V}$. We have the following description of $\Bbbk$-points

[TABLE]

where every $q_{0}$, $q_{1}$, $q_{2}$, …is nonzero and the sequence concludes (after at most $d=\dim_{\Bbbk}V$ steps) with a full rank quadric. We will abbreviate the sequence $([q_{0}],[q_{1}],\ldots)$ to $[q_{\bullet}]$. For indexing convenience we assume $q_{-1}=0$, so that $\ker q_{-1}=V^{\vee}$ and $\operatorname{coker}q_{-1}=V$.

Definition 3.2**.**

A $\Bbbk$-point $[q_{\bullet}]$ is called a broken quadric. For $[q_{\bullet}]\in\operatorname{CQ}(V)$, let $F_{i}:=\ker q_{i-1}$, so that $F_{0}=V^{\vee}$. The sequence

[TABLE]

is the flag associated to $[q_{\bullet}]$.

For a point $[q_{\bullet}]$ with flag $F_{\bullet}$, for each $i\geq 0$ the quadric $q_{i}$ induces a full rank quadric on $F_{i}/F_{i+1}$.

The $\Bbbk$-points above lie on the spectrum between the following two extreme cases presented below.

Example 3.3** (main stratum).**

If $q_{0}$ has full rank, then there are no other quadrics and the flag is trivial. In this way, the variety $(\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}})(\Bbbk)$ of full rank quadrics up to rescaling is contained in $\operatorname{CQ}(V)(\Bbbk)$. We will see in §3.2 that this inclusion comes from an open embedding.

Example 3.4** (most degenerate stratum).**

Suppose now that $q_{0}$, $q_{1}$, …all have rank one, so that the last quadric is $q_{d-1}$. In this case, the classes $[q_{0}]$, $[q_{1}]$, …are determined by the flag. For example if $F_{1}:=\ker q_{0}$, then the exact sequence

[TABLE]

dualises to

[TABLE]

so $\operatorname{coker}q_{0}=F_{1}^{\vee}$ and so $q_{0}$ is the unique (up to rescaling) $\Bbbk$-linear map $q_{0}\colon V^{\vee}\to V$ with kernel $F_{1}$ and cokernel $F_{1}^{\vee}$. Let $F_{i+1}:=\ker q_{i}$ for $i=0,1,\ldots$, then the associated flag

[TABLE]

is a full flag on $V$. It will follow from the construction that in this way the flag variety becomes a closed subvariety of $\operatorname{CQ}(V)$.

We now describe sets which will be the $\Bbbk$-points of the compatible and anticompatible bundles. In contrast with the above, this part is new; it is not present in the literature.

We keep the notation that $V$ is a rank $d$ vector space over $\Bbbk$.

Definition 3.5** (compatibility).**

Let $x\in\operatorname{End}(V)$ and $[q_{\bullet}]\in\operatorname{CQ}(V)$. Let $F_{\bullet}:=\ker q_{\bullet-1}$ be the flag associated to $[q_{\bullet}]$ and let $x^{\top}\in\operatorname{End}(V^{\vee})$ be the dual endomorphism. We say that $x$ and $[q_{\bullet}]$ are compatible if both conditions below hold

1. (1)

the endomorphism $x$ preserves the flag $F_{\bullet}$, that is, $x^{\top}(F_{i})\subseteq F_{i}$ for every $i\geq 0$. 2. (2)

for every $i\geq 0$, the induced endomorphism $\bar{x}:=x^{\top}|_{F_{i}/F_{i+1}}\in\operatorname{End}(F_{i}/F_{i+1})$ is symmetric with respect to the full rank quadric $\bar{q}:=q_{i}|_{F_{i}/F_{i+1}}$, that is

[TABLE]

Compatible endomorphisms for a given $[q_{\bullet}]$ form a $\Bbbk$-vector subspace of $\operatorname{End}(V)$.

Definition 3.6**.**

Keeping the notation from Definition 3.5, we say that $x$ and $[q_{\bullet}]$ are anticompatible, if the conditions from the definition hold with symmetric replaced by antisymmetric.

We define the $\Bbbk$-points of the (anti)compacible bundle $\mathcal{C}\to\operatorname{CQ}(V)$, $a\mathcal{C}\to\operatorname{CQ}(V)$ to be the following subsets of $\operatorname{End}(V)\times\operatorname{CQ}(V)$:

[TABLE]

Example 3.9** (main stratum, continued).**

If $q_{0}$ has full rank, as in Example 3.3, then $x$ is compatible with $[q_{0}]$ if and only if $x$ is symmetric with respect to it. Same holds for anticompatibility. Hence $\mathcal{C}(\Bbbk)|_{[q_{0}]}$ is a vector space of dimension $\binom{d+1}{2}$, while $a\mathcal{C}(\Bbbk)|_{[q_{0}]}$ has dimension $\binom{d}{2}$.

Example 3.10** (most degenerate stratum, continued).**

When $q_{0}$, $q_{1}$, …have rank one, as in Example 3.4, an endomorphism $x$ is compatible with $[q_{\bullet}]$ if and only if it preserves the flag. Therefore, the fibre of $\mathcal{C}(\Bbbk)$ over $[q_{\bullet}]$ is the Borel Lie subalgebra in $\operatorname{End}(V)$. In down to earth terms, up to a choice of basis, the fibre identifies with upper triangular matrices. In particular, the dimension is $\binom{d+1}{2}$.

For an anticompatible $x$ however, the antisymmetry condition is not vacuous and such endomorphisms form the radical of the Borel Lie subalgebra, or, in down to earth terms, the strictly upper triangular matrices. The dimension is $\binom{d}{2}$.

Example 3.11**.**

Let us give an example where $[q_{\bullet}]$ varies. Let $d=2$ and $V=\left\langle e_{1},e_{2}\right\rangle$. Take the family $[q_{\bullet}(\lambda)]$ indexed by $\lambda\in\Bbbk$, given by $q_{0}=e_{0}^{2}+\lambda e_{1}^{2}$ for $\lambda\in\Bbbk$. For $\lambda\neq 0$ this is full rank, while at $\lambda=0$ we need to put $q_{1}=e_{1}^{2}$, which is unique up to scalars. Compatible operators $\mathcal{C}|_{[q_{\bullet}(\lambda)]}$ yield a one-parameter family of subspaces

[TABLE]

From the above description of the (would be) fibers of the compatible bundle, we conclude that this vector bundle (once constructed) provides degenerations of the space of symmetric matrices to the upper triangular ones. The construction itself is given in Proposition 3.30 below.

Now we describe the fibres $\mathcal{C}|_{[q_{\bullet}]}$, $a\mathcal{C}|_{[q_{\bullet}]}$ over an arbitrary $\overline{\Bbbk}$-point $q_{\bullet}$. Take a basis $(e_{1},\ldots,e_{d})$ of $V$ such that

[TABLE]

Proposition 3.13**.**

For $[q_{\bullet}]$ given in (3.12), the fibres $\mathcal{C}|_{[q_{\bullet}]}$ and $a\mathcal{C}|_{[q_{\bullet}]}$ are given by the block matrices

[TABLE]

where $*$ denote arbitrary matrices, while $A_{0},\ldots,A_{k}$ are symmetric (for the compatible fibre) or antisymmetric (for the anticompactible fibre), respectively. In particular, the fibres $\mathcal{C}|_{[q_{\bullet}]}$ and $a\mathcal{C}|_{[q_{\bullet}]}$ are orthogonal complements with respect to the trace pairing on $\operatorname{End}(V)$.

Proof.

The description is straightforward from the definition. Consider now matrices $x\in\mathcal{C}|_{[q_{\bullet}]}$ that has diagonal blocks $A_{0},\ldots,A_{k}$ and $y\in a\mathcal{C}|_{q_{\bullet}}$, that has diagonal blocks $A_{0}^{\prime},\ldots,A_{k}^{\prime}$. To prove that $\operatorname{tr}(xy)=0$, it is enough to prove that $\operatorname{tr}(A_{i}A_{i}^{\prime})=0$ for every $i=0,1,\ldots,k$.

We have $(A_{i}A_{i}^{\prime})^{\top}=-A_{i}^{\prime}A_{i}$, so that

[TABLE]

and so the trace is zero (since $1/2\in\Bbbk$). It remains to prove that

[TABLE]

but this is immediate: in the diagonal blocks, the symmetric and antisymmetric matrices have complementary dimensions, while each off-diagonal block appears twice: above and below the diagonal. ∎

3.2. Construction of $\operatorname{CQ}(\mathcal{V})$

We use the convention that the vector bundle associated to $\mathcal{V}$ is $\operatorname{Spec}_{B}\operatorname{Sym}\mathcal{V}^{\vee}$, so that the global sections of this vector bundle and of sheaf $\mathcal{V}$ are the same. By abuse of notation, we will use $\mathcal{V}$ to denote the vector bundle as well. Via standard constructions, we may in particular form the projective bundle $\mathbb{P}\operatorname{Sym}^{2}\mathcal{V}^{\vee}$ of quadrics.

There are few equivalent constructions of $\operatorname{CQ}(\mathcal{V})$, all involved. We recommend [TK88, Tha99] for a thorough discussion. For us it is most convenient to use the explicit description as a closed subscheme. We summarise it below. For any section $q\in\operatorname{Sym}^{2}\mathcal{V}$ let $\Lambda^{\bullet}q$ denote the tuple

[TABLE]

where $\Lambda^{i}q\in\operatorname{Sym}^{2}\Lambda^{i}\mathcal{V}$ is the $i$-th wedge power of $q$ viewed as a symmetric linear map $q\colon\mathcal{V}^{\vee}\to\mathcal{V}$. In local coordinates, the form $\Lambda^{i}q$ is given by $(\Lambda^{i}q)(e_{1}^{*}\wedge\ldots\wedge e_{i}^{*})^{2}=\det[q(e_{j}^{*}e_{k}^{*})]_{1\leq j,k\leq i}$.

Consider now the product

[TABLE]

of projective spaces, where the last factor is canonically isomorphic to $B$. Let $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V})^{\mathrm{full\,rk}}$ inside $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V})$ be the open subset of quadrics of full rank (that is, rank $d$). The map $q\mapsto\Lambda^{\bullet}q$ gives an embedding of $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V})^{\mathrm{full\,rk}}$ into $\mathcal{P}$. The points in the image are called exterior [TK88, p.258]. The main theorem, which may serve as a definition, is the following.

Proposition 3.15** ([Lak87, TK88]).**

The variety of completed quadrics $\operatorname{CQ}(\mathcal{V})$ is the closure of the variety $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V})^{\mathrm{full\,rk}}$ in $\mathcal{P}$.

From the description, it is not clear why $\operatorname{CQ}(\mathcal{V})$ is smooth. On the positive side, the variety is projective by construction, so we may compute some examples of limits.

Example 3.16**.**

Let $V$ be a $\Bbbk$-vector space of dimension $4$ with basis $e_{1},e_{2},e_{3},e_{4}$. Consider a family of quadrics $q_{\lambda}:=e_{1}^{2}+\lambda e_{2}^{2}+\lambda^{2}(e_{3}^{2}+e_{4}^{2})$, for $\lambda\in\Bbbk$.

For $\lambda\neq 0$ the quadric $q_{\lambda}$ has full rank, so it yields an exterior point of $\operatorname{CQ}(V)$. Explicitly, this point is the triple

[TABLE]

Let us take the limit with $\lambda\to 0$ in $\mathcal{P}$. It is computed coordinate-wise. In our case the limit is the triple

[TABLE]

The first factor is the “usual” limit in $\mathbb{P}\operatorname{Sym}^{2}V$, but the remaining ones contain much more information.

