Instability and Singularity of Projective Hypersurfaces

TL;DR

This paper investigates the properties of projective hypersurfaces, demonstrating the invariance of Hesselink stratification and establishing a link between virtual 1-parameter subgroup length and hypersurface multiplicity.

Contribution

It proves the independence of Hesselink stratification from Plücker coordinates and relates the worst virtual 1-parameter subgroup length to hypersurface multiplicity.

Findings

Hesselink stratification is coordinate-independent.

A positive correlation exists between virtual subgroup length and multiplicity.

The results enhance understanding of hypersurface stability and singularity.

Abstract

In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Pl\"ucker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface.

Equations

$$P(t)=\left(\genfrac{}{}{0ex}{}{r+t}{r}\right)-\left(\genfrac{}{}{0ex}{}{r+t-d}{r}\right)$$

$$\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})\rightarrow\mathbb{P}\bigg{(}\bigwedge^{Q(d+t)}H^{0}(\mathbb{P}^{r}_{k},\mathcal{O}_{\mathbb{P}^{r}_{k}}(d+t))\bigg{)}$$

$${\text{Hilb}}^P({\mathbb{P}}_k^r{)}_{d+t}^{\text{us}}=\coprod _{[\lambda ],\delta >0}{E}_{d+t,[\lambda ],\delta }$$

$$\phi_{d}:\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})\rightarrow\mathbb{P}\bigg{(}\bigwedge^{Q(d)}H^{0}(\mathbb{P}^{r}_{k},\mathcal{O}_{\mathbb{P}^{r}_{k}}(d))\bigg{)}.$$

$$Q(d)=h^0({\mathbb{P}}_k^r,{\mathcal{O}}_{{\mathbb{P}}_k^r}(d))-P(d).$$

$${\varphi }_d\left({\text{Hilb}}^P({\mathbb{P}}_k^r)\right)=\{V\in \text{Gr}(S_d,Q(d))|{\dim }_k{S}_{1}V\le {\dim }_k{S}_{1}U,\forall U\in \text{Gr}(S_d,Q(d))\}.$$

$$\lambda_{i}(t)_{ab}=\left\{\begin{array}[]{ll}t&a=b=i\\ 1&a=b\neq i\\ 0&a\neq b\end{array}\right..$$

$$V=\underset{i\in \mathbb{Z}}{⨁}{V}_i$$

$$\mu (v,{\lambda }^{\prime })=\min \{i\in \mathbb{Z}|v_i\ne 0\}.$$

$$\Lambda_{x,d}=\bigg{\{}\lambda\in\Gamma(H)\bigg{|}\lambda\textrm{ is indivisible and }\frac{\mu(\phi_{d}(x),\lambda)}{\|\lambda\|}=\max_{\lambda^{\prime}\in\Gamma(H)}\frac{\mu(\phi_{d}(x),\lambda^{\prime})}{\|\lambda^{\prime}\|}.\bigg{\}}.$$

$$E_{d,[\lambda],\delta}=\bigg{\{}x\in\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})\bigg{|}\Lambda_{x,d}\cap[\lambda]\neq\emptyset,\quad\frac{\mu(\phi_{d}(x),\lambda_{\textup{max}})}{\|\lambda_{\textup{max}}\|}=\delta,\textrm{ }\forall\lambda_{\textup{max}}\in\Lambda_{x,d}.\bigg{\}}$$

$${\text{Hilb}}^P({\mathbb{P}}_k^r{)}_d^{\text{us}}=\coprod _{[\lambda ],\delta >0}{E}_{d,[\lambda ],\delta }$$

$$V_d^P=\underset{m\in M_{dQ(d)}}{⨁}{V}_{m,d}^P$$

$$\bigg{\{}\bigwedge_{i=1}^{Q(d)}m_{i}\bigg{|}m_{i}\in M_{d}\textrm{ for all }1\leq i\leq Q(d)\textrm{ and }\prod_{i=1}^{Q(d)}m_{i}=m\bigg{\}}.$$

