Robin Hartshorne
We review the present state of the problem, for each degree $d$ and genus $g$, is the Hilbert scheme of locally Cohen--Macaulay curves in ${\mathbb P}^3$ connected?
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Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.
$\mathbb{A}^1$-connectedness of moduli of semistable bundles and symplectic bundles on a curve
本文证明无限域上光滑射影曲线半稳定向量丛（固定行列式）及辛向量丛模栈的 $\mathbb{A}^1$-连通性，条件依赖于有理点存在，并推广至拟抛物情形。
$\mathbb{A}^1$-connectedness of moduli of vector bundles on a stacky curve
本文证明了无限域上栈式曲线具有固定行列式和重数的$\mathcal{E}$-半稳定向量丛模栈的$\mathbb{A}^1$-连通性，并推论出抛物$\alpha$-半稳定向量丛模栈的$\mathbb{A}^1$-连通性。
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.
$\mathbb{A}^1$--connectedness of moduli stack of semi-stable and parabolic semi-stable vector bundles over a curve
本文研究了曲线上的半稳定向量丛及抛物半稳定向量丛模堆的$\mathbb{A}^1$-连通性，证明了其在固定秩和确定性下的连通性，以及在特定条件下抛物数据下的连通性。
We study the relationship between several notions of connectedness arising in ${\mathbb A}^1$-homotopy theory of smooth schemes over a field $k$: ${\mathbb A}^1$-connectedness, stable ${\mathbb A}^1$-connectedness and motivic connectedness, and we discuss the relationship between these notations and rationality properties of algebraic varieties. Motivically connected smooth proper $k$-varieties are precisely those with universally trivial $CH_0$. We show that stable ${\mathbb A}^1$-connectedness coincides with motivic connectedness, under suitable hypotheses on $k$. Then, we observe that there…
Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness
本文研究了积分点的Log Manin猜想及其与$\mathbb A^1$-连通性的联系，证明了Campana有理曲线和分裂托里除数上的Log Manin猜想，并结合Cox环描述和Batyrev的启发式计数方法给出几何解释。
Naive $\mathbb{A}^1$-connectedness of retract rational varieties
本文在无限域情形下改进收缩有理簇的 $\mathbb{A}^1$-连通性结果，证明光滑紧致收缩有理簇具有朴素 $\mathbb{A}^1$-连通性。
We show that $\mathbb A^1$-connectedness of a large class of varieties over a field $k$ can be characterized as the condition that their generic point can be connected to a $k$-rational point using (not necessarily naive) $\mathbb A^1$-homotopies. We also show that symmetric powers of $\mathbb A^1$-connected varieties (over an arbitrary field), as well as smooth proper models of them (over an algebraically closed field of characteristic $0$), are $\mathbb A^1$-connected. As an application of these results, we show that the standard norm varieties over a field $k$ of characteristic 0 become $\m…
We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C^2)^6 as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure how to realise the calculation of invariant Hilbert schemes, which have been introduced by Alexeev and Brion. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.
It is a result of Gruson and Peskine that the invariants of a set points in $\ptwo$ in general position are connected. Associated to a space curve there are sequences of invariants which generalize the invariants of points in $\ptwo$. The main result of this paper is to show that the invariants of reduced, irreducible, non-degenerate curves in $\pthree$ also satisfy a connectedness property. This result greatly restricts the kinds of Borel-fixed monomial ideals which can occur as generic initial ideals of such curves and thus gives us more control over their Hilbert functions.
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness via the $\mathbb{A}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all $\mathbb{A}^1$-curves. As an …
On rational connectedness and parametrization of finite Galois extensions
该研究围绕有理数域的$G$-伽罗瓦扩张的$R$-等价关系展开，确定了基本群族$G$的$R$-等价类，在$G$无通用扩张的情况下实现了对应伽罗瓦扩张的参数化。
In this paper we determine a class of admissible matrices which are the Hilbert functions of some 0-dimensional schemes in $\mathbb P^1\times\mathbb P^1$.
In this paper, we study $\mathbb{A}^1$-connected varieties from log geometry point of view, and prove a criterion for $\mathbb{A}^1$-connectedness. As applications, we provide many interesting examples of $\mathbb{A}^1$-connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne's conjecture characterizing projective spaces and affine spaces.
In this paper we show that the Hilbert scheme $H(3,g)$ of locally Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected. In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many irreducible components (roughly $-g/3$ of them). To show connectedness, we classify the curves (giving particular attention to the triple lines), determine the irreducible components, and give flat families over $\Aone$ to show that the components meet. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.
We give an explicit stratification of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the Grothendieck group of varieties. As an application, we calculate the classes of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^3$ and the classes of the Hilbert schemes of $n$ points of $\mathbb{A}^3$ in the Grothendieck of varieties for $n\leq 5$.
