Proof. We begin by constructing a morphism $\theta : {\mathcal {H}}^{\mathfrak {h}}(S) \to {\mathcal {H}}^{\overline {{\mathfrak {h}}}}(S)$ .

Let $ K $ be an arbitrary field. By the Bigatti-Hulett-Pardue Theorem [Reference PardueP96, Theorem 31], there exists an integer $D_0 \in \mathbb {N}$ , depending only on $ m $ and ${\mathfrak {h}}$ , such that $\mathrm {reg}(I) \leq D_0$ for all homogeneous ideals $I \subseteq K[x_0,\ldots , x_m]$ whose quotient algebra has Hilbert function ${\mathfrak {h}}$ .

By [Reference Bayer and MumfordBM93, Proposition 3.8] or [Reference Caviglia and SbarraCS05, Corollary 2.7], there exists an integer $D_1\in \mathbb {N}$ , depending only on $D_0$ and $ m $ , such that we have $\mathrm {reg}(I) \leq D_1$ for all homogeneous ideals $I \subseteq K[x_0,\ldots , x_m]$ generated by polynomials of degree at most $D_0$ .

Choose an integer $D \geq D_1+m+1$ , and let $\overline {{\mathfrak {h}}}$ be the truncation of ${\mathfrak {h}}$ in degree $ D $ , defined as in the statement of the theorem. The Hilbert scheme ${\mathcal {H}}^{\mathfrak {h}}(S)$ represents the Hilbert functor $\underline {\mathrm {Hilb}}^{\mathfrak {h}}_S$ , from the category of $\Bbbk $ -algebras to the category of sets, defined by

$$ \begin{align*}R \mapsto \big\{\text{homogeneous } I \subseteq R\otimes_\Bbbk S \, \big| \, [(R\otimes_\Bbbk S)/I]_d \text{ locally free over } R \text{ of rank } {{\mathfrak{h}}}(d)\,\, \forall \,\, d \big\}, \end{align*} $$

and likewise for $\overline {{\mathfrak {h}}}$ ; see [Reference Haiman and SturmfelsHS04] for details. Let $\mathbf {m} = (x_0, \ldots , x_m)$ denote the homogeneous maximal ideal of $R \otimes _\Bbbk S$ . For a given $I \in \underline {\mathrm {Hilb}}^{\mathfrak {h}}_S(R)$ , the formula

(3.1) $$ \begin{align} \theta(I) = I + \mathbf{m}^D \end{align} $$

yields an ideal $\theta (I) \subseteq R\otimes _\Bbbk S$ such that $ [(R\otimes _\Bbbk S)/\theta (I)]_d= [(R\otimes _\Bbbk S)/I]_d$ if $d < D$ and $ [(R\otimes _\Bbbk S)/\theta (I)]_d=0$ otherwise. In particular, $ [(R\otimes _\Bbbk S)/\theta (I)]_d$ is locally free of rank $\overline {{\mathfrak {h}}}(d)$ for all $ d $ . Thus, $\theta $ defines a natural transformation $ \underline {\mathrm {Hilb}}^{\mathfrak {h}}_S(R) \to \underline {\mathrm {Hilb}}^{\overline {{\mathfrak {h}}}}_S(R)$ , and a morphism of schemes $\theta : {\mathcal {H}}^{\mathfrak {h}}(S) \to {\mathcal {H}}^{\overline {{\mathfrak {h}}}}(S)$ .

We show that $\theta $ is bijective on $ K $ -points, where $\Bbbk \subseteq K$ is any field extension. Observe that a $ K $ -point of ${\mathcal {H}}^{\mathfrak {h}}(S)$ is a homogeneous ideal

$$ \begin{align*}I \subseteq K \otimes_\Bbbk S = K[x_0,\ldots,x_m] =:T \end{align*} $$

whose quotient has Hilbert function ${\mathfrak {h}}$ , that is, an ideal $I\in {\mathcal {H}}^{\mathfrak {h}}(T)$ ; likewise for a $ K $ -point of ${\mathcal {H}}^{\overline {{\mathfrak {h}}}}(S)$ . We use the symbol $\theta $ also to denote the induced map $\theta : {\mathcal {H}}^{\mathfrak {h}}(T) \to {\mathcal {H}}^{\overline {{\mathfrak {h}}}}(T)$ , which is defined by the same formula (3.1). Consider the map $\rho \colon {\mathcal {H}}^{\overline {{\mathfrak {h}}}}(T) \to {\mathcal {H}}^{\mathfrak {h}}(T)$ defined, at the level of $ K $ -points, by the formula $ \rho (J) = J_{\leq D_0} $ . We claim that $\rho $ and $\theta $ are inverses as set-theoretic maps.