$${\varphi }_d(x)=\left[\sum _{m\in M_{dQ(d)}}{\varphi }_d(x{)}_m\right]$$

$${\Xi }_{x,d}=\{m\in M_{dQ(d)}|{\varphi }_d(x{)}_m\ne 0\}.$$

$$f=\sum _{m\in M_d}{f}_{m}m.$$

$${\varphi }_d(x)\in E_{d,[{\lambda }_{g.x,d}],|{\Delta }_{g.x,d}|}$$

$$n_{p,X}=\min\bigg{\{}t\in\mathbb{N}\bigg{|}x_{0}^{d-t}|f\bigg{\}}.$$

$$m<_{\lambda}m^{\prime}\Longleftrightarrow\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m\bigg{\rangle}<\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m^{\prime}\bigg{\rangle}\textrm{ or}$$

$$\left[\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m\bigg{\rangle}=\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m\bigg{\rangle}\textrm{ and }m<_{\textrm{lex}}m^{\prime}\right].$$

$$|\Delta_{x,t}|=\max_{\lambda\in\Gamma(T)}\min_{m\in\Xi_{x,t}}\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m\bigg{\rangle}.$$

$$\min_{m\in\Xi_{x,t}}\bigg{\langle}\eta(\lambda)\otimes\frac{1}{\|\lambda\|},m\bigg{\rangle}$$

$$\frac{\mu ({\varphi }_t(x),\lambda )}{\Vert \lambda \Vert }.$$

$$P(t)=\left(\genfrac{}{}{0ex}{}{r+t}{r}\right)-\left(\genfrac{}{}{0ex}{}{r+t-d}{r}\right).$$

$${\text{Hilb}}^P({\mathbb{P}}_k^r{)}_t^{\text{us}}=\coprod _{[\lambda ],\delta >0}{E}_{t,[\lambda ],\delta }$$

$$\tau (\delta ,D)=|M_D|\delta$$

$$E_{d+D,[\lambda ],\tau (\delta ,D)}=E_{d,[\lambda ],\delta }$$

$$g.f=\sum _{i=1}^{|M_d|}(g.f{)}_i{m}_{i,d}.$$

$$\min_{m\in\Xi_{g.x,d}}\bigg{\langle}\eta(\gamma)\otimes\frac{1}{\|\gamma\|},m\bigg{\rangle}=\bigg{\langle}\eta(\gamma)\otimes\frac{1}{\|\gamma\|},m_{\alpha,d}\bigg{\rangle}$$

Full text

This work was supported by IBS-R003-D1.

1. Introduction

Let $k$ be an algebraically closed field. The table in [8, p. 80] says that a hypersurface over $k$ is unstable if and only if it has a singular point of some special tangent cone for some possibly small degree and dimension.

If there is a singular point $p$ of a hypersurface $X$, we can measure its magnitude of singularity by its multiplicity $n_{p,X}$, which is defined to be the degree of the tangent cone $TC_{p}X$ as a subscheme of the tangent space $T_{p}X$([2, p. 258]). In particular, $n_{p,X}\geq 2$ if and only if $p$ is a singular point of $X$. The number $n_{X}=\max_{p\in X}n_{p,X}$ is a geometric invariant.

On the other hands, we can measure how much a hypersurface is unstable. A hypersurface of a projective space is represented by a $k$ point of a Hilbert scheme $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$, when $P$ is of the form

[TABLE]

for some $r$ and $d$. There is a canonical action of $\textup{SL}_{r+1}(k)$ on $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$, possibly with a choice of Plücker coordinate

[TABLE]

for some $t\geq 0$. For each choice of $t$, we have the unstable locus of a Hilbert scheme and Hesselink stratification of the unstable locus

[TABLE]

where each $E_{d+t,[\lambda],\delta}$ is a constructible subset of the Hilbert scheme([3]).

In this paper, we will show that every Hesselink stratification in (1) is actually independent of the choice of the natural number $t$(Theorem 3.1). We will also show that there is a positive relation between $n_{H}$ and the pair $([\lambda],\delta)$ where $x\in E_{d,[\lambda],\delta}$ for the Hilbert point $x$ which represents $H$, for every projective hypersurface $H$ over $k$(Theorem 4.3). This is given by (6), which is a sharp inequality.

2. Preliminaries and Notations

A sequence of closed immersions which map a Hilbert Scheme $\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})$ into projective spaces is given in [1, (3.4)]. If $g_{P}$ is the Gotzmann number corresponding to $P$, then for any $d\geq g_{P}$, there is a closed immersion

[TABLE]

Here,

[TABLE]

Let $S=k[x_{0},\ldots,x_{r}]$ be graded by the degree of polynomial. Then we have natural isomorphism $S\cong\oplus_{i\geq 0}H^{0}(\mathbb{P}^{r}_{k},\mathcal{O}_{\mathbb{P}^{r}_{k}}(i))$ as graded rings over $k$.