Let $Q = \mathbb P^1 x \mathbb P^1$ and let $X\subset Q$ be a 0-dimensional scheme. This paper is a first step towards the characterization of Hilbert functions of 0- dimensional schemes in $Q$. In particular we show how, under some conditions on $X$, its Hilbert function changes when we add points to $X$ lying on a $(1,0)$ or $(0, 1)$-line. As a particular case we show also that if $X$ is ACM this result holds without any additional hypothesis.
We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.
Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$ boils down to being the Hilbert polynomial of $(X,H)$, hence it is totally reducible over $\mathbb C$; moreover, some of the linear factors appearing in the factorization have rational coefficients, e.g. if $X$ has index $\geq 2$. It is natural to ask when the same happens for all linear factors. Here the total reducibility over $\mathbb Q$ of the Hilbert polynomial is investigated for three special kinds of Fano manifolds: Fano manifolds of lar…
Relying on a monadic description of the moduli space of framed sheaves on Hirzebruch surfaces, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\mathcal O(-n)$ on $\mathbb P^1$.
The present paper shall provide a framework for working with Gröbner bases over arbitrary rings $k$ with a prescribed finite standard set $Δ$. We show that the functor associating to a $k$-algebra $B$ the set of all reduced Gröbner bases with standard set $Δ$ is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a $k$-algebra $B$ the set of all border bases with standard set $Δ$ and give reasonably small sets of equations defining t…
The aim of this paper is to give an explicit extension of the classical elliptic integrals to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We study a family of Kummer surfaces corresponding to the Humbert surface of invariant $5$ with two complex parameters. Our Kummer surface is given by a double covering of the weighted projective space $\mathbb{P}(1:1:2)$ branched along a parabola and a quintic curve. The period mapping for our family is given by double integrals of an algebraic function on chambers coming from an arrangement of a parabola and a quintic curve in $\mathbb{C}^2$.
On the connectedness of the singular set of holomorphic foliations
本文研究全纯叶形奇点集的连通性，证明其在特定条件下必连通，并由此得到关于全纯分布可积性的拓扑障碍，回答了Cerveau关于三维射影空间上一维叶形的问题。
We regard the pre-hyperkähler structure on the moduli space of framed flags of sheaves $\mathcal{F}(1,3,1)$ on the projective plane $\mathbb{P}^2$ via an adaptation of the ADHM construction of framed sheaves. Then, we study and categorize the degenerate points of the holomorphic pre-symplectic form presented in the moduli space.
In this paper we introduce an effective method to construct rational deformations between couples of Borel-fixed ideals. These deformations are governed by flat families, so that they correspond to rational curves on the Hilbert scheme. Looking globally at all the deformations among Borel-fixed ideals defining points on the same Hilbert scheme, we are able to give a new proof of the connectedness of the Hilbert scheme and to introduce a new criterion to establish whenever a set of points defined by Borel ideals lies on a common component of the Hilbert scheme. The paper contains a detailed alg…
In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert poly…
Extremal divisors in the Hilbert scheme of points on $\mathbb{P}^{2}$ are preserved under residuality
针对平面 $n$ 点 Hilbert 概型中极值除子的性质问题，在 $n=\frac{r(r+1)}{2}$ 或 $n=r(r+1)$ 时，证明其在剩余性下保持不变。
We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we show the existence (and non-existence as well in some sporadic cases) of every Hilbert scheme of linearly normal curves with $i(\mathcal{D})=4$. We also determine the irreducibility of $\mathcal{H}^\mathcal{L}_{2r+4,r+8,r}$ for $3\le r\le 8$, which are rather pec…
Virtual K-theoretic invariants of the nested Hilbert scheme on $\mathbb{C}^2$
通过构造非交换Hilbert概形的嵌套版本，将$\mathbb{C}^2$上嵌套Hilbert概形表示为向量丛截面的零轨迹，赋予其完美障碍理论，并利用遗忘映射的推出和局部化，推导出等变虚拟欧拉示性数的多元生成级数闭公式。
Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the intersection of C and F, and the curve R residual in C to this intersection. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes. In our paper "Hilbert Schemes of Degree Four Curves" we use this result to prove the connectedness of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curv…
Let $M(n,ξ)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $ξ$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,ξ)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,ξ))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove th…
Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$
本文研究 $\mathbb{P}^5$ 中低次数光滑曲线的 Hilbert 概型，确定 $d\le 14$ 时的不可约性，并分析一般成员的 Hilbert 函数、到模空间的自然映射纤维及射影正规性。
Birational Algebraic Topology
本文在qcqs概形S上研究动机∞-范畴的双有理局部化，证明其与杠构造交换并保持连通性；在完美域k上，刻画了双有理群层为强A1-不变且Gm-收缩平凡；建立了双有理局部化与S^{2,1}-零化函子的等价，并证明了关于proper概形的若干双有理不变量性质。
On Hilbert scheme of complete intersection on the biprojective
本文通过在双齐次形式双次数上定义偏序构造双射影空间中完全交的希尔伯特概型，显式计算特定亏格曲线的希尔伯特概型，并构造$\mathbb{P}^1\times\mathbb{P}^1$中完全交的粗模空间。
We prove a generalization of the Fulton-Hansen connectedness theorem, where ${\mathbb P}^n$ is replaced by a normal variety on which an algebraic group acts with a dense orbit.