For every ideal $I \in {\mathcal {H}}^{\mathfrak {h}}(T)$ , since $ I $ is generated in degrees at most equal to $D_0$ and since $D_0<D$ , we have

$$ \begin{align*}I = I_{\leq D_0} = (I+\mathbf{m}^{D})_{\leq D_0}\, ;\end{align*} $$

therefore, $\rho (\theta (I)) = I$ . Consider, in particular, the lexsegment ideal with Hilbert function ${\mathfrak {h}}$

$$ \begin{align*}L =\mathrm{Lex}({\mathfrak{h}}) \in {\mathcal{H}}^{\mathfrak{h}}(T).\end{align*} $$

Then,

$$ \begin{align*}\overline{L}= \theta(L) = L+\mathbf{m}^{D}\in {\mathcal{H}}^{\overline{{\mathfrak{h}}}}(T)\end{align*} $$

is again a lexsegment ideal, with Hilbert function equal to $\overline {{\mathfrak {h}}}$ . Since $ L $ is generated in degrees up to $D_0$ , $\overline {L}$ is generated in degrees up to $D_0$ and in degree $ D $ . Using the Bigatti-Hulett-Pardue Theorem again, we conclude that every ideal $J \subseteq T$ with Hilbert function equal to ${\overline {{\mathfrak {h}}}}$ is generated, at most, in degrees up to $D_0$ and in degree $ D $ .

Consider now any such ideal $J \subseteq T$ with Hilbert function equal to ${\overline {{\mathfrak {h}}}}$ . By the conclusion of the previous paragraph, the graded components of the ideals $ J $ and $\rho (J) =J_{\leq D_0}$ agree up to degree $D-1$ . In other words,

$$ \begin{align*}\mathrm{HF}(T/\rho(J);d) = \overline{{\mathfrak{h}}} (d)= {\mathfrak{h}} (d)\end{align*} $$

for all $d\leq D-1$ (in particular, for all $d \leq D_1+m$ ). Since $\rho (J)$ is generated in degrees up to $D_0$ , it follows, by choice of $D_1$ , that the regularity of $\rho (J)$ is at most equal to $D_1$ . Therefore,

$$ \begin{align*}\mathrm{HP}(T/\rho(J);d) = \mathrm{HF}(T/\rho(J);d)\end{align*} $$

for all $d \geq D_1$ . Let ${\mathfrak {p}}(z)$ be the Hilbert polynomial of the Hilbert function ${\mathfrak {h}}$ . By choice of $D_0$ , we have ${\mathfrak {p}}(d) = {\mathfrak {h}}(d)$ for all $d \geq D_0$ . In conclusion, we have

$$ \begin{align*}\mathrm{HP}(T/\rho(J);d) = \mathrm{HF}(T/\rho(J);d) = \overline{{\mathfrak{h}}} (d)= {\mathfrak{h}} (d) = {\mathfrak{p}}(d) \end{align*} $$

for all $d = D_1, D_1+1, \ldots , D_1+m$ . Since $\mathrm {HP}(T/\rho (J);d) $ and $ {\mathfrak {p}}(d)$ are univariate polynomials of degree at most $ m $ , they must be the same polynomial. We deduce that

$$ \begin{align*}\mathrm{HF}(T/\rho(J);d) ={\mathfrak{p}}(d) = {\mathfrak{h}}(d)\end{align*} $$

for all $d \geq D_1$ ; in other words, $T/\rho (J)$ has Hilbert function equal to $ {\mathfrak {h}} $ , and $\rho (J) \in {\mathcal {H}}^{{{\mathfrak {h}}}}(T)$ .