3.2.1. Points of completed quadrics in the exterior power

construction

In this subsection we compare the description of $\Bbbk$-points given in §3.1 and the one coming from the construction. This is mostly for completeness and to be able to compute limits on $\operatorname{CQ}(\mathcal{V})$. Again, we reduce from $\operatorname{CQ}(\mathcal{V})$ to $\operatorname{CQ}(V)$, where $V$ is a $d$-dimensional $\Bbbk$-vector space.

For a broken quadric $[q_{\bullet}]$ we define a $\Bbbk$-point of $\mathcal{P}$ defined in (3.14), as the tuple

[TABLE]

where $\det q_{i}:=\Lambda^{\operatorname{rk}q_{i}}q_{i}$ for every $i\geq 0$.

Lemma 3.18**.**

The tuple (3.17) lies in $\operatorname{CQ}(V)$.

Proof.

Fix a basis $e_{1},\ldots,e_{d}$ of $V$ such that every $F_{i}=\ker q_{i-1}$ is a subspace of the form

[TABLE]

and every $q_{i}$ is diagonal. Lift every $q_{i}$ to a quadric on $V$ such that $q_{i}$ is zero on elements $e_{j}$ which do not lie in $F_{i}$ or which do lie in $F_{i+1}$. Consider the quadric

[TABLE]

which is of full rank. We leave to the reader the check that (3.17) is the limit of $\Lambda^{\bullet}q(t)$ as $t$ approaches zero. ∎

Lemma 3.19**.**

Every $\Bbbk$-point of $\operatorname{CQ}(V)$ has the form (3.17), hence comes from a unique broken quadric.

Proof.

Let $p\in\operatorname{CQ}(V)$ be a $\Bbbk$-point and let $\overline{\Bbbk}$ be an algebraic closure of $\Bbbk$. Since full rank quadrics are dense in $\operatorname{CQ}(V)$, there is a curve germ

[TABLE]

such that the generic point $\eta$ maps to the full rank locus $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}\subseteq\operatorname{CQ}(V)$, and the special point maps to $p$. The induced map $\eta\colon\operatorname{Spec}\overline{\Bbbk}(\!(t)\!)\to\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ is a class of a quadric. We rescale it to the form $q(t)\in\operatorname{Sym}^{2}V\otimes_{\Bbbk}\overline{\Bbbk}[\![t]\!]$ with nonzero reduction modulo $t$. Since $\overline{\Bbbk}[\![t]\!]$ is a discrete valuation ring, we can diagonalise this quadric $q(t)$ using matrices in $\operatorname{GL}(V[\![t]\!])$, to obtain the form

[TABLE]

for $q_{0}=e_{1}^{2}+e_{2}^{2}+\ldots+e_{\operatorname{rk}q_{0}}^{2}$, $q_{1}=e_{1+\operatorname{rk}q_{0}}^{2}+\ldots+e_{\operatorname{rk}q_{1}+\operatorname{rk}q_{0}}^{2}$, …, where $0<\nu_{1}<\nu_{2}<\ldots$ is a finite, possibly empty, indexing sequence. Computing the limit of $q(t)$ in $\mathcal{P}$ we obtain the desired form (3.17) with quadrics having coefficients in $\overline{\Bbbk}$. Since $p$ is a $\Bbbk$-point, by the form (3.17), the quadric $q_{0}$ has coefficients in $\Bbbk$. Then, since the quadric $\det q_{0}\wedge q_{1}$ has coefficients in $\Bbbk$, also $q_{1}$ has coefficients in $\Bbbk$ and by induction we conclude that all quadrics have coefficients in $\Bbbk$. ∎

Example 3.20**.**

In the setup of Example 3.16, the limiting point corresponds to the broken quadric with a flag

[TABLE]

and quadrics $q_{0}=e_{1}^{2}$, $q_{1}=e_{2}^{2}$, $q_{2}=e_{3}^{2}+e_{4}^{2}$.

Example 3.21**.**

Let $q_{0}+t^{\nu}q_{1}\in\operatorname{Sym}^{2}(V)[\![t]\!]$ be a quadric on $\Bbbk[\![t]\!]$, which is generically of full rank. Lemma 3.19 might suggest that every $q_{0}+t^{\nu}q_{1}+t^{\nu+1}(\ldots)$ has the same limit as $q_{0}+t^{\nu}q_{1}$, but this is wrong. Indeed, take $V=\left\langle e_{1},e_{2},e_{3},e_{4}\right\rangle$ four-dimensional and $q(t):=e_{1}^{2}+e_{2}^{2}+t^{2}(e_{1}e_{3}+e_{2}e_{4})\in\operatorname{Sym}^{2}V[\![t]\!]$. This has limit $([e_{1}^{2}+e_{2}^{2}],\,[e_{3}^{2}+e_{4}^{2}])$. In contrast, the limit of the quadric $q(t)+t^{3}e_{3}^{2}$ is $([e_{1}^{2}+e_{2}^{2}],\,[e_{3}^{2}],\,[e_{4}^{2}])$.

Remark 3.22**.**

It follows from the explicit description (3.17) that not only $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ embeds into $\operatorname{CQ}(V)$, but even the bigger open locus $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{corank}\leq 1}$ of quadrics of rank $\geq\!\dim V-1$ embeds into $\operatorname{CQ}(V)$.

3.3. Tyrrell’s affine patches

Tyrrell [Tyr56] produces a beautiful explicit open cover of $\operatorname{CQ}(V)$ by affine spaces $\mathbb{A}^{\binom{d+1}{2}}$. The cover in practice is much more convenient that the full construction.

Each patch in the cover depends on a chosen full flag

[TABLE]

We use the notation $G_{\bullet}$ to stress that this will not be the flag associated to a broken quadric. We also fix a suitable basis $e_{\bullet}:=(e_{1},\ldots,e_{d})$ of $V$, such that $G_{d-i}=\left\langle e_{1}^{*},\ldots,e_{i}^{*}\right\rangle$ for every $i=1,2,\ldots,d-1$. The coordinates on the patch will depend on the choice of basis. The patch itself is independent of the choice of the basis, but depends on the choice of the flag.

Consider the open subset $\operatorname{Tyr}_{e_{\bullet}}\subseteq\operatorname{CQ}(V)$ which consists of the elements $(p_{1},\ldots,p_{d})\in\operatorname{CQ}(V)$ such that

[TABLE]

is nonzero for every $i=1,\ldots,d$. In other words, in bases, we consider symmetric matrices with nonvanishing top left principal minors. Fix an affine space of dimension $\binom{d+1}{2}-1$ with coordinates $y_{1},\ldots,y_{d-1}$, $x_{12}$, $\ldots$ ,$x_{1d}$, $x_{23}$, $\ldots$, $x_{2d}$, $\ldots$, $x_{d-1d}$. Let $U$ be the unipotent radical of the Borel group in $\operatorname{GL}(V^{\vee})$ which fixes the flag (3.23). In coordinates, we have

[TABLE]

that is, a generic strictly upper-triangular matrix. Let $D$ be the diagonal matrix with entries

[TABLE]

Consider the $i$-th wedge power $\Lambda^{i}D$. It is diagonal and, by construction of $D$, it has all entries divisible by the entry $y_{1}^{i-1}y_{2}^{i-2}\ldots y_{i-1}$ corresponding to $(e_{1}^{*}\wedge\ldots\wedge e_{i}^{*})^{2}$. Let $D^{(i)}\in\operatorname{Sym}^{2}\Lambda^{i}V$ be given by

[TABLE]

so that the entry of $D^{(i)}$ corresponding to $(e_{1}^{*}\wedge\ldots\wedge e_{i}^{*})^{2}$ is $1$. Finally, let

[TABLE]

Thanks to the unit entry, this gives a well defined map from $\operatorname{Spec}\left(\Bbbk[y_{1},\ldots,y_{d-1},x_{ij}\ |\ i<j]\right)$ to the product. When $y_{1}$, $y_{2}$, …, $y_{d-1}$ are all nonzero, the image is in the locus of full rank quadrics, hence the full image is contained in $\operatorname{CQ}(V)$.

Proposition 3.25** **(Tyrrell [Tyr56],

see also [TK88, Lemma (7.8)]).

The map

[TABLE]

is an open immersion and its image is exactly $\operatorname{Tyr}_{e_{\bullet}}$. Moreover, iterating over all the flags (3.23) this yields an open cover of $\operatorname{CQ}(V)$.

We can identify $\operatorname{Spec}\left(\Bbbk[y_{1},\ldots,y_{d-1},x_{ij}\ |\ i<j]\right)$ with $U\times\operatorname{Spec}(\Bbbk[y_{1},\ldots,y_{d-1}])$. In this way, the open immersion is $U$-equivariant, where $U$ is the group of strictly upper-triangular matrices.

The closure of $\operatorname{Spec}(\Bbbk[y_{1},\ldots,y_{d-1}])$ in $\operatorname{CQ}(V)$ is the permuhedral variety, which appears prominently in Huh’s work on Chow rings of matroids, see for example [Eur24], see also §1.2.2.

Each Tyrrell’s affine patch can be reconstructed as a dominant cell of the Białynicki-Birula decomposition [BB73] of $\operatorname{CQ}(V)$ for a $\mathbb{G}_{\mathrm{m}}$-action which is diagonal in coordinates $e_{1},\ldots.,e_{d}$ and has suitable weights. The patches appear also as translates of the image of map $\Gamma$ in [DCS99, Proposition 3.8].

3.4. Representation theory and the construction of

(anti)compatible bundles

Let $V$ be a vector space of dimension $d$, as usual. The variety $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ is homogeneous under the $\operatorname{PGL}(V)$-action and for a full rank quadric $q$, its stabiliser $O(q)\subseteq\operatorname{PGL}(V)$ is equal to $\operatorname{PGL}(V)^{\sigma}$ for an involution $\sigma$ of $\operatorname{PGL}(V)$. Indeed, if we use $q$ to identify $V$ and $V^{\vee}$, so that $q$ is diagonal, then

[TABLE]

so the involution is $\sigma(g)=(g^{\top})^{-1}$. In such a setup, the variety $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ admits a wonderful compactification constructed by de Concini and Procesi [DCP83]. A great survey of wonderful compactifications is [Pez18]. A source very well-adapted to our needs is a survey by Bălibanu [Băl23]. Both surveys assume that $\Bbbk$ has characteristic zero. Below, we explicitly construct the structures appearing in the wonderful compactification, hence we do not impose characteristic zero (but always assume characteristic $\neq 2$). Outside characteristic zero, a nice source from a more representation-theoretic perspective is [DCS99].

Let $\partial\!\operatorname{CQ}(V)=\operatorname{CQ}(V)\setminus\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ be the boundary divisor, with its reduced structure. This divisor splits into $d-1$ components $D_{1},\ldots D_{d-1}$, where $D_{i}$ parameterises broken quadrics $[q_{\bullet}]\in\operatorname{CQ}(V)$ such that “there is a break at $i$”, that is, there is an index $j$ such that

[TABLE]

In terms of Tyrrell’s patches (§3.3) the divisor $D_{i}$ is locally given by $y_{i}=0$. In particular, $\partial\!\operatorname{CQ}(V)$ is Cartier and has simple normal crossings.

It is natural to consider the logarithmic tangent sheaf $T_{\operatorname{CQ}(V),\,\partial\!\operatorname{CQ}(V)}$ on $\operatorname{CQ}(V)$, whose sections are vector fields tangent to $\partial\!\operatorname{CQ}(V)$, see [Băl23, §3]. Let $\operatorname{\mathfrak{sl}}V=\left\{x\in\operatorname{End}(V)\ |\ \operatorname{tr}(x)=0\right\}$ be the Lie algebra of $\operatorname{PGL}(V)$. The vector fields $\operatorname{\mathfrak{sl}}V\subseteq H^{0}(T_{\operatorname{CQ}(V)})$ preserve $\partial\!\operatorname{CQ}(V)$, hence we obtain a map

[TABLE]

Proposition 3.26** **(anticompactible bundle via

representations).