In a previous paper, a realization of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that result to construct ADHM data for the Hilbert scheme of points of the total space of the line bundles $\mathcal O(-n)$ on $\mathbb P^1$, for $n \ge 1$, i.e., the resolutions of the singularities of type $\frac1n(1,1)$. Basically by implementing a version of the special McKay correspondence, this ADHM description is in turn used to realize these Hilbert schemes as irreducible connected components of quiver varieties. We obtain in this way …
Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the connectedness of the Hilbert scheme of curves in projective 3 space" Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric su…
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Seve…
We give an ADHM description of the Quot scheme of points ${\rm Quot}_{\mathbb{C}^{n}}(c,r),$ of length $c$ and rank $r$ on affine spaces $\mathbb{C}^{n}$ which naturally extends both Baranovsky's representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces $\mathbb{C}^{n},$ described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of ${\rm Quot}_{\mathbb{C}^{n}}(c,r)$ and its punctual version ${\rm…
On Hilbert's 16th Problem
本文通过将实奇点构造为 Milnor 纤维化上解析函数层在 𝒟-模范畴中的前推微分方程组，为研究平面多项式动力系统的希尔伯特第十六问题提供新工具。
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\mathcal{H}_…
For a variety $X$ separated over a perfect field of characteristic $p>0$ which admits an embedding into a smooth variety, we establish an anti-equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb Z/p \mathbb Z$-sheaves on the étale site $X_{\text{ét}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal O_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisi…
Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular form…
We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gröbner degenerations with respect to the given term order. Then, we introduce…
Disconnected multigraded Hilbert schemes on $\mathbb{P}^2\times\mathbb{P}^1$
本研究针对双射影空间P²×P¹上的多重分次Hilbert概型连通性问题，通过构造特定多项式对应的概型族，证明存在无穷多个不连通的该类概型，拓展了低变量多项式环下的相关结论。
We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat poi…
Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert Irreducibility
本文通过非单位理Hilbert不可约性方法，证明了PSL₂(F_{p²})在特定同余条件下满足正则反Galois问题。
We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.
Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of $\mathbb A ^2$
本文利用Hilbert-Burch矩阵，研究了Hilbert方案上的Białynicki-Birula细胞，并给出了固定点位留正维时的有理映射构造，同时探讨了形式变形的显式描述。
In this article we study the period map for a family of $K3$ surfaces which is given by the anticanonial divisor of a toric variety. We determine the period differential equation and its monodromy group. Moreover we show the exact relation between our period differential equation and the unifomizing differential equation of the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$.
Initial Ideals, Low-degree Foliations and the Hilbert Scheme of Points on $\mathbb{CP}^{2}$: A First Approach
该研究针对复射影平面上点的希尔伯特概型，以$d=2$次全纯叶状结构对应的7个点的希尔伯特函数为对象分析覆盖该概型的仿射簇，同时给出$d=3$次的相关结果，是相关领域的初步探索。
We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.
Hilbert-Burch virtual resolutions for points in $\mathbb{P}^1\times\mathbb{P}^1$
本文证明了 $\mathbb{P}^1\times\mathbb{P}^1$ 中有限点集多重度正则性极小元素的虚拟分解大多属 Hilbert-Burch 类型，给出仅依赖点数的明确描述，并指出其并非总是短分解，提供了长度为 3 的充分条件。
Hilbert's 13th problem in prime characteristic
该研究针对素特征域场景，证明了希尔伯特关于六次、七次（第13问题）、八次一般多项式预解次数的三个猜想在正特征基域下均不成立，为正特征域代数方程可解性研究提供了新结论。
A higher dimensional Hilbert irreducibility theorem
本文在弱Bombieri-Lang猜想下将希尔伯特不可约性定理推广到几何mordellic簇族，并由此证明不存在从Q×Q到Q的多项式双射。
For $G$ a finite group acting linearly on $\mathbb{A}^2$, the equivariant Hilbert scheme $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ is a natural resolution of singularities of $\operatorname{Sym}^r(\mathbb{A}^2/G)$. In this paper we study the topology of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $GL_2$. This is done by using the Bialynicki-Biru…
We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.
In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the $q$-ho…
We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$-homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{F…
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