Since $ J $ is generated in degrees up to $D_0$ and in degree $ D $ , and $J_{\geq D} = \mathbf {m}^{D}$ , we obtain

$$ \begin{align*}J = J_{\leq D_0} + \mathbf{m}^{D}\, ;\end{align*} $$

therefore, $\theta (\rho (J))=J$ as desired. Therefore, we have proven that $\theta : {\mathcal {H}}^{\mathfrak {h}}(S) \to {\mathcal {H}}^{\overline {{\mathfrak {h}}}}(S)$ is bijective on $ K $ -points.

Next, we show that $\theta $ is an open immersion. By [Reference ErmanE12, Proposition 3.1], for each $I \in {\mathcal {H}}^{\mathfrak {h}}(S)$ , the morphism $\theta $ induces an isomorphism between the functors of Artin rings which send a local Artinian $\Bbbk $ -algebra $ R $ to the set of homogeneous deformations over $ R $ of $S/I$ and of $S/\theta (I)$ , respectively. These two functors of Artin rings are pro-represented by the completed local rings $\widehat {\mathscr {O}}_{{\mathcal {H}}^{\mathfrak {h}}(S),I}$ and $\widehat {\mathscr {O}}_{{\mathcal {H}}^{\overline {{\mathfrak {h}}}}(S),\theta (I)}$ of the two Hilbert schemes at the respective $\Bbbk $ -points (see, e.g., [Reference SernesiS07, Section 2.2]). Thus, $\theta $ induces isomorphisms of completed local rings at $\Bbbk $ -points and therefore is an étale morphism (since $\Bbbk $ is algebraically closed).

Since $\theta $ is injective on $ K $ -points for any field $\Bbbk \subseteq K$ , it follows from [Sta, Lemma 29.10.2] that it is universally injective. Since $\theta $ is universally injective and étale, it is an open immersion by [Sta, Lemma 35.25.2].

Finally, since $\theta $ is an open immersion which is bijective on $\Bbbk $ -points, and $\Bbbk $ is algebraically closed, it follows that it must be an isomorphism.

Proof. Let $\mathscr {M}_{g}$ be the moduli space of curves of genus $ g $ . By [Reference Eisenbud and HarrisEH87, Reference Harris and MumfordHM82, Reference Farkas, Jensen and PayneFJP25], $\mathscr {M}_{g}$ is of general type if $g \geq 22$ . We will prove the theorem by constructing a dominant rational map

$$ \begin{align*}\mathcal{C}\dashrightarrow \mathscr{M}_{g}\, ,\end{align*} $$

where $\mathcal {C}\subseteq \mathrm {Hilb}(\mathbb {A}^{12})$ is an irreducible component. We will realize the map as a composition of a series of dominant morphisms $({\mathsf {H}}_{i+1},I_{i+1}) \to ({\mathsf {H}}_{i},I_{i})$ , where each ${\mathsf {H}}_i$ is an irreducible open subscheme of a corresponding Hilbert scheme and $I_i\in {\mathsf {H}}_i$ is an ideal parametrized by that Hilbert scheme.

Fix integers $g \geq 22$ and $d\geq \frac {3g+12}{4}$ , and consider smooth curves in $\mathbb {P}^3$ of genus $ g $ and degree $ d $ . The inequality $d\geq \frac {3g+12}{4}$ amounts to the nonnegativity of the Brill–Noether number $\rho (g, 3,d)=g-4(g-d+3) $ . It follows from the main results of Brill–Noether theory that there exists a unique irreducible component

$$ \begin{align*}Y \subseteq \mathrm{Hilb}_{dz+1-g}(\mathbb{P}^3)\end{align*} $$

which dominates the moduli space $\mathscr {M}_{g}$ ; see, for example, [Reference Eisenbud and HarrisEH87-2] and references therein or [Reference Eisenbud and HarrisEH24, Theorem 19.9]. For a general curve $C\subseteq \mathbb P^3$ parametrized by $ Y $ , the Hilbert function of the coordinate ring is determined by the Maximal Rank Theorem [Reference Ballico and ElliaBE85, Reference LarsonL17] (see also [Reference Eisenbud and HarrisEH24, Theorem 12.10]), asserting that the multiplication maps