For any closed point $x\in\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})$ which represents a saturated homogeneous ideal $I\subset S$, $\phi_{d}(x)=[I^{\wedge Q(d)}]$. We can describe the set of closed points in the image of $\phi_{d}$ explicitly:

[TABLE]

Here we consider $\textrm{Gr}(S_{d},Q(d))$ as a closed subscheme of $\mathbb{P}\left(\bigwedge^{Q(d)}S_{d}\right)$ via Plücker embedding.

Now $\mathbb{P}\left(\bigwedge^{Q(d)}S_{d}\right)$ admits a canonical $G:=\textrm{GL}_{r+1}(k)$-action which stabilizes the image of $\phi_{d}$ so that we can discuss about the unstable locus of a Hilbert Scheme via induced $H=\textup{SL}_{r+1}(k)$ action.[8, Theorem 1.19 and Theorem 2.1, 2nd Chpater]. We call $\phi_{d}(x)$ a $d$’th Hilbert point for any $x\in\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})$ and $d\geq g_{P}$.

2.1. Hesselink Stratification of a Hilbert Scheme and State Polytope

We can define a real-valued norm $\|\cdot\|$ on the group of every 1-parameter subgroups $\Gamma(G)$ of $G$, which satisfies conditions in [5, p.305]. Let $T_{0}$ be the maximal torus of $G$ which consists of all diagonal matrices. Then $\Gamma(T_{0})\cong\mathbb{Z}^{r+1}$ so there is the Euclidean norm on $\Gamma(T_{0})\otimes\mathbb{R}$, associated with the standard basis given by $\{\lambda_{i}|0\leq i\leq r\}$ satisfying

[TABLE]

For any $g\in G$ and $\lambda\in\Gamma(G)$, let $g\star\lambda\in\Gamma(G)$ be the 1-parameter subgroup which maps $t\in\mathbb{G}_{m}$ to $g\lambda(t)g^{-1}$. The norm $\|\cdot\|$ is invariant under the action of the Weyl group of $T_{0}$ in $G$. Thus this norm can be extended to $\Gamma(G)$ via conjugation by [4, Corollary A, p. 135]. Let $T=H\cap T_{0}$, which is a maximal torus of $H$.

Suppose $\lambda^{\prime}\in\Gamma(G)$ and $V$ be a $G$-representation. There is a decomposition

[TABLE]

where $V_{i}=\{w\in V|\lambda^{\prime}(t).w=t^{i}w\}$([7, Proposition 4.7.])and there is a decomposition $v=\sum_{i\in\mathbb{Z}}v_{i}$ with $v_{i}\in V_{i}$ so that we can explain the value $\mu(v,\lambda^{\prime})$ as follows:

[TABLE]

For any unstable $d$’th Hilbert point $\phi_{d}(x)$, any two indivisible worst 1-parameter subgroups of $\phi_{d}(x)$ are conjugate so that we can measure the magnitude of instability of $\phi_{d}(x)$ by the conjugacy classes of indivisible 1-parameter subgroups[5, Theorem 3.4]. Let $\Lambda_{x,d}$ be the set of all indivisible worst 1-parameter subgroups of $\phi_{d}(x)$. That is,

[TABLE]

The set $\Lambda_{x,d}$ is non-empty. Moreover,

[TABLE]

is a constructible subset of $\textrm{Hilb}^{P}(\mathbb{P}^{r}_{k})$ for a conjugacy class $[\lambda]$ containing an indivisible 1-parameter subgroup $\lambda$ of $T$[3]. Consequently, there is a stratification

[TABLE]

for each integer $d\geq g_{P}$.