The map $\rho$ is surjective. Its kernel is a locally free sheaf of rank $\binom{d}{2}$ which corresponds to a bundle with $\Bbbk$-points as described in (3.8). This bundle is called the anticompatible bundle on $\operatorname{CQ}(V)$ and denoted by $a\mathcal{C}$, so that we have an exact sequence

[TABLE]

Proof.

We will check surjectivity on Tyrrell’s patches. We keep the notation from §3.3. By definition, the patch depends on a basis $e_{\bullet}=(e_{1},\ldots,e_{d})$ of $V$ and it has the form $\operatorname{Tyr}_{e_{\bullet}}=U\times\mathbb{A}^{d-1}$, where the coordinates on $U$ are $(x_{ij})_{1\leq i<j\leq d}$ and the coordinates on $\mathbb{A}^{d-1}$ are $y_{1},\ldots,y_{d-1}$. The restriction of the logarithmic tangent sheaf $(T_{\operatorname{CQ}(V),\,\partial\!\operatorname{CQ}(V)})|_{\operatorname{Tyr}_{e_{\bullet}}}$ is free, generated by $(\partial_{x_{ij}})_{1\leq i<j\leq d}$ and $(y_{i}\partial_{y_{i}})_{1\leq i\leq d-1}$. The free $U$-action on $\operatorname{Tyr}_{e_{\bullet}}$ yields an injection $\mathfrak{u}\hookrightarrow H^{0}(\operatorname{Tyr}_{e_{\bullet}},T_{\operatorname{CQ}(V),\,\partial\!\operatorname{CQ}(V)})$, whose image is the span

[TABLE]

The $U$-action is part of the $\operatorname{PGL}(V)$-action, so the $\mathfrak{u}\otimes_{\Bbbk}\mathcal{O}_{\operatorname{Tyr}_{e_{\bullet}}}$ part of the logarithmic tangent sheaf is in the image of $\rho$. Consider the Lie algebra $\mathfrak{t}$ of the diagonal torus in $\operatorname{PGL}(V)$. Fix the basis of $\mathfrak{t}$ given by

[TABLE]

for $i=1,\ldots,d-1$. To prove surjectivity of $\rho$, we can work modulo $\mathfrak{u}$, so it is enough to understand the action of $\mathfrak{t}$ on the universal quadric

[TABLE]

on $\mathbb{A}^{d-1}$. The element $(\lambda,1,\ldots,1,\lambda^{-1},1,\ldots,1)$ in the diagonal torus acts on (3.28) by multiplying the entry $y_{1}\cdot\ldots\cdot y_{i}\cdot(e_{i})^{2}$ by $\lambda$ and leaving all other entries intact. This implies that we have

[TABLE]

and surjectivity of $\rho$ follows. Thus the kernel $\ker\rho$ is locally free of rank $d^{2}-1-(\binom{d+1}{2}-1)=\binom{d}{2}$. We identify it with the corresponding vector bundle $\ker\rho\subseteq\operatorname{CQ}(V)\times\operatorname{End}(V)$. It remains to prove that $\Bbbk$-points of this bundle are exactly the anticompatible operators from Definition 3.6. The map $\rho$ and the notion of anticompatibility are both $U$-equivariant, so it is enough to prove this on $\mathbb{A}^{d-1}\hookrightarrow\operatorname{Tyr}_{e_{\bullet}}$. Take variables $a_{ij}$ for $1\leq i\leq j\leq d$, which will be the coordinates on our bundle and consider the subbundle of $\mathbb{A}^{d-1}\times\operatorname{End}(V)$ given by the transpose of the matrix

[TABLE]

Denote the matrix by $M_{a\mathcal{C}}\in\mathbb{A}^{d-1}\times\operatorname{End}(V^{\vee})$. Consider the quadric (3.28) and let $Q$ be the associated symmetric matrix. We have $M_{a\mathcal{C}}^{\top}Q=-QM_{a\mathcal{C}}$, since the $(i,j)$ entry of both sides is

[TABLE]

This identifies the $\ker\rho$ restricted to $\mathbb{A}^{d-1}$. To check that the subbundle above describes exactly the anticompatible endomorphisms, pick a broken quadric $[q_{\bullet}]$ in $\mathbb{A}^{d-1}$. Thanks to $M_{a\mathcal{C}}^{\top}Q=-QM_{a\mathcal{C}}$, we see that the endomorphisms preserve $\ker q_{0}$ and yield antisymmetric matrices in $V/\ker q_{0}$. Suppose that $q_{0}$ has rank $r<d$. This means that $y_{r}=0$. The matrix $M_{a\mathcal{C}}|_{y_{r}=0}$ is a $2\times 2$ block matrix. The lower-right block is the analogue of $M_{a\mathcal{C}}$ on the space $\left\langle e_{r+1},\ldots,e_{d}\right\rangle$. By induction, it is anticompatible with $\left([q_{1}],[q_{2}],\ldots\right)$ and the claim follows. ∎

Remark 3.29**.**

In characteristic zero, for every wonderful compactification, the corresponding map $\rho$ is surjective by a result of Pezzini [Pez18, Proposition 4.2], and the kernel of $\rho$ is described by Brion [Bri07, Proposition 2.1.2].

Proposition 3.30** (Compatible bundle).**

Consider the dual of the exact sequence (3.27) with an added trivial factor coming from $\operatorname{\mathfrak{gl}}V=\operatorname{\mathfrak{sl}}V\oplus\Bbbk$:

[TABLE]

The vector bundle corresponding to $T_{\operatorname{CQ}(V),\,\partial\!\operatorname{CQ}(V)}^{\vee}\oplus\mathcal{O}_{\operatorname{CQ}(V)}$ has rank $\binom{d+1}{2}$ and $\Bbbk$-points as described in (3.7). We call it the compatible bundle and denote it by $\mathcal{C}$.

Proof.

The kernel of the surjection $\mathcal{O}_{\operatorname{CQ}(V)}\otimes_{\Bbbk}\operatorname{\mathfrak{gl}}V\to a\mathcal{C}^{\vee}$ is, fibre by fibre, the orthogonal complement of $a\mathcal{C}$ with respect to the trace pairing. This is equal to the fibre of $\mathcal{C}$ by Proposition 3.13. ∎

Example 3.31**.**

In the case $\dim V=2$, we see that $\operatorname{CQ}(V)$ is isomorphic to $\mathbb{P}^{2}=\mathbb{P}\operatorname{Sym}^{2}V$. Fix coordinates $V=\left\langle e_{1},e_{2}\right\rangle$. For a quadric $q=\lambda_{11}e_{1}^{2}+2\lambda_{12}e_{1}e_{2}+\lambda_{22}e_{2}^{2}$, the element

[TABLE]

is anticompatible with $q$, so that $a\mathcal{C}$ corresponds to the sheaf $\mathcal{O}_{\mathbb{P}^{2}}(-1)$ and $T_{\operatorname{CQ}(V),\,\partial\!\operatorname{CQ}(V)}$ is $T_{\mathbb{P}^{2}}(-1)$. Using Proposition 3.30, we obtain that $\mathcal{C}$ comes from the sheaf $\mathcal{O}_{\mathbb{P}^{2}}\oplus\Omega_{\mathbb{P}^{2}}(1)$.

Remark 3.32**.**

The construction of $a\mathcal{C}$ and $\mathcal{C}$ generalise immediately to the relative setting of $\operatorname{CQ}(\mathcal{V})$: the surjectivity of $\rho$ is checked locally, hence we can assume that $\mathcal{V}$ is trivial and the proof above applies.

3.5. Duality

It may seem that the theory involves a choice: one can work on $\operatorname{CQ}(V)$ or on $\operatorname{CQ}(V^{\vee})$. In this section we prove that both viewpoints are canonically isomorphic. This is important, since each of the two choices is sometimes preferable.

A full rank quadric on a section of $\mathcal{V}$ induces one on $\mathcal{V}^{\vee}$ and vice versa, so the open subsets $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V})^{\mathrm{full\,rk}}$ and $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V}^{\vee})^{\mathrm{full\,rk}}$ are isomorphic. More explicitly, if $q\in\operatorname{Sym}^{2}\mathcal{V}^{\vee}$ is of full rank and we write it as a symmetric matrix $q\colon\mathcal{V}^{\vee}\to\mathcal{V}$, then $q^{-1}\colon\mathcal{V}\to\mathcal{V}^{\vee}$ is a symmetric matrix and it gives the corresponding element of $\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V}^{\vee})^{\mathrm{full\,rk}}$.

The main duality theorem (see [TK88, Introduction]) is that this isomorphism extends to an isomorphism

[TABLE]

The isomorphism is constructed as follows. Recall that $\mathcal{V}$ has rank $d$. Let $\det\mathcal{V}:=\Lambda^{d}\mathcal{V}$, then for $i=0,1\ldots,d$ we have isomorphisms

[TABLE]

given by the contraction of wedges and, correspondingly, an isomorphism

[TABLE]

This induces an isomorphism of ambient varieties

[TABLE]

The isomorphism extends the map $q\mapsto q^{-1}$ on full rank quadrics and $\operatorname{CQ}(\mathcal{V})$, $\operatorname{CQ}(\mathcal{V}^{\vee})$ are by definition the closures of this loci, so we obtain the following result.

Proposition 3.33** (duality, [TK88, (7.23)]).**

The isomorphism above restricts to an isomorphism $\iota_{\mathcal{V}}\colon\operatorname{CQ}(\mathcal{V}^{\vee})\to\operatorname{CQ}(\mathcal{V})$ such that $\iota_{\mathcal{V}^{\vee}}\circ\iota_{\mathcal{V}}$ is the identity.

Example 3.34**.**

In the setup of Examples 3.16, 3.20, the duality interchanges the flags

[TABLE]

and maps the quadrics $q_{0}=e_{1}^{2}$, $q_{1}=e_{2}^{2}$, $q_{2}=e_{3}^{2}+e_{4}^{2}$ to their inverses $(e_{1}^{*})^{2}$, $(e_{2}^{*})^{2}$, $(e_{3}^{*})^{2}+(e_{4}^{*})^{2}$, arranged in reversed order.

3.6. The functor of points

In this section we recall the functor of points of $\operatorname{CQ}(\mathcal{V})$. We will not use it below, however we believe that it should be stated for completeness. We follow Thorup-Kleiman [TK88, §7]; in their notation $\operatorname{CQ}(\mathcal{V})$ is $B_{\dim V}^{\mathrm{sym}}(\mathcal{V},\mathcal{V})$. They were the first to describe the functor of points of $\operatorname{CQ}(\mathcal{V})$. We recall their result below. As usual, let $\mathcal{V}$ be of rank $d$.

In [TK88] the authors consider also the nonsymmetric case and as result they use $u$ rather $q$ to denote their forms. In this subsection we keep their notation. The work [TK88] uses $S$ for the scheme which is the argument of the functor. We used $S$ above, so we use $T$. Let $T$ be a $B$-scheme and let $\mathcal{V}_{T}$ be the pullback of $\mathcal{V}$ via $T\to B$.