$$ \begin{align*}\text{Sym}^a H^0(C, \mathscr{O}_C(1))\rightarrow H^0(C, \mathscr{O}_C(a))\end{align*} $$

are of maximal rank for all integers $a\geq 1$ . In particular, there exists a nonempty open subset ${\mathsf {H}}_1 \subseteq Y$ where the Hilbert function of $S/I$ is constant for all $I \in {\mathsf {H}}_1$ . By comparing dimensions, curves $ C $ parametrized by ${\mathsf {H}}_1$ never lie on a quadric surface. Let ${\mathfrak {h}}$ denote this Hilbert function, and let $I_1 \in {\mathsf {H}}_1$ be an ideal that belongs to no other component of $\mathrm {Hilb}_{dz+1-g}(\mathbb {P}^3)$ .

Let $S= \mathbb {C}[x_0, x_1,x_2, x_3]$ and consider the saturated ideal $I_2 = I_1$ as a point in the graded Hilbert scheme ${\mathcal {H}}^{\mathfrak {h}}(S)$ . The natural map ${\mathcal {H}}^{\mathfrak {h}}(S) \to \mathrm {Hilb}_{dz+1-g}(\mathbb {P}^3)$ restricts to an isomorphism $({\mathsf {H}}_2,I_2) \cong ({\mathsf {H}}_1,I_1)$ for some ${\mathsf {H}}_2\subseteq {\mathcal {H}}^{\mathfrak {h}}(S)$ open and irreducible.

Let $P = S[x_{4}, x_{5}] = \mathbb {C}[x_0, \ldots , x_{5}]$ , let $I_3 = I_2 \cdot P$ be the extended ideal, and let ${\mathfrak {h}}' =\mathrm {HF}(P/I_3)$ . We claim that there is a local retraction

(4.1) $$ \begin{align} ({\mathcal{H}}^{{\mathfrak{h}}'}(P),I_3) \to ({\mathcal{H}}^{{\mathfrak{h}}}(S),I_2)\,. \end{align} $$

The local retraction (4.1) can be constructed by means of a suitable Białynicki–Birula decomposition, proceeding very similarly to [Reference JelisiejewJ20, Proof of Theorem 1.3]. Specifically, let $\mathbb {G}_m = \mathbb {C}^*$ act on $ P $ with weight 1 on the variables $x_4, x_5$ and weight 0 on the remaining variables. Let $\delta _1$ denote the standard grading $\delta _1(x_i) = 1$ for all $ i $ , and let $\delta _2$ denote the $\mathbb {Z}$ -grading on $ P $ induced by the $\mathbb {G}_m$ -action,

$$ \begin{align*}\delta_2(x_0)= \delta_2(x_1)= \delta_2(x_2)=\delta_2(x_3)= 0\, , \ \ \ \ \delta_2(x_4)=\delta_2(x_5) =1. \end{align*} $$

By general theory [Reference DrinfeldD13, Reference JelisiejewJ19, Reference Jelisiejew and SienkiewiczJS19], there exists a $\mathbb {C}$ -scheme ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^-$ , called the negativeFootnote ² Białynicki–Birula decomposition for ${\mathcal {H}}^{{\mathfrak {h}}'}(P)$ . The scheme ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^-$ represents the functor of $\mathbb {G}_m$ -equivariant families that admit a limit at $\infty $ : the functor assigns to a $\mathbb {C}$ -algebra $ R $ the ideals of $R [t^{-1},x_0, \ldots , x_{5}]$ that are homogeneous with respect to $\delta _1$ , with locally free components of ranks ${\mathfrak {h}}'(d)$ , and are also homogeneous with respect to $\delta _2$ , where

$$ \begin{align*}\delta_1(t^{-1})=0 \ \text{ and }\ \delta_2(t^{-1})=-1\, .\end{align*} $$