The $T_{0}$ action induced by the G action on $V_{d}^{P}=\wedge^{Q(d)}S_{d}$ has decomposition

[TABLE]

where $V_{m,d}^{P}$ is a $k$-subspace of $\wedge^{Q(d)}S_{d}$ which is spanned by

[TABLE]

Here $M_{d}$ is the set of monomials of degree $d$ in $S$. This decomposition coincides with weight decomposition of the $T_{0}$ action because wedges of monomials are exactly the eigenvectors of the action. Multiplicative semi-group of monomials in $S$ is isomorphic to $\mathbb{N}^{r+1}$ so that it is naturally embedded in $X(T_{0})\cong\mathbb{Z}^{r+1}$ via morphism sending an eigenvector to its weight, which is given by a character of $T_{0}$. There is also a natural isomorphism $\eta:\Gamma(T_{0})\rightarrow X(T_{0})$ defined by a basis $\{\lambda_{i}\}_{i=0}^{r}$ of $\Gamma(T_{0})$ and its dual basis with respect to the pairing in[5, p.304]. For arbitrary $d$’th Hilbert point $\phi_{d}(x)$,

[TABLE]

for unique choice of sequence (up to scalar multiplication) $\{\phi_{d}(x)_{m}\}_{m\in M_{dQ(d)}}$ satisfying $\phi_{d}(x)_{m}\in V_{m,d}^{P}$. Let

[TABLE]

This set is called the state of the Hilbert point $\phi_{d}(x)$([5] and [6]). For $f\in S_{d}$ let $\{f_{m}\}_{m\in M_{d}}$ be the sequence satisfying

[TABLE]

Let $\Delta_{x,d}$ be the convex hull of $\Xi_{x,d}\otimes\mathbb{R}$ in $X(T_{0})\otimes\mathbb{R}\cong\mathbb{R}^{r+1}$ and $|\Delta_{x,d}|$ be the distance between $\Delta_{x,d}$ and $\xi_{d}:=\frac{dQ(d)}{r}\mathbbm{1}\in X(T_{0})\otimes\mathbb{R}$. $\mathbbm{1}\in\mathbb{R}^{r+1}$ is the vector whose coefficients are 1. There is also a unique point $h_{x,d}$ in $\Delta_{x,d}$ satisfying $\|h_{x,d}-\xi_{d}\|=|\Delta_{x,d}|$ and unique indivisible $\lambda_{x,d}\in\Gamma(T)$ satisfying $\eta(\lambda_{x,d})\otimes\mathbb{R}=qh_{x.d}$ for some $q\in\mathbb{R}^{+}$. Now we are ready to state

Theorem 2.1**.**

Suppose $\lambda\in\Gamma(H)$ is an indivisible worst 1-parameter subgroup of $\phi_{d}(x)$ for some $d\geq g_{P}$ and $x\in\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})^{\textup{us}}_{d}$. Then,

[TABLE]

For every $g\in G$ satisfying the image of $g\star\lambda$ is a subset of $T$. Such a $g\in G$ exists for any choice of $x$ and $d$. In particular, $\lambda\in[\lambda_{g.x,d}]$ and $|\Delta_{g.x,d}|=\max_{h\in G}{|\Delta_{h.x,d}|}$.

Proof.

[5, Lemma 3.2., Theorem 3.4.] and [6, Criterion 3.3]. ∎

Theorem 2.1 means that a problem finding $E_{d,[\lambda],\delta}$ containing $\phi_{d}(x)$ is an optimization problem. That is, it is equivalent to the problem finding $g\in G$ maximalizng $|\Delta_{g.x,d}|$.

2.2. The Multiplicity of a Projective Hypersurface at a point

Suppose $p=[1:0:\ldots:0]\in\mathbb{P}^{r}_{k}$ where $X$ is a projective hypersurface corresponding to the homogeneous ideal generated by the single generator $f\in S_{d}$, without loss of generality. Multiplicity of $X$ at a point $p$ is

[TABLE]

This is an extrinsic definition. However, we have also an intrinsic definition. $n_{p,X}$ is the degree of the tangent cone of $X$ at $p$ as a subscheme of the tangent space $T_{p}(X)$ of $X$ at $p$.([2, p. 258])

3. Computation of Worst State Polytope

Let $\langle,\rangle$ be a standard inner-product on $X(T)\otimes\mathbb{R}$ with respect to the basis $\{\eta(\lambda_{i})\}_{i=0}^{r}$. For $\lambda\in\Gamma(T)$, let’s define a monomial order $<_{\lambda}$ as follows:

[TABLE]

[TABLE]

Here $<_{\textrm{lex}}$ is a lexicographic order with respect to the term order $x_{i}<x_{i+1}$. We can describe $\Delta_{x,t}$ for $t\geq g_{P}$ with these notations.