A section $u\in\mathbb{P}(\operatorname{Sym}^{2}\mathcal{V}_{T})$ corresponds to a map $\operatorname{Sym}^{2}\mathcal{V}_{T}^{\vee}\twoheadrightarrow L$, where $L$ is an invertible sheaf on $T$. For every $i\geq 1$ this induces a form

[TABLE]

which is locally given by $(\Lambda^{i}u)(e_{1}^{*}\wedge\ldots\wedge e_{i}^{*})^{2}=\det[u(e_{j}^{*}e_{k}^{*})]_{1\leq j,k\leq i}$. The image of $\Lambda^{i}u$ is the ideal of size $i$ minors of $u$. When the image of (3.35) is an invertible sheaf on $T$ for every $1\leq i\leq d$, then we call the sequence $(u,\Lambda^{2}u,\ldots,\Lambda^{d}u)$ a $d$-divisorial symmetric form on $T$, see [TK88, (3.1)]. If $T$ and $B$ are both equal to $\operatorname{Spec}\Bbbk$, then the condition boils down to $u$ having full rank and we recover the open embedding $\mathbb{P}(\operatorname{Sym}^{2}V)^{\mathrm{full\,rk}}$ described in §3.2.

Finally, following [TK88, (7.15), (4.1)], a projectively $d$-complete symmetric form is a sequence $u_{1},\ldots,u_{d}$, where $u_{i}\colon\operatorname{Sym}^{2}\Lambda^{i}\mathcal{V}^{\vee}_{T}\twoheadrightarrow\mathcal{M}_{i}$ is a surjection of $\mathcal{O}_{T}$-vector bundles onto a line bundle $\mathcal{M}_{i}$ on $T$ and such that for every point of $T$, there is its open neighbourhood $U\subseteq T$, a map $\sigma\colon U\to T^{\prime}$ of schemes and a $d$-divisorial symmetric form $v$ on $T^{\prime}$ such that $u_{i}|U\simeq\sigma^{*}(\Lambda^{i}v)$ for every $i$.

Theorem 3.36** **([TK88, (7.15),

(4.3)]).

The variety $\operatorname{CQ}(\mathcal{V})$ represents the functor of projectively $d$-complete symmetric forms.

4. Compatible families and the Iarrobino scheme

With appropriately developed theory of $\operatorname{CQ}(\mathcal{V})$, the definition of compatible families and the Iarrobino scheme of a scheme $X$ becomes natural, but still quite involved technically, especially since it makes sense to work in maximal generality, similarly to how one introduces the Quot scheme to define the Hilbert scheme [FGI*+*05].

4.1. Points of compatible quadrics

Let $S$ be a $\Bbbk$-algebra. Let $V$ be a $d$-dimensional vector space that is an $S$-module. Suppose that $[q_{\bullet}]\in\operatorname{CQ}(V)$ is compatible with the action of every element of $S$. Let $F_{\bullet}=\ker q_{\bullet-1}$ be the flag associated to $[q_{\bullet}]$. As a consequence of compatibility, the subspaces and quadrics become $S$-linear, as follows.

Proposition 4.1**.**

The following are equivalent for a quadric $[q_{\bullet}]\in\operatorname{CQ}(V)$.

1. (1)

$[q_{\bullet}]$* is compatible with (every element of) the ring $S$.* 2. (2)

For every $i\geq 0$, the subspace $F_{i}$ is an $S$-submodule of $V^{\vee}$ and the quadric $q_{i}$ induces an isomorphism

[TABLE]

of $S$-modules that satisfies $q_{i}=q_{i}^{\top}$. Therefore, the associated graded $S$-module $\bigoplus_{i\geq 0}F_{i}/F_{i+1}$ is self-dual via $q_{\bullet}$.

Proof.

By Definition 3.5, the spaces $F_{\bullet}$ are preserved by the action of any $x\in S$, hence are submodules. Fix $i\geq 0$. From the same definition, we see that $q_{i}$ satisfies $q_{i}(x\cdot(-),-)=q_{i}(-,x\cdot(-))$ for every $x\in S$. This is tantamount to saying that the linear isomorphism (4.2) is $S$-linear. ∎

Example 4.3** (The case $S=\Bbbk[x]/(x^{d})$).**

This example will be useful later. Let $S=\Bbbk[x]/(x^{d})$ and $V=S^{\oplus 1}$ so that $V^{\vee}\simeq S^{\oplus 1}$. Take any $[q_{\bullet}]$ compatible with $x$. By Proposition 4.1 each subspace $F_{i}\subseteq\Bbbk[x]/x^{d}$ is an ideal, so it has the form $(x^{\nu_{i}})/(x^{d})$ for $\nu_{i}:=\dim V^{\vee}/F_{i}$.

A quadric $q_{i}$ is defined on $(x^{\nu_{i}})/(x^{\nu_{i+1}})\simeq\Bbbk[x]/(x^{\nu_{i+1}-\nu_{i}})$, where $\nu_{0}=d$. The algebra $A_{i}:=\Bbbk[x]/(x^{\nu_{i+1}-\nu_{i}})$ is Gorenstein and $q_{i}$ has full rank, so, by Example 2.2, the quadric $q_{i}$ on $A_{i}$ has the form

[TABLE]

for a functional $\alpha_{i}$ which is nonzero on the class of $x^{\nu_{i+1}-\nu_{i}-1}$. Since we consider $q_{i}$ up to scalars, we may even assume that $\alpha(x^{\nu_{i+1}-\nu_{i}-1})=1$. In this case we obtain a $(\nu_{i+1}-\nu_{i}-1)$-dimensional affine space parameterising possible $q_{i}$. This completes the set-theoretic description. We will see in Theorem 4.18 that the whole compatible locus is irreducible. A willing reader can do this now by hand.

Example 4.4**.**

Let $S=\Bbbk\times\ldots\times\Bbbk$ be the algebra which is the product of $d$ copies of the field $\Bbbk$. Let $V=S^{\oplus 1}$ with basis $e_{1},\ldots,e_{d}$. From Example 2.2, we see that the full rank quadrics compatible with $S$ correspond to full rank diagonal matrices. The closure of such matrices in $\operatorname{CQ}(V)$ is the permutohedral toric variety. It will follow from Theorem 4.18 that this closure coincides with the variety of all broken quadrics compatible with $S$.

4.2. Compatible quadrics scheme-theoretically, affine version

The compatibility of a single operator $x\in\operatorname{End}(V)$ with a given $[q_{\bullet}]\in\operatorname{CQ}(V)$ was given in Definition 3.5. By Proposition 3.30, it can be rephrased by saying that $x\in\operatorname{End}(V)$ lies in $\mathcal{C}|_{[q_{\bullet}]}\subseteq\operatorname{End}(V)$. The latter formulation immediately generalises to more operators and vector bundles.

Let $X=\operatorname{Spec}(S)$ be an affine $\Bbbk$-scheme and $\mathcal{V}$ be a coherent $\mathcal{O}_{X\times B}$-module which is locally free of rank $d$ as a $\mathcal{O}_{B}$-module. The $\mathcal{O}_{X\times B}$-module structure induces a morphism $\alpha\colon S\times B\to\operatorname{End}(\mathcal{V})$ of vector bundles over $B$, where $\operatorname{End}(\mathcal{V})$ is the vector bundle corresponding to the endomorphisms of the locally free $\mathcal{O}_{B}$-module $\mathcal{V}$ and $S=H^{0}(\mathcal{O}_{X})$ so that $S\times B$ is a trivial vector bundle (usually of infinite rank) on $B$. The morphism pulls back to a morphism

[TABLE]

of vector bundles on $\operatorname{CQ}(\mathcal{V})$.

Definition 4.5** **(Compatibility, relative affine

version).

The scheme of compatible quadrics on $\mathcal{V}$ is a closed subscheme $\operatorname{CCQ}(\mathcal{V})\hookrightarrow\operatorname{CQ}(\mathcal{V})$ given by the condition that $\alpha_{\operatorname{CQ}}$ factors through the compatible bundle $\mathcal{C}\hookrightarrow\operatorname{End}(\mathcal{V})\times_{B}\operatorname{CQ}(\mathcal{V})$.

Formally, the scheme structure on $\operatorname{CCQ}(\mathcal{V})$ is given by the vanishing of the following morphism of vector bundles over $\operatorname{CQ}(\mathcal{V})$:

[TABLE]

The right-hand-side is a vector bundle of rank $\binom{d}{2}$ on $\operatorname{CQ}(\mathcal{V})$, see Proposition 3.30. Locally on $\operatorname{CQ}(\mathcal{V})$, it is trivial, so that the right hand side is isomorphic to $\operatorname{CQ}(\mathcal{V})\times\mathbb{A}^{\binom{d}{2}}$ and the $(\dim_{\Bbbk}S)\cdot\binom{d}{2}$ coordinate functions cut out the scheme $\operatorname{CCQ}(\mathcal{V})$. The zero sets of these local equations agree on overlaps, giving the scheme structure globally. If $S$ is a $\Bbbk$-algebra generated by $n$ elements, then only the multiplication by these elements needs to be taken into account and we reduce to $n\cdot\binom{d}{2}$ equations.

Definition 4.6**.**

Let $\mathcal{U}\to\operatorname{Hilb}_{d}(X)$ be the Hilbert scheme of $d$ points on $X$ with its universal family. The scheme $\operatorname{CCQ}(\mathcal{U})\to\operatorname{Hilb}_{d}(X)$ is called the Iarrobino scheme of $d$ points on $X$ and denoted $\operatorname{Iar}_{d}(X)$.

Proposition 4.7** (Points of the Iarrobino scheme).**

The $\Bbbk$-points of $\operatorname{Iar}_{d}(X)$ correspond bijectively to pairs $[Z_{\bullet}],[q_{\bullet}]$, where $Z_{\bullet}$ is a flag

[TABLE]

of zero-dimensional subschemes of $X$ such that $\deg Z_{0}=d$, and $[q_{\bullet}]=\left([q_{0}],[q_{1}],\ldots\right)$ are classes of $\mathcal{O}_{X}$-module isomorphisms

[TABLE]

which satisfy $q_{i}^{\top}=q_{i}$.

We find it remarkable that both the length of the tuple $Z_{0}\supsetneq Z_{1}\supsetneq\ldots$ and the degrees of $Z_{1},Z_{2},\ldots$ are allowed to vary.

Proof.

We write $\mathcal{O}_{Z_{i}}$ for $H^{0}(\mathcal{O}_{Z_{i}})$ for $i=0,1\ldots$.

By construction of $\operatorname{Iar}_{d}(X)$, its $\Bbbk$-points correspond bijectively to broken quadrics $[q_{\bullet}]$ on $\mathcal{O}_{Z_{0}}$ compatible with $\mathcal{O}_{Z_{0}}$-module structure, where $Z_{0}\subseteq X$ ranges over zero-dimensional subschemes of degree $d$.

We have a sequence of cokernels

[TABLE]

and we define, for $i\geq 1$, the subscheme $Z_{i}$ as given by the cokernel $\mathcal{O}_{Z_{0}}\to\operatorname{coker}q_{i-1}$, so that we have

[TABLE]

and every quadric $q_{i}$ yields a symmetric isomorphism $(I_{i+1}/I_{i})^{\vee}\simeq\omega_{Z_{i}}/\omega_{Z_{i+1}}\to I_{i+1}/I_{i}$. ∎

Just as the Hilbert schemes of points generalise to Quot schemes of points, the Iarrobino scheme generalises to its Quot counterpart.

Definition 4.8**.**

Let $\mathcal{E}$ be a coherent sheaf on $X$ and consider the Quot scheme of points $\operatorname{Quot}_{d}(\mathcal{E})$ with its universal quotient sheaf

[TABLE]

The self-dual Quot scheme of $d$ points is $\operatorname{CQuot}_{d}(\mathcal{E})$ given by the compatibility locus $\operatorname{CCQ}(\mathcal{M})\to\operatorname{Quot}_{d}(\mathcal{E})$.

There is a Iarrobino-to-Hilbert map

[TABLE]

which forgets about $[q_{\bullet}]$. The fibre $\tau_{X}^{-1}(Z\subseteq X)$ is the scheme of quadrics compatible with $\mathcal{O}_{Z}$, hence is independent of the embedding of $Z$ into $X$.