There is a natural map ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^-\to {\mathcal {H}}^{{\mathfrak {h}}'}(P)$ by evaluating at $t=1$ . By taking the limit $t \to \infty $ , there is a natural map to the $\mathbb {G}_m$ -fixed locus,

(4.2) $$ \begin{align} {\mathcal{H}}^{{\mathfrak{h}}'}(P)^-\to {\mathcal{H}}^{{\mathfrak{h}}'}(P)^{\mathbb{G}_m}\,. \end{align} $$

Since there is a section ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^{\mathbb {G}_m} \to {\mathcal {H}}^{{\mathfrak {h}}'}(P)^- $ defined by taking constant families, the map (4.2) is a retraction. See [Reference JelisiejewJ20, Diagram (1.1)] and the related discussion for more details.

Since $I_3$ is homogeneous with respect to both $\delta _1$ and $\delta _2$ , we can consider $I_3$ as a $\mathbb {C}$ -point of all three schemes ${\mathcal {H}}^{{\mathfrak {h}}'}(P)$ , ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^-$ , and ${\mathcal {H}}^{{\mathfrak {h}}'}(P)^{\mathbb {G}_m}$ . The tangent space to $ {\mathcal {H}}^{{\mathfrak {h}}'}(P)$ at the point $I_3$ is the vector space $[\mathrm {Hom}_P(I_3,P/I_3)]_0$ of $ P $ -linear homomorphisms that are of $\delta _1$ -degree 0. The tangent space to $ {\mathcal {H}}^{{\mathfrak {h}}'}(P)^-$ at $I_3$ is the subspace consisting of homomorphisms that are also $\delta _2$ -graded, of nonnegative degree. Since the generators of $I_3$ have $\delta _2$ -degree 0, and $P/I_3$ is concentrated in nonnegative $\delta _2$ -degrees, the two tangent spaces coincide; that is, $[\mathrm {Hom}_P(I_3,P/I_3)]_0$ is already concentrated in nonnegative $\delta _2$ -degrees. As a consequence, the map

(4.3) $$ \begin{align} {\mathcal{H}}^{{\mathfrak{h}}'}(P)^-\to {\mathcal{H}}^{{\mathfrak{h}}'}(P) \end{align} $$

is an open immersion around $I_3$ ; see [Reference Jelisiejew and SienkiewiczJS19, Example 1.3 and Proposition 1.6] or the argument of [Reference JelisiejewJ19, Theorem 4.5]. After combining the open immersion (4.3) with the retraction (4.2), we obtain a local retraction

$$ \begin{align*}({\mathcal{H}}^{{\mathfrak{h}}'}(P),I_3)\to ({\mathcal{H}}^{{\mathfrak{h}}'}(P)^{\mathbb{G}_m},I_3)\,.\end{align*} $$

For the last step in the construction of (4.1), we observe that $({\mathcal {H}}^{{\mathfrak {h}}'}(P)^{\mathbb {G}_m},I_3)$ is isomorphic to $({\mathcal {H}}^{{\mathfrak {h}}}(S),I_2)$ , with isomorphisms given by extension and contraction of ideals along the ring map $S \to P$ . The ideals of $ P $ that are homogeneous with respect to both $\delta _1$ and $\delta _2$ and have the same bigraded Hilbert function as $I_3 = I_2 \cdot P$ are precisely the extensions of homogeneous ideals of $ S $ with the same Hilbert function as $I_2$ .

Using the local retraction (4.1), we conclude that there is a dominant morphism

$$ \begin{align*}({\mathsf{H}}_3, I_3) \to ({\mathsf{H}}_2,I_2)\end{align*} $$

for some open irreducible ${\mathsf {H}}_3 \subseteq {\mathcal {H}}^{{\mathfrak {h}}'}(P)$ . Since $I_2$ is saturated, we have $\mathrm {depth}(S/I_2) \geq 1$ and, therefore, $\mathrm {depth}(P/I_3) \geq 3 $ . Moreover, $I_3$ contains no quadrics by construction.