[TABLE]

The value

[TABLE]

is equal to

[TABLE]

3.1. Stability of Hesselink Stratifications of a Hilbert Scheme of Hypersurfaces

In this subsection, assume that

[TABLE]

Considering a lex-segment homogeneous ideal whose Hilbert polynomial is $P$, we get $g_{P}=d$([1]). This means that $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$ is a parameter space of every hypersurface of $\mathbb{P}^{r}_{k}$ defined by a homogeneous polynomial of degree $d$. Also, we have a Hesselink stratification

[TABLE]

for each integer $t\geq d$. Before we state a new theorem, let

[TABLE]

for $D\in\mathbb{N}$ and $\delta\in\mathbb{R}^{+}$.

Theorem 3.1**.**

For above stratifications, there are identities

[TABLE]

as constructible subsets of $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$. Consequently, all Hesselink stratifications of $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})^{\textrm{us}}_{t}$ corresponding to $t\geq d$ coincide. In particular, $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})^{\textup{us}}_{d+D}=\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})^{\textup{us}}_{d}$ as sets.

Proof.

We will show that $E_{d+D,[\lambda],\tau(\delta,D)}=E_{d,[\lambda],\delta}$ as a subset of $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$. Suppose that $x\in E_{d,\left[\lambda\right],\delta}\cap E_{d+D,\left[\lambda^{\prime}\right],\delta^{\prime}}$ for some $D\in\mathbb{N}\setminus\{0\}$, $\lambda,\lambda^{\prime}\in\Gamma(T)$ and $\delta,\delta^{\prime}>0$. Fix $\gamma\in\Gamma(T)$. For each $t\geq d$, we have a unique sequence $\{m_{i,t}\}_{i=1}^{|M_{t}|}$ satisfying $m_{i,t}\in M_{t}$ for all $1\leq i\leq|M_{t}|$ and $m_{i,t}>_{\gamma}m_{i+1,t}$ for all $1\leq i<|M_{t}|$. Let $f$ be the unique generator (up to scalar) of the saturated ideal represented by $x$. Then we can write $[g.f]=g.\phi_{d}(x)$ for arbitrary $g\in G$ as follows:

[TABLE]

Therefore,

[TABLE]

where $\alpha=\max\{1\leq i\leq|M_{d}|\textrm{ }|(g.f)_{i}\neq 0\}$.

$V_{d+D}^{P}$ has a basis $\{\wedge_{j=1}^{|M_{D}|}m_{a_{j},d+D}|a_{j}<a_{j+1},1\leq a_{j}\leq|M_{D}|\}$. Now the coefficient of $\wedge_{j=1}^{|M_{D}|}m_{a_{j},d+D}$ in $g.\phi_{d+D}(x)$ with this basis can be written as follows:

[TABLE]

[TABLE]

[TABLE]

where $P_{r+1}$ is the permutation group on $r+1$. This means that

[TABLE]

where $A_{g.f,D}^{a_{1},\ldots,a_{|M_{D}|}}$ is a $|M_{D}|\times|M_{D}|$-minor of matrix $A_{g.f,D}$ with the choice of $a_{i}$’th columns. Here $[A_{g.f,D}]_{ij}=[m_{i,D}(g.f)]_{m_{j,d+D}}$ for all $1\leq i\leq|M_{D}|$ and $1\leq j\leq|M_{d+D}|$. If a strictly increasing sequence of integers $\{a_{j}\}_{j=1}^{|M_{D}|}$ satisfies $1\leq a_{i}\leq|M_{d+D}|$ and

[TABLE]

then $\{i\in\mathbb{Z}|1\leq i\leq|M_{D}|,\textrm{ }m_{a_{i},d+D}<_{\gamma}m_{\alpha,d}m_{i,D}\}\neq\emptyset$. Let

[TABLE]

By the definition of $\alpha$,

[TABLE]

for all $1\leq i\leq\beta$ and $\beta\leq j\leq|M_{D}|$. This means that

[TABLE]

Therefore,

[TABLE]

[TABLE]

by (5). Taking a $\gamma\in\Gamma(T)$ maximalizing above values, we can verify that

[TABLE]

by (3). Taking $g\in G$ maximalizing these values, we have $[\lambda^{\prime}]=[\lambda]=[\lambda_{g.x,d}]$ and $\delta^{\prime}=\tau(\delta,D)$ by Theorem 2.1. ∎

3.2. Optimization Preserving Property of Lower-Triangular matrix

Once we choose a solution $g\in G$ to the optimization problem arising from Theorem 2.1, we can check that $lg$ is also a solution for an arbitrary lower-triangular matrix $l$ if we assume that

[TABLE]

and $a_{i}\leq a_{i+1}$ for all $0\leq i<r$. Furthermore, $h_{g.x,d}=h_{lg.x,d}$. Actually, this is a consequence of [5, Theorem4.2.].