Below, we investigate $\operatorname{Iar}_{d}(C)$ for a curve $C$ much more precisely. Before that, we extend the definition from the affine to quasi-projective setup.

4.2.1. Compatible quadrics, from affine to quasi-projective version

If $X$ is quasi-affine, open in an affine $\overline{X}$, then $\operatorname{Hilb}_{d}(X)$ is open in $\operatorname{Hilb}_{d}(\overline{X})$, hence we obtain $\operatorname{Iar}_{d}(X)$ open in $\operatorname{Iar}_{d}(\overline{X})$.

Let $X$ now be a quasi-projective $\Bbbk$-scheme and $\mathcal{V}$ a coherent $\mathcal{O}_{X\times B}$-module which is locally free of rank $d$ as a $B$-module. For every point $b\in B$, the support of $\mathcal{V}|_{b}$ is finite as a subset of $X\times b$, hence by quasi-projectivity, there exists a hypersurface $H\subseteq X$ such that $H\subseteq X$ does not intersect the image of the support of $\mathcal{V}|_{b}$. It follows that for some open subset $B^{\prime}\subseteq B$ the support of $\mathcal{V}$ is disjoint from $X\times B^{\prime}$ and hence $\mathcal{V}$ can be viewed as a coherent $\mathcal{O}_{X\setminus H\times B^{\prime}}$-module. The scheme $X\setminus H$ is quasi-affine. Applying the construction from above to $\mathcal{V}$, we obtain $\operatorname{CCQ}(\mathcal{V}|_{B^{\prime}})$, which is independent up to isomorphism of the choice of $H$. Gluing such families for all $b\in B$, we obtain the desired scheme. Similarly one obtains the self-dual Quot scheme for any coherent sheaf on a quasi-projective $X$.

4.2.2. Stacks

As a general reference for stacks, we use [Ols16, sta25]. Let $X$ be a quasi-projective scheme. The stack $\operatorname{Coh}_{d}X$ is a fibered category over $\operatorname{Spec}(\Bbbk)$-schemes, whose fibre over a scheme $B$ is the groupoid of coherent $\mathcal{O}_{X\times B}$-modules $\mathcal{M}$ such that $\mathcal{M}$ is finite flat of degree $d$ over $B$. The universal family on the Quot scheme yields a map $\operatorname{Quot}_{d}(\mathcal{O}_{X}^{\oplus d})\to\operatorname{Coh}_{d}X$ that is a smooth cover and so $\operatorname{Coh}_{X}$ is an Artin stack.

Definition 4.9**.**

The Artin stack $\operatorname{CQCoh}_{d}X\to\operatorname{Coh}_{d}X$ is the compatible locus $\operatorname{CCQ}(\mathcal{M})$ for the universal family $\mathcal{M}$ on the stack $\operatorname{Coh}_{d}X$.

In other words, for a scheme $B$, the fibre of $\operatorname{CQCoh}_{d}X$ is the groupoid of pairs $(M,[q_{\bullet}])$, where is a coherent $\mathcal{O}_{X\times B}$-module, locally free of rank $d$ as an $\mathcal{O}_{B}$-module, and $[q_{\bullet}]\colon B\to\operatorname{CCQ}(M)$ is a morphism, where $\operatorname{CCQ}(-)$ is the scheme of compatible quadrics as in Definition 4.5, so that $[q_{\bullet}]$ corresponds to a family of compatible broken quadrics over $B$. The morphism $\operatorname{CQCoh}_{d}X\to\operatorname{Coh}_{d}X$ forgets about $[q_{\bullet}]$.

Completed quadrics and compatibility commute with base change, so we have a cartesian diagram

[TABLE]

which proves that $\operatorname{CQCoh}_{d}X$ is indeed an Artin stack.

We now pass to the stack of finite algebras. The Artin stack $\operatorname{Alg}_{d}$ is the fibered category over $\operatorname{Spec}(\Bbbk)$ whose fiber over a scheme $B$ is the groupoid of finite flat degree $d$ maps $Y\to B$. It admits a smooth cover $\operatorname{Hilb}_{d}(\mathbb{A}^{d-1})\to\operatorname{Alg}_{d}$.

The Artin stack $\operatorname{CQAlg}_{d}\to\operatorname{Alg}_{d}$ is defined by considering pairs $(\pi\colon Y\to B,[q_{\bullet}])$, where $[q_{\bullet}]\in\operatorname{CQ}(\pi_{*}\mathcal{O}_{Y})$ is compatible with the action of $Y$. This requires a slight extension of the definition of compatibility.

4.2.3. Functorial interpretation

As for $\operatorname{CQ}(\mathcal{V})$, one can describe the functor of points for $\operatorname{Iar}_{d}(X)$ as follows. The functor will not be used below.

Proposition 4.10**.**

The scheme $\operatorname{Iar}_{d}(X)\to\operatorname{Hilb}_{d}(X)$ represents the functor which assigns to a morphism $T\to\operatorname{Hilb}_{d}(X)$ the set of compatible projectively $d$-complete symmetric forms $(u_{1},\ldots,u_{d})$, where

[TABLE]

Proof.

By the relative version of Theorem 3.36, the scheme $\operatorname{CQ}(\mathcal{U})$ represents the functor of projectively $d$-complete symmetric forms as above. The closed subscheme $\operatorname{Iar}_{d}(X)$ is cut out of $\operatorname{CQ}(\mathcal{U})$ exactly by the compatibility condition, hence it represents compatible such forms. ∎

4.3. The unbroken locus and the main component of the Iarrobino

scheme

For an irreducible $X$, the Hilbert scheme has a distinguished irreducible component, the smoothable component $\operatorname{Hilb}_{d}^{\operatorname{sm}}(X)$, that parameterises tuples of points and more general smoothable schemes, see [Ber12]. We now describe a similar irreducible component in $\operatorname{Iar}_{d}(X)$, again for an irreducible $X$. Recall that $\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(X)\to\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$ is a smooth morphism (with $(d-1)$-dimensional fibres) and the space upstairs parameterises pairs $(Z\subseteq X,[q])$, where $q\colon\omega_{Z}\to\mathcal{O}_{Z}$ is an orientation.

Definition 4.11**.**

The unbroken locus $\operatorname{Iar}_{d}^{\operatorname{Gor}}(X)$ in $\operatorname{Iar}_{d}(X)$ is the locus where the broken quadric $[q_{\bullet}]$ is not broken, that is, where $q_{0}$ has full rank.

Lemma 4.12**.**

The unbroken locus $\operatorname{Iar}_{d}^{\operatorname{Gor}}(X)$ is open in $\operatorname{Iar}_{d}(X)$ and isomorphic to $\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(X)$. If $X$ is smooth and $\dim X\leq 3$, then this locus is smooth.

Proof.

Openness follows for the openness of the locus of full rank quadrics in $\operatorname{CQ}(\mathcal{V})$ for every $\mathcal{V}$, see §3.2. By Proposition 4.7, the $\Bbbk$-points of this locus correspond to pairs $(Z,[q])$, for $Z\subseteq X$ a zero-dimensional Gorenstein subscheme of degree $d$ and $q\colon\omega_{Z}\to\mathcal{O}_{Z}$ a orientation, hence $\operatorname{Iar}_{d}(X)\to\operatorname{Hilb}_{d}^{\operatorname{Gor}}(X)$ is smooth and surjective. It follows that $\operatorname{Iar}_{d}^{\operatorname{Gor}}(X)$ is isomorphic to $\operatorname{Hilb}_{d}^{\operatorname{Gor},\mathrm{or}}(X)$.

For $\dim X\leq 3$, smoothness follows from the Buchsbaum-Eisenbud theorem [BE77], as proved in [KMR98]. ∎

Definition 4.13**.**

The locus of tuples of points $\left(\operatorname{Iar}_{d}(X)\right)^{\circ}$ is the intersection of $\operatorname{Iar}_{d}^{\operatorname{Gor}}(X)$ with the locus $Z\subseteq X$ such that $Z$ is reduced. The locus $\left(\operatorname{Iar}_{d}(X)\right)^{\circ}$ is irreducible, open in $\operatorname{Iar}_{d}(X)$ and has dimension $(1+\dim X)d-1$. The closure of $\left(\operatorname{Iar}_{d}(X)\right)^{\circ}$ is called the smoothable component and denoted $\operatorname{Iar}^{\operatorname{sm}}_{d}(X)$.

4.4. Iarrobino scheme of a smooth curve is smooth

In this section, we discuss $\operatorname{Iar}_{d}(C)$ for $C$ a smooth curve. It is an interesting case, since $\operatorname{Iar}_{d}(C)$ is smooth and so is the symmetric product $\operatorname{Hilb}_{d}(C)$, yet the natural map $\operatorname{Iar}_{d}(C)\to\operatorname{Hilb}_{d}(C)$ is not smooth.

Similarly as for the Hilbert scheme, the case of affine line $C=\mathbb{A}^{1}$ is concrete and serves as a local model. Recall [Fog68] that $\operatorname{Hilb}_{d}(\mathbb{A}^{1})$ is isomorphic to $\mathbb{A}^{d}$, the isomorphism sends a finite subscheme $V(x^{d}+a_{d-1}x^{d-1}+\ldots+a_{0})$ to $(a_{d-1},\ldots,a_{0})$. The Iarrobino scheme $\operatorname{Iar}_{d}(\mathbb{A}^{1})$ has more interesting geometry; we will see that it is fibred over $\operatorname{Hilb}_{d}(\mathbb{A}^{1})$ in permutohedral varieties (Example 4.4) and their degenerations.

It is convenient to prove smoothness of $\operatorname{Iar}_{d}(C)$ in the generality of the Quot schemes.

Theorem 4.14**.**

Let $C$ be a smooth irreducible curve and $\mathcal{E}$ be a locally free sheaf on $C$. Then the scheme $\operatorname{CQuot}_{d}(\mathcal{E})$ is smooth connected of dimension $d\cdot(1+\operatorname{rk}\mathcal{E})-1$.

Proof.

Let us begin with the proof of smoothness. After passing to $\Bbbk$ algebraically closed, it is enough to prove smoothness on $\Bbbk$-points. Take a $\Bbbk$-point $(M,[q_{\bullet}])$ of $\operatorname{CQuot}_{d}(\mathcal{E})$, where $p_{C}\colon\mathcal{E}\twoheadrightarrow M$ is a surjection of $\mathcal{O}_{C}$-modules. Let $\{c_{1},\ldots,c_{e}\}\subseteq C(\Bbbk)$ be the support of $M$. Passing to an open neighbourhood $U$ of the support, we may assume that $\mathcal{E}|_{U}=\mathcal{O}_{U}^{\oplus r}$ is free.

Fix any pairwise different $\Bbbk$-points $p_{1},\ldots,p_{e}\in\mathbb{A}^{1}$. By Cohen’s structure theorem, there is an isomorphism of products of complete local rings

[TABLE]

Let $\mathcal{E}^{\prime}:=\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r}$. The $\mathcal{O}_{C}$-module structure of $M$ is completely determined by the action of complete local rings, hence there exists a surjection $p_{\mathbb{A}^{1}}\colon\mathcal{E}^{\prime}\twoheadrightarrow M^{\prime}$ of $\mathcal{O}_{\mathbb{A}^{1}}$-modules, such that $\iota^{*}p_{C}=p_{\mathbb{A}^{1}}$, in particular, $\iota^{*}M=M^{\prime}$.