Let $D\in \mathbb {N}$ be an integer obtained by applying Theorem 2 to the graded Hilbert scheme ${\mathcal {H}}^{{\mathfrak {h}}'}(P)$ , and let

$$ \begin{align*}I_4 = I_3 + (x_0, \ldots, x_{5})^{D} \subseteq P\, .\end{align*} $$

By Theorem 2, there exists an isomorphism $({\mathsf {H}}_4,I_4) \cong ({\mathsf {H}}_3,I_3)$ for some open irreducible ${\mathsf {H}}_4 \subseteq {\mathcal {H}}^{\overline {{\mathfrak {h}}}'}(P)$ , where $\overline {{\mathfrak {h}}}' = \mathrm {HF}(P/I_4)$ .

Let $T = P[y_0, \ldots , y_{5}] = \mathbb {C}[x_0, \ldots , x_{5 }, y_0,\ldots , y_{5}]$ , and consider the ideal

$$ \begin{align*}I_5 = I_3 T + (x_0, \ldots, x_{5})^D+ (y_0, \ldots, y_{5})^2 + \left( \sum_{i=0}^5 x_i y_i\right) \subseteq T. \end{align*} $$

Then, $T/I_5$ is a finite $\mathbb {C}$ -algebra, so $I_5 \in \mathrm {Hilb}(\mathbb {A}^{12})$ . In [Reference JelisiejewJ20, Equation (1.3)], the ideal $I_5$ is called a TNT frame of size $a=D-1$ for $I_3$ . Since $\mathrm {depth}(P/I_3) \geq 3 $ and $I_3$ contains no quadrics, we can apply [Reference JelisiejewJ20, Proposition 1.4] and obtain a local retraction

$$ \begin{align*}(\mathrm{Hilb} (\mathbb{A}^{12}), I_5) \to ({\mathcal{H}}^{\overline{{\mathfrak{h}}}'}(P), I_4)\, .\end{align*} $$

Therefore, there exists a dominant morphism $({\mathsf {H}}_5,I_5) \to ({\mathsf {H}}_4,I_4)$ for some open irreducible $ {\mathsf {H}}_5 \subseteq \mathrm {Hilb} (\mathbb {A}^{12})$ .

Finally, consider the irreducible component $\mathcal {C}= \overline {{\mathsf {H}}}_5 \subseteq \mathrm {Hilb} (\mathbb {A}^{12})$ . The composition of the maps

$$ \begin{align*}({\mathsf{H}}_5, I_5) \to ({\mathsf{H}}_4,I_4)\cong ({\mathsf{H}}_3,I_3) \to ({\mathsf{H}}_2,I_2) \cong ({\mathsf{H}}_1,I_1) \end{align*} $$

together with the dominant rational map $Y \dashrightarrow \mathscr {M}_{g}$ yields a dominant rational map

$$ \begin{align*}\mathcal{C}\dashrightarrow \mathscr{M}_{g}\, .\end{align*} $$

We obtain a dominant rational map $\mathcal {C}_{red}\dashrightarrow \mathscr {M}_{g}$ on the reduced structure. Since the moduli space $\mathscr {M}_{g}$ is of general type for $g \geq 22$ , we conclude that $\mathcal {C}_{red}$ is not rationally connected. In fact, by functoriality of MRC-fibrations [Reference KollárK96, Theorem IV.5.5], we obtain a dominant rational map

$$ \begin{align*}MRC(\mathcal{C}_{red})\dashrightarrow MRC(\mathscr{M}_{g})\, .\end{align*} $$

Since $\mathscr {M}_{g}$ is of general type, we have $MRC(\mathscr {M}_{g})= \mathscr {M}_{g}$ . Thus, the MRC dimension of $\mathcal {C}_{red}$ is at least $\dim \mathscr {M}_{g} = 3g-3$ .