The same argument applies to show that the deformations functors of $p_{\mathbb{A}^{1}}$ and $p_{C}$ over Artin rings are isomorphic. In particular, the complete local rings at $p_{C}=[\mathcal{E}\twoheadrightarrow M]\in\operatorname{Quot}_{d}(\mathcal{E})$ and $p_{\mathbb{A}^{1}}=[\mathcal{E}^{\prime}\twoheadrightarrow M^{\prime}]\in\operatorname{Quot}_{d}(\mathcal{E}^{\prime})$ are isomorphic in a way compatible with the restrictions of universal families:

[TABLE]

Let $[q_{\bullet}^{\prime}]$ be a compatible quadric on $M^{\prime}$ obtained from $[q_{\bullet}]$ using the isomorphism $\operatorname{CQ}(M^{\prime})\simeq\operatorname{CQ}(M)$ given by $\iota$. The isomorphism $i$ yields an isomorphism

[TABLE]

which shows that the complete local rings as $(M,[q_{\bullet}])$ and $(M^{\prime},[q_{\bullet}]^{\prime})$ are isomorphic. Hence, we can and do reduce to the case $C=\mathbb{A}^{1}$, $\mathcal{E}=\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r}$, in particular we drop the primes below.

We will use the infinitesimal lifting criterion. Let $\operatorname{Spec}(A)\hookrightarrow\operatorname{Spec}(B)$ be a small extension, where $B\twoheadrightarrow A$ are Artinian local $\Bbbk$-algebras. We are to prove that any deformation $(\widetilde{M},[\widetilde{q}_{\bullet}])$ of $(M,[q_{\bullet}])$ over $A$ can be extended to a deformation over $B$.

The $A$-module $\widetilde{M}$ is free. Take a free $B$-module $\widetilde{M}^{\prime}$ with surjection $\widetilde{M}^{\prime}\twoheadrightarrow\widetilde{M}$. The element $[\widetilde{q}_{\bullet}]$ is a map $\operatorname{Spec}(A)\to\operatorname{CQ}(\widetilde{M})\hookrightarrow\operatorname{CQ}(\widetilde{M}^{\prime})$. The variety of completed quadrics $\operatorname{CQ}(\widetilde{M}^{\prime})$ is smooth over $B$, hence we take a lift $[q_{\bullet}]$ to a map $\operatorname{Spec}(B)\to\operatorname{CQ}(\widetilde{M}^{\prime})$ which corresponds to a broken quadric $[\widetilde{q}_{\bullet}^{\prime}]$.

Let $\mathcal{C}\to\operatorname{CQ}(\widetilde{M}^{\prime})$ be the compatible bundle. The broken quadric $[q_{\bullet}]$ is compatible with the action of $S=H^{0}(\mathcal{O}_{\mathbb{A}^{1}})=\Bbbk[x]$, we have a structure morphism

[TABLE]

of $A$-schemes. Restricting it to the action of $x\in S$, we obtain a map

[TABLE]

The vector bundle $\mathcal{C}$ is smooth over $\operatorname{CQ}(\widetilde{M}^{\prime})$, hence (4.16) lifts to a morphism $\operatorname{Spec}(B)\to\mathcal{C}$, which yields a coherent $S_{B}$-module structure on $\widetilde{M}^{\prime}$, compatible with the broken quadric $[\widetilde{q}_{\bullet}^{\prime}]$. Since $\mathcal{E}$ is free, the surjection $\mathcal{E}|_{\operatorname{Spec}(A)}\twoheadrightarrow\widetilde{M}$ lifts to a surjection $\mathcal{E}|_{\operatorname{Spec}(B)}\twoheadrightarrow\widetilde{M}^{\prime}$ and so we obtain a map $\operatorname{Spec}(B)\to\operatorname{CQuot}_{d}(\mathcal{E})$, as required. This concludes the proof of smoothness.

To prove connectedness of $\operatorname{CQuot}_{d}(\mathcal{E})$, consider first the open locus $V\subseteq\operatorname{Quot}_{d}(\mathcal{E})$ that parameterises modules $\mathcal{E}\twoheadrightarrow M$, where $M$ is supported at exactly $d$ points. The $\mathcal{O}_{C}$-modules $M$ and $\mathcal{O}_{C}/\!\operatorname{Ann}(M)$ are isomorphic. This implies that the fibre of $\operatorname{CQuot}_{d}(\mathcal{E})\to\operatorname{Quot}_{d}(\mathcal{E})$ over $[\mathcal{E}\twoheadrightarrow M]$ is isomorphic to the fibre of $\operatorname{Iar}_{d}(C)\to\operatorname{Hilb}_{d}(C)$ over $V(\operatorname{Ann}(M))\subseteq C$, hence it is isomorphism to the permutohedral variety, and so it is irreducible. Hence, the open locus $V$ is irreducible itself.

To prove connectedness, it is enough to show that $V$ is dense. This amounts to producing, for every point, a germ $\operatorname{Spec}(\Bbbk[\![t]\!])\to\operatorname{Quot}_{d}(\mathcal{E})$ such that the special point maps to a given point, while the generic points maps to $V$. As in the proof of smoothness, we reduce to the case $C=\mathbb{A}^{1}$, $\mathcal{E}=\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r}$. At this point $V$ is no longer relevant for the proof.

We will link any $\Bbbk$-point $(M_{0},[q_{\bullet}])$ to a special $\Bbbk$-point which is $M_{0}=\Bbbk[x]/(x^{d})$ with an unbroken quadric coming from the functional $(x^{d-1})^{*}$ as in Example 2.2.

For the first step, consider the $\mathbb{G}_{\mathrm{m}}$-action on $\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r})$, which is the composition of the usual action on $\mathbb{A}^{1}$ and an action with general weights assigned to $r$ generators of $\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r}$. The only torus-fixed points are quotients of the form

[TABLE]

For the second step, observe that the infinitesimal lifting criterion above gives more than just smoothness: it shows that for any infinitesimal deformation of $q_{\bullet}$, the remaining structure deforms as well. This in particular implies that any $(M,[q_{\bullet}])$ lies in the closure of the locus $(M,[q_{0}])$ where $q_{\bullet}=[q_{0}]$ is unbroken. This locus itself is irreducible, because it is open in the vector subspace $\operatorname{Sym}^{2}_{A}M\subsetneq\operatorname{Sym}^{2}_{\Bbbk}M$, which is the symmetric square of an $A$-module, not a $\Bbbk$-module.

For the third step, it remains to connect any module of the form (4.17) with a full rank quadric given by the functional $\sum_{i=1}^{r}(x^{\nu_{i}-1}e_{i})^{*}$ to the module $M_{0}$ above, which is a special case, with $0=\nu_{2}=\ldots=\nu_{r}$. This is an elementary construction, which we leave to the reader. This concludes the proof of connectedness. The dimension estimate follows, since this is the dimension of the open locus $V$. ∎

Theorem 4.18**.**

Let $C$ be a smooth connected curve. The scheme $\operatorname{Iar}_{d}(C)$ is smooth irreducible of dimension $2d-1$. The morphism $\tau_{C}\colon\operatorname{Iar}_{d}(C)\to\operatorname{Hilb}_{d}(C)$ is flat and projective with integral fibres of dimension $d-1$ which are reduced complete intersections in $\operatorname{Iar}_{d}(C)$. It satisfies $(\tau_{C})_{*}\mathcal{O}_{\operatorname{Iar}_{d}(C)}=\mathcal{O}_{\operatorname{Hilb}_{d}(C)}$.

Proof.

Smoothness, irreducibility, and $\dim\operatorname{Iar}_{d}(C)=2d-1$ all follow from Theorem 4.14 for $\mathcal{E}:=\mathcal{O}_{C}$. Projectivity of $\tau_{C}$ is clear, since $\tau_{C}$ is a restriction of $\tau_{\overline{C}}$ for a projective curve $\overline{C}$. We will show below that the fibres of $\tau_{C}$ are reduced, irreducible of dimension $d-1$. Once this is done, Miracle Flatness shows that $\tau_{C}$ is flat and that the fibres are complete intersections. Stein factorisation yields $(\tau_{C})_{*}\mathcal{O}_{\operatorname{Iar}_{d}(C)}=\mathcal{O}_{\operatorname{Hilb}_{d}(C)}$.

Take a subscheme $Z\subseteq C$ and embed it into $\mathbb{A}^{1}$. The fibres $\tau_{C}^{-1}(Z\subseteq C)$ and $\tau_{\mathbb{A}^{1}}^{-1}(Z\subseteq\mathbb{A}^{1})$ are isomorphic, hence we reduce to the case $C=\mathbb{A}^{1}$. By Example 4.3 the fibre over $\operatorname{Spec}(\Bbbk[x]/x^{d})$ has dimension $d-1$. The morphism $\tau_{\mathbb{A}^{1}}$ is equivariant with respect to the $\mathbb{G}_{\mathrm{m}}$-action on $\mathbb{A}^{1}$. Every fiber of $\tau_{\mathbb{A}^{1}}$ is cut out by $d$ equations, hence has dimension $\geq 2d-1-d$ at every point. By semicontinuity using the torus action, every fiber has dimension $d-1$ everywhere and is a complete intersection. Finally, to prove reducedness of fibres, recall that they are Cohen-Macaulay, so [sta25, Tag 031R] it is enough to prove generic reducedness. But each of the fibres is over a Gorenstein subscheme, hence admits a point corresponding to an unbroken quadric, see Lemma 4.12, which is smooth. ∎

Example 4.19**.**

The fibre of $\operatorname{Iar}_{3}(\mathbb{A}^{1})\to\operatorname{Hilb}_{3}(\mathbb{A}^{1})$ over $\operatorname{Spec}(\Bbbk[x]/x^{3})$ is singular, as discovered by Reinier F. Schmiermann. Indeed, by definition, the fibre is the locus in $\operatorname{CQ}(\Bbbk[x]/x^{3})$ of broken quadrics compatible with the action of $x$. We identify $R=\Bbbk[x]/x^{3}$ with its dual $R^{\vee}$. One compatible quadric is the one corresponding to the full flag

[TABLE]

This quadric can be deformed according to the description from Example 4.3. Namely, we can consider the quadrics with flag $R\supseteq(x^{2})\supseteq(0)$ and a full rank quadric on $R/(x^{2})\simeq\Bbbk[x]/(x^{2})$ given by

[TABLE]

where at the end we apply an arbitrary functional that is nonzero on $x$. This yields one tangent vector. Performing an analogous procedure for the flag $R\supseteq(x)\supseteq(0)$ yields a mirrored situation and another tangent vector.

The third tangent vector is not so easy to describe and it is best to employ Tyrrell’s coordinates, as in §3.3. Namely, consider the ring $\Bbbk[\varepsilon]/(\varepsilon^{2})$ and the broken quadric which in Tyrrell’s coordinates corresponds to $y_{1}=\varepsilon$, $y_{2}=-\varepsilon$, $x_{13}=\varepsilon$ and all other coordinates zero. In the exterior forms notation from §3.2 this broken quadric is given by

[TABLE]

This is compatible with $x$ and yields the third tangent vector.

Example 4.20**.**

The morphism $\tau_{\operatorname{Quot}}\colon\operatorname{CQuot}_{d}(\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r})\to\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{1}}^{\oplus r})$ is not flat for $r\geq 2$. Indeed, for example take the module $M=\Bbbk[x]/(x)\oplus\Bbbk[x]/(x^{d-1})$. Take any surjection that makes $M$ a point of the Quot scheme, and consider full rank compatible quadrics on $M$. The space of such quadrics is open in $\operatorname{Sym}^{2}_{\Bbbk[x]}M\subsetneq\operatorname{Sym}^{2}M$, which is a vector subspace of dimension $d+1$. We consider quadrics up to rescaling, but still this implies that the fibre of $\tau_{\operatorname{Quot}}$ is at least $d$-dimensional, while the fibre over a cyclic module $M$ is $(d-1)$-dimensional.

4.4.1. Applications to characteristic numbers

In this section we reprove one of the results of [JPS25], as explained in §1.2.2.

For any rank $d$ locally free sheaf $\mathcal{V}$ on a scheme $B$, the embedding of $\operatorname{CQ}(\mathcal{V})$ into the product $\mathcal{P}$ given in (3.14) equips $\operatorname{CQ}(\mathcal{V})$ with $d-1$ line bundles $\mathcal{L}_{1},\ldots,\mathcal{L}_{d-1}$: the pullbacks of $\mathcal{O}(1)$ from the first, second, …, $(d-1)$-th factor. For every $b\in B$ with residue field of characteristic zero, the restrictions $L_{1}=\mathcal{L}_{1}|_{b}$, …, $L_{d-1}=\mathcal{L}_{d-1}|_{b}$ generate the Picard group of $\operatorname{CQ}(\mathcal{V}|_{b})$, see [Mas20].

Proposition 4.21**.**

Let $\Bbbk$ have characteristic different from two (we just repeat here the global assumption to stress that we do not assume characteristic zero). Let $b_{1},\ldots,b_{d-1}$ be any integers summing up to $d-1$. Let $V=\Bbbk[x]/(f)$ be a finite $\Bbbk$-algebra with $\dim_{\Bbbk}V=d$. Let $X_{V}\subseteq\operatorname{CQ}(V)$ be the associated variety, as described in §1.2.2. Then the intersection number

[TABLE]

is independent of $f$, that is, constant on the whole Hilbert scheme $\operatorname{Hilb}_{d}(\mathbb{A}^{1})$.

Proof.

We will prove that the variety $X_{V}$ is the fibre of the Iarrobino-to-Hilbert map

[TABLE]

The variety $X_{V}$ is obtained as follows. The multiplication on $V$ is a map $\operatorname{Sym}^{2}V\to V$. Its transpose is a map $V^{\vee}\to\operatorname{Sym}^{2}V^{\vee}$. We consider elements $\Bbbk\to V^{\vee}\to\operatorname{Sym}^{2}V^{\vee}$ of its image which are full rank quadrics and we take the closure of this locus in $\operatorname{CQ}(V^{\vee})$. Observe that $\Bbbk\to V^{\vee}\to\operatorname{Sym}^{2}V^{\vee}$ is of full rank if and only if the dual map $\operatorname{Sym}^{2}V\to V\to\Bbbk$ is of full rank. By Example 2.2 the locus above consists exactly of $q^{-1}$ for a full rank compatible quadric $q\in\operatorname{Sym}^{2}V$.

Recall from §3.5 that there is an isomorphism $\operatorname{CQ}(V^{\vee})\to\operatorname{CQ}(V)$ which maps a full rank quadric $q$ to its inverse $q^{-1}$ and interchanges $\mathcal{L}_{j}$ with $\mathcal{L}_{d-j}$. Taking the closures of the corresponding loci, we obtain an isomorphism of $X_{V}$ and $\tau^{-1}_{\mathbb{A}^{1}}([f])$.

Now, the desired characteristic number is the intersection number

[TABLE]

for a $\Bbbk$-point $[f]\in\operatorname{Hilb}_{d}(\mathbb{A}^{1})$. Since $\tau_{\mathbb{A}^{1}}$ is flat, this is independent of $[f]$, see [Ful98, §10.2], since the intersection number is obtained from Euler characteristics [Kle66]. ∎

4.4.2. Commuting matrices

In this section we give an alternative, slightly less technical description of $\operatorname{Iar}_{d}(\mathbb{A}^{1})$. It is well-known that $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ can be presented as a global quotient of the space of commuting matrices together with a generating vector and stability condition, see [Nak99]. We present below the corresponding construction of $\operatorname{Iar}_{d}(\mathbb{A}^{n})$ in the case $n=1$.

Let $\mathcal{W}$ be the moduli spaces of finite rank $d$ quotients $\Bbbk[x]/I$ with a fixed basis $V=\Bbbk^{\oplus d}\to\Bbbk[x]/I$. The scheme $\mathcal{W}$ identifies with an open subset of $\operatorname{End}(V)\times V$. There is a natural map

[TABLE]

which is a quotient by a free $\operatorname{GL}(V)$-action. We can construct $\operatorname{Iar}_{d}(\mathbb{A}^{1})$ mimicking this construction, without any reference to completed quadrics of vector bundles.

Let $\mathcal{U}\to\operatorname{Hilb}_{d}(\mathbb{A}^{1})$ be the universal bundle. Consider the pullback of $\operatorname{CQ}(\mathcal{U})\to\operatorname{Hilb}_{d}(\mathbb{A}^{1})$ via the map $\pi\colon\mathcal{W}\to\operatorname{Hilb}_{d}(\mathbb{A}^{1})$ constructed in (4.22). By functoriality of $\operatorname{CQ}(-)$, this pullback is isomorphic to $\operatorname{CQ}(\pi^{*}\mathcal{U})$. The scheme $\mathcal{W}$ by definition comes with a universal (in particular, canonical) trivialisation of $\pi^{*}\mathcal{U}$, so we obtain an isomorphism

[TABLE]

Recall the subbundle of compatible operators $\mathcal{C}$ defined in Proposition 3.30. The compatibility condition pulls back as well and we obtain a pullback diagram

[TABLE]

The scheme $\mathcal{C}$ is smooth irreducible as a bundle over a smooth scheme $\operatorname{CQ}(V)$, so $\mathcal{C}|_{\operatorname{CQ}(V)\times\mathcal{W}}$ is smooth irreducible as well. The map $\mathcal{C}|_{\operatorname{CQ}(V)\times\mathcal{W}}\to\operatorname{Iar}_{d}(\mathbb{A}^{1})$ is a pullback of a quotient by a free $\operatorname{GL}(V)$-action, so it is such a quotient as well, hence it is smooth and surjective. The source of this map is smooth irreducible, hence so is the target. A dimension count shows that the dimension of $\operatorname{Iar}_{d}(\mathbb{A}^{1})$ is $2d-1$.

4.5. Torus limits on $\mathbb{A}^{n}$ and Iarrobino’s symmetric

decomposition

In this section we explain the origin of the name Iarrobino scheme, based on Anthony’s Iarrobino work recalled in §2.2.

We consider $\operatorname{Iar}_{d}(\mathbb{A}^{n})$ and fix the action of the torus $\mathbb{G}_{\mathrm{m}}$ on $\mathbb{A}^{n}$ by scalar multiplication:

[TABLE]

In particular, the fixed points will not be isolated. We begin with the Hilbert scheme case.

4.5.1. Białynicki-Birula decompositions

Let us recall the abstract apparatus of Białynicki-Birula decompositions [BB73, JS19]. Let $\mathcal{M}$ be a separated $\mathbb{G}_{\mathrm{m}}$-scheme (below, $\mathcal{M}=\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ or $\mathcal{M}=\operatorname{Iar}_{d}(\mathbb{A}^{n})$). For a $\Bbbk$-point $[Z]\in\mathcal{M}$, we can consider the orbit map

[TABLE]

given by $g\mapsto g\cdot[Z]$ and ask whether it extends to a map $\overline{\mu}_{[Z]}\colon\mathbb{P}^{1}\setminus\{0\}\to\mathcal{M}$. If an extension exists, it is unique and the image $\overline{\mu}_{[Z]}(\infty)\in\mathcal{M}(\Bbbk)$ is called the limit of $[Z]$ at $t\to\infty$. In this situation, the Białynicki-Birula decomposition $\mathcal{M}^{-}$ is a scheme with maps

[TABLE]

The theory is subtle, especially when $\mathcal{M}$ is not smooth. Below, we will be mostly interested in the $\Bbbk$-points, so we only observe that

[TABLE]

If every point admits a limit, then $\mathcal{M}^{-}(\Bbbk)=\mathcal{M}(\Bbbk)$ as sets (nevertheless, the scheme $\mathcal{M}^{-}$ is not isomorphic to $\mathcal{M}$!). This happens when $\mathcal{M}$ is projective, or, more generally, semiprojective, see [HRV15].

4.5.2. Białynicki-Birula decompositions for the Hilbert scheme of

points

The Białynicki-Birula decompositions were classically employed by [ES87] to compute the homology of $\operatorname{Hilb}_{d}(\mathbb{A}^{2})$ and $\operatorname{Hilb}_{d}(\mathbb{P}^{2})$. More recently, they have become a standard tool to exhibit singularities of $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ for $n\geq 3$, see [Jel19, Jel20, SS23, GGGL24, FPS25].

Example 4.23**.**

Consider $\operatorname{Hilb}_{1}(\mathbb{A}^{n})=\mathbb{A}^{n}$. The only point admitting a limit at $t\to\infty$ is the point $(0,\ldots,0)$.

It follows from the example that $(\operatorname{Hilb}_{d}(\mathbb{A}^{n}))^{-}(\Bbbk)$ consists only of subschemes supported only at the origin. Conversely, the punctual Hilbert scheme $\operatorname{Hilb}_{d}(\mathbb{A}^{n},0)$ is projective, so that every $Z\subseteq\mathbb{A}^{n}$ supported only at the origin admits a limit with $t\to\infty$. Therefore, we have

[TABLE]

Let us compute the actual limit. Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, so that $\mathbb{A}^{n}=\operatorname{Spec}(S)$. For $Z=\operatorname{Spec}(S/I)$ supported only at the origin, we have [Jel19] the formula

[TABLE]

where $I(Z_{0})$ is the initial ideal $\operatorname{in}(I)$. The map $\mathbb{A}^{1}=\mathbb{P}^{1}\setminus\{0\}\to\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ associated to the limit corresponds to the ideal $I^{\mathrm{hom}}\subseteq S[u]$, where $I^{\mathrm{hom}}$ is the homogenisation of $I$ with respect to a parameter $u=t^{-1}$ on the torus, in particular $\deg(u)=-1$.

Example 4.25**.**

Let $Z\subseteq\mathbb{A}^{2}$ be given by $I(Z)=(xy,x^{2}-y^{3})$. The homogenisation is given by $(xy,x^{2}-uy^{3},y^{4})$. The initial ideal is generated by lowest degree forms of polynomials in $I(Z)$. We have $I(Z_{0})=\operatorname{in}I(Z)=(xy,x^{2},y^{4})$.

Abstractly, $S/I$ is a local algebra $(A,\mathfrak{m})$ then $S/\operatorname{in}(I)$ is isomorphic to the associated graded $\operatorname{gr}A$. The observation is easy, yet foundational for applications of Białynicki-Birula decomposition to Hilbert schemes [Jel19].

4.5.3. Białynicki-Birula decompositions for the Iarrobino

scheme

Now we pass to the Iarrobino scheme. The construction of $\operatorname{Iar}_{d}(-)$ is functorial, so the $\mathbb{G}_{\mathrm{m}}$-action on $\mathbb{A}^{n}$ yields a $\mathbb{G}_{\mathrm{m}}$-action on $\operatorname{Iar}_{d}(\mathbb{A}^{n})$ and the forgetful map

[TABLE]

is $\mathbb{G}_{\mathrm{m}}$-equivariant and projective.

Consider a finite Gorenstein subscheme $Z=\operatorname{Spec}(A)\subseteq\mathbb{A}^{n}$ of degree $d$ supported only at the origin, and fix an orientation $q\colon\omega_{Z}\to\mathcal{O}_{Z}$, as in Example 2.2. The pair $([Z],[q])$ is a $\Bbbk$-point of the unbroken locus $\operatorname{Iar}_{d}^{\operatorname{Gor}}(\mathbb{A}^{n})$ from §4.3.