Proof. We continue to use the notation of the proof of Theorem 3. The argument is, again, based on a suitable Białynicki–Birula decomposition, and is very similar to the one used in the proof of Theorem 3. Specifically, let $Q = T[z_{13}, \ldots , z_n]$ and consider the ideal

$$ \begin{align*}I_6 = I_5 \cdot Q+ (z_{13}, \ldots, z_n) \subseteq Q\, .\end{align*} $$

Let $\mathbb {G}_m = \mathbb {C}^*$ act on $ Q $ with weight 1 on the variables $z_i$ and weight 0 on the remaining variables. The action induces a $\mathbb {Z}$ -grading $\delta _3$ on $ Q $ with

$$ \begin{align*}\delta_3(x_i) = \delta_3(y_i) =0\, ,\ \ \delta_3(z_j) =1\, .\end{align*} $$

We consider now the positive Białynicki–Birula decomposition $\mathrm {Hilb}(\mathbb {A}^n)^+$ for $\mathrm {Hilb}(\mathbb {A}^n)$ , that is, the decomposition associated with the compactification $\overline {\mathbb {G}}_m = \mathbb {G}_m\cup \{0\}$ associated to the limit $t \to 0$ .

Since $I_6$ is generated in $\delta _3$ -degrees 0 and 1, and $Q/I_6$ is concentrated in $\delta _3$ -degree 0, the tangent space $\mathrm {Hom}(I_6,Q/I_6)$ to $\mathrm {Hilb}(\mathbb {A}^n)$ at the point $I_6$ is concentrated in nonpositive $\delta _3$ -degrees. Therefore, the tangent spaces to $\mathrm {Hilb}(\mathbb {A}^n)^+$ and to $\mathrm {Hilb}(\mathbb {A}^n)$ at the point $I_6$ coincide. As before, it follows from [Reference Jelisiejew and SienkiewiczJS19, Proposition 1.6] that the natural map

$$ \begin{align*}\mathrm{Hilb}(\mathbb{A}^n)^+ \to \mathrm{Hilb}(\mathbb{A}^n)\end{align*} $$

is an open immersion near $I_6$ , and that we have a local retraction to the fixed locus

$$ \begin{align*}(\mathrm{Hilb}(\mathbb{A}^n),I_6) \to (\mathrm{Hilb}(\mathbb{A}^n)^{\mathbb{G}_m},I_6)\, .\end{align*} $$

The connected component of $\mathrm {Hilb}(\mathbb {A}^n)^{\mathbb {G}_m}$ containing $I_6$ parametrizes $\delta _3$ -homogeneous ideals of $ Q $ with the same Hilbert function as $I_6$ . Since they define quotients of $ Q $ that are concentrated in $\delta _3$ -degree 0, they are exactly the ideals of the form

$$ \begin{align*}I \cdot Q + (z_{13}, \ldots, z_n) \subseteq Q\, ,\end{align*} $$

where $I \in \mathrm {Hilb}(\mathbb {A}^{12})$ has the same colength as $I_5$ . The connected component of $\mathrm {Hilb}(\mathbb {A}^n)^{\mathbb {G}_m}$ containing $I_6$ is therefore isomorphic to the connected component of $\mathrm {Hilb}(\mathbb {A}^{12})$ containing $I_5$ . We thus obtain a local retraction

$$ \begin{align*}(\mathrm{Hilb}(\mathbb{A}^n),I_6) \to (\mathrm{Hilb}(\mathbb{A}^{12}),I_5)\, ,\end{align*} $$

and hence a dominant morphism $({\mathsf {H}}_6,I_6) \to ({\mathsf {H}}_5,I_5)$ for some open irreducible $ {\mathsf {H}}_6 \subseteq \mathrm {Hilb} (\mathbb {A}^{n})$ . The conclusion follows as in the last paragraph of the proof of Theorem 3 by taking the composition with the dominant rational map $\mathcal {C}\dashrightarrow \mathscr {M}_{g}$ .

Question 6. Are any of the irreducible components of $\mathrm {Hilb}(\mathbb {A}^{12})$ constructed in the proof of Theorem 3 reduced?

Question 7. Does $\mathrm {Hilb}(\mathbb {A}^{3})$ contain irrational components?

Question 8. Does $\mathrm {Hilb}(\mathbb {A}^n)$ contain components of unbounded degree of irrationality?

Question 9. Do all birational types of MRC-quotients of varieties defined over $\overline {\mathbb {Q}}$ occur for irreducible components of $\mathrm {Hilb}(\mathbb {A}^n)$ over $\overline {\mathbb {Q}}$ ?