Since the limit $\lim_{t\to\infty}t\cdot[Z]$ in $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ exists, and $\tau$ is projective, also the limit

[TABLE]

exists in $\operatorname{Iar}_{d}(\mathbb{A}^{n})$, say it is equal to $([Z_{0}],[q_{\bullet}])$. From the previous section we know that $Z_{0}$ is abstractly the spectrum of the associated graded algebra $\operatorname{gr}(A)$. We will see below that $[q_{\bullet}]$ equips this algebra exactly with Iarrobino’s symmetric decomposition.

Example 4.26**.**

Let us verify the claim in the situation from Example 4.25. This will be the promised “geometrisation” of Example 2.4.

Recall that we consider $Z=V(xy,x^{2}-y^{3})\subseteq\mathbb{A}^{2}$. The corresponding algebra $A=\frac{\Bbbk[x,y]}{(xy,x^{2}-y^{3})}$ is a complete intersection, hence it is Gorenstein. The algebra has a basis $1,x,y,y^{2},y^{3}$. Let $\alpha\colon A\to\Bbbk$ be the functional dual to $y^{3}$ in this basis and let $\varphi\in\operatorname{Sym}^{2}A^{\vee}$ be the associated quadric: $\varphi(a,a^{\prime}):=\alpha(aa^{\prime})$. It has full rank and as a map $\varphi\colon A\to A^{\vee}$ it is given by

[TABLE]

We need to compute the dual quadric $q\in\operatorname{Sym}^{2}A$. As a map $q\colon A^{\vee}\to A$, it is equal to $\varphi^{-1}$, so it is given by

[TABLE]

Consider now the $\mathbb{G}_{\mathrm{m}}$-orbit of $[Z]$. We will be interested in the limit at $t\to\infty$, so we use the coordinate $u:=t^{-1}$. The action on $Z$ is given by

[TABLE]

for $u_{0}\in\mathbb{G}_{\mathrm{m}}(\Bbbk)$. We see that for every nonzero $u_{0}$ the resulting quotient algebra

[TABLE]

is again spanned by $1,y,y^{2},y^{3},x$ and equipped with a functional $\alpha_{u_{0}}$ defined by $\alpha_{u_{0}}(a):=u_{0}^{3}\cdot\alpha(u_{0}^{-1}a)$ for every $a\in A_{u_{0}}$. It follows that $\alpha_{u_{0}}$ is dual to $y^{3}$ in the above basis. From $\alpha_{u_{0}}$, we obtain $\varphi_{u}$ and $q_{u}$, given by

[TABLE]

To compute $\lim_{t\to\infty}t\cdot([Z],[q])$, it remains to compute $\lim_{u\to 0}q_{u}$. This is most conveniently done by applying §3.2.1 to $u\cdot q_{u}$, which yields the same class in $\mathbb{P}\operatorname{Sym}^{2}A_{u}$ as $q_{u}$. It follows that $\lim_{u\to 0}q_{u}$ is the broken quadric corresponding to the flag

[TABLE]

dual to the flag

[TABLE]

and the (unique up to scalars) quadric $q_{0}:=(uq_{u})|_{u=0}$ on $\left\langle x^{*}\right\rangle\simeq\Bbbk x$ together with the residual quadric $q_{1}$ on $\frac{\Bbbk[x,y]}{(x^{2},xy,y^{4},x)}$ given by the functional $(y^{3})^{*}$. This is in accordance with Example 2.4: the quadric $q_{1}$ yields self-duality of $\mathcal{Q}(0)\simeq\frac{\Bbbk[x,y]}{(x,y^{4})}$ and the quadric $q_{0}$ does it for $\Bbbk x\simeq\frac{\Bbbk[x,y]}{(x,y)}$.

Theorem 4.27** **(Iarrobino decomposition is a torus limit on the Iarrobino

scheme).

Let $Z=\operatorname{Spec}(A)\subseteq\mathbb{A}^{n}$ be supported only at the origin, where $A$ is local with maximal ideal $\mathfrak{m}$ and $s=\max\{i\colon\mathfrak{m}^{i}\neq 0\}$.

Let $q\colon\omega_{Z}\to\mathcal{O}_{Z}$ be an orientation. The limit $\lim_{t\to\infty}t\cdot[Z]$ is given by $([Z_{\bullet},[q_{\bullet}])$, where $Z_{\bullet}$ is a flag of subschemes

[TABLE]

with $Z_{0}\simeq\operatorname{Spec}(\operatorname{gr}(A))$, and for $\delta\geq 1$, the scheme $Z_{\delta}$ is given by the ideal

[TABLE]

and

[TABLE]

proves that the subquotient

[TABLE]

is self-dual.

Proof.

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$ and let $S^{\vee}\simeq\Bbbk[x_{1}^{*},\ldots,x_{n}^{*}]$ be the graded dual vector space. We have $\mathcal{O}_{Z}\simeq S/I$ for an ideal $I$. Since $Z$ is supported only at the origin, the ideal $I$ contains $(x_{1},\ldots,x_{n})^{d}$, so we have $\omega_{Z}=\mathcal{O}_{Z}^{\vee}\hookrightarrow S^{\vee}$. Let $\varphi:=q^{-1}$ and $\alpha:=\varphi(1_{\mathcal{O}_{Z}},-)\colon\mathcal{O}_{Z}\to\Bbbk$ be a functional, then $(q^{-1})(a_{1},a_{2})=\alpha(a_{1}a_{2})$, see Example 2.2 and $\alpha\in\omega_{Z}\subseteq S^{\vee}$.

From the definition of $s$, we have $\alpha=\alpha_{s}+\ldots+\alpha_{0}$. Let us see how $\varphi$ interacts with $\mathcal{Q}(\bullet)$. Fix any $i,\delta$. Take a basis of $(\mathcal{Q}(\delta))_{i}$ from (4.28) and lift it to a subset $\mathcal{B}_{i,\delta}\subseteq S_{i}$ of homogeneous elements of $S$ of degree $i$. For $b\in\mathcal{B}_{i,\delta}$ we have

[TABLE]

This is an element annihilated by $\mathfrak{m}^{s+1-i-\delta}$. Being annihilated $\mathfrak{m}^{s+1-i-\delta}$ is the same as being annihilated by $(x_{1},\ldots,x_{n})^{s+1-i-\delta}$, which is the same as being an element of $S^{\vee}_{\leq s-i-\delta}$. By homogeneity, for each $j$, the element $\alpha_{j}(b\cdot(-))$ lies in $S^{\vee}_{j-i}$, hence for $j\geq s-\delta+1$ we obtain

[TABLE]

Let us denote

[TABLE]

By construction, the images of elements in $\mathcal{B}_{i,\delta}$ in $\operatorname{gr}(A)$ are linearly independent modulo the space $\mathfrak{m}^{i+1}\cap(0:\mathfrak{m}^{s+1-i-\delta})+\mathfrak{m}^{i}\cap(0:\mathfrak{m}^{s-i-\delta})$. This translates to saying that $W_{i,\delta}$ are linearly independent modulo

[TABLE]

Let us now understand how the torus action on $S$ and $S^{\vee}$ fits into the picture. For the torus action, we have $\deg x_{\bullet}^{*}=-1$ and $\deg(x_{i})=1$. Take the parameter $u:=t^{-1}$ and consider the morphism $\operatorname{Spec}(\Bbbk[u])\to\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ that yields the limit at $t\to\infty$ and the corresponding subscheme $\mathcal{Z}\subseteq\mathbb{A}^{n}\times\operatorname{Spec}(\Bbbk[u])$. As explained in §4.5.2, the ideal $I(\mathcal{Z})\subseteq S[u]$ is the homogenisation of $I$ using $u$. Correspondingly, for $u\neq 0$, the functional $\alpha_{u}\in\mathcal{O}_{\mathcal{Z}|_{u\neq 0}}^{\vee}$ that yields the orientation is the homogenisation of $\alpha$ with respect to $u$:

[TABLE]

It is homogeneous of degree $-s$ and yields the quadric

[TABLE]

and $q_{u}=\varphi_{u}\colon\mathcal{O}_{\mathcal{Z}|_{u\neq 0}}^{\vee}\to\mathcal{O}_{\mathcal{Z}|_{u\neq 0}}$, both homogeneous of degree $-s$. For any $i$, $\delta$, and $b\in\mathcal{B}_{i,\delta}$, it follows from the discussion above that

[TABLE]

and that $\alpha_{s-\delta}(b)\in W_{i,\delta}$ are linearly independent modulo $\left\langle W_{i+j,\delta-j}\ |\ j=1,\ldots,\delta\right\rangle$. Let $\mathcal{B}_{\delta}:=\bigcup_{i=0}^{s}\mathcal{B}_{i,\delta}$ and $W_{\delta}:=\bigcup_{i=0}^{s}W_{i,\delta}$. In the bases $\bigcup_{\delta}\mathcal{B}_{\delta}$ and $\bigcup_{\delta}W_{\delta}$, the matrix of $\varphi_{u}$ becomes

[TABLE]

where $u^{\delta}\Phi_{\delta}$ is a symmetric matrix corresponding to the map $\mathcal{B}_{\delta}\to W_{\delta}$; it is divisible exactly by $u^{\delta}$. (It is well-known, but irrelevant here, that $\Phi_{s}=0$, $\Phi_{s-1}=0$ whenever $s\geq 2$.)

By [TK88, Theorem 7.20], we can compute the limit of (4.29) even though in our bases the matrix is non symmetric. This is because we compute the limit in each $\mathbb{P}(\Lambda^{\bullet}\operatorname{gr}(A))^{\otimes 2}$ and obtain a result in $\mathbb{P}\operatorname{Sym}^{2}(\Lambda^{\bullet}\operatorname{gr}(A))=\left(\mathbb{P}(\Lambda^{\bullet}\operatorname{gr}(A))^{\otimes 2}\right)^{\mathrm{sym}}$. Since the matrix (4.29) is lower-triangular, the computation is straightforward: the resulting broken quadric $[q_{\bullet}]$ is a chain of quadrics $[\Phi_{0}],\,[\Phi_{1}],\,\ldots$ as claimed. Finally, by Duality §3.5, the limiting quadric for $q_{u}$ is obtained from the limiting quadric for $\varphi_{u}=(q_{u})^{-1}$. ∎

Example 4.30**.**

In this example we show that there might be no broken Gorenstein structure, as defined in §1.2.1, in the limit of Gorenstein algebras. This has to happen for a singular point of $\operatorname{Hilb}_{d}(\mathbb{A}^{3})$. We consider $d=4$ and

[TABLE]

Let us take a family of Gorenstein subschemes converging to it. We choose a family

[TABLE]

which is the passage to the associated graded algebra of the complete intersection $(x^{2},\,y^{2},\,z-xy)$. Let us choose the functional $\alpha:=z^{*}$ as an orientation on this algebra. It the usual bases $V=\left\langle 1,x,y,z\right\rangle$ and $V^{\vee}=\left\langle 1^{*},x^{*},y^{*},z^{*}\right\rangle$, we have, for $t\neq 0$, the following

[TABLE]

so we obtain from $\varphi_{t}$ the broken quadric in $\operatorname{CQ}(V^{\vee})$ with the associated flag

[TABLE]

and quadrics $[1^{*}\cdot x^{*}]$, $[y^{*}\cdot z^{*}]$ on subquotients. Correspondingly, the broken quadric $\lim_{t\to 0}\varphi_{t}^{-1}$ on $\operatorname{CQ}(V)$ has the associated flag

[TABLE]

and quadrics $[y\cdot z]$, $[1\cdot x]$ on subquotients. We see that $\left\langle y,z\right\rangle$ is not a principal ideal, hence (4.31) does not yield a broken Gorenstein algebra structure.