Lemma 3.2**.**

Suppose $g$ satisfies

[TABLE]

[TABLE]

and $a_{i}\leq a_{i+1}$ for all $0\leq i<r$. Then, $\lambda_{lg.x,d}=\lambda_{g.x,d}\in\Lambda_{g.x,d}$ for every lower-triangular matrix $l\in H$.

Proof.

Let $\lambda=\lambda_{g.x,d}$. For every lower triangular matrix $l\in H$,

[TABLE]

Thus,

[TABLE]

by [6, Lemma 4.2.]. Since the norm $\|\cdot\|$ on $\Gamma(G)$ is invariant under the conjugate action, $\lambda\in\Lambda_{lg.x,d}\cap\Gamma(T)$ so that $\lambda_{lg.x,d}=\lambda_{g.x,d}$ by [5, Theorem 4.2.b)(4)]. ∎

4. A Relation between Instability and Singularity

From now on, let’s assume (4) for Hilbert polynomial $P$. In previous sections, we proved that Hesselink stratifications of a Hilbert Scheme $\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$ of hypersurfaces is unique in some sense. In this section, we will show that for $x\in\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})^{\textup{us}}_{d}$, there is a positive relation between the multiplicity $n_{H_{x}}$ of the hypersurface $H_{x}$ represented by $x$ and the Hesselink strata $E_{d,[\lambda],\delta}$ which contains $x$. We see that the multiplicity of the hypersurface $H_{x}$ at $e=[1:0:\ldots:0]\in\mathbb{P}^{r}_{k}$ determines a supporting hyperplane of $\Delta_{x,d}$. Here we choose the homogeneous coordinate $[x_{0}:x_{1}:\ldots:x_{r}]$ of $\mathbb{P}^{r}_{k}$.

Lemma 4.1**.**

In the above situation,

[TABLE]

and

[TABLE]

Proof.

It is trivial by the definition of $\Xi_{x,d}$ and (2). ∎

Lemma 4.1 means that there is a relation between a state and the multiplicity of a fixed point. We know that the set $\{\Xi_{g.x}|g\in G\}$ determines the Hesselink strata $E_{d,[\lambda],\delta}$ which contains $x$ for any unstable Hilbert point $x$ and the subgroup of $G$ generated by upper triangular matrices and permutation matrices acts on $\mathbb{P}^{r}_{k}$ transitively, whose dual action is induced by the canonical action on $V_{d}^{P}$. $n_{H_{x}}$ is a geometric invariant, so it is independent under the choice of coordinate. These facts lead us to

Lemma 4.2**.**

For any choice of unstable $x\in\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$, there is $g_{x}\in G$ satisfying

[TABLE]

and

[TABLE]

Proof.

By Lemma 2.1, there is a $g^{\prime}$ satisfying

[TABLE]

Without loss of generality, we can guess that

[TABLE]

satisfies $a_{i}\leq a_{i+1}$ for all $0\leq i<r$. If it doesn’t, we may take $wg^{\prime}$ instead of $g^{\prime}$ for some permutation matrix $w$. There is $y=[b_{0}:b_{1}:\ldots:b_{r}]\in H_{g^{\prime}.x}$ satisfying $n_{H_{g^{\prime}.x}}=n_{y,H_{g^{\prime}.x}}$. We can choose a lower-triangular matrix $l$ such that some row of $l$ is equal to a representative vector $(b_{0},b_{1},\ldots,b_{r})$ of $y$. By Lemma 3.2,

[TABLE]

There is a permutation matrix $q$ satisfies $e.(ql)=(eq).l=y$ by the construction. Now $g_{x}=qlg^{\prime}$ has every desired property. ∎

Choosing $g_{x}\in G$ as in Lemma 4.2 for a fixed Hilbert point $x\in E_{d,[\lambda],\delta}$, we can compare $n_{H_{x}}$ and the pair $([\lambda],\delta)$.

Theorem 4.3**.**

Suppose $x\in E_{d,[\lambda],\delta}$ for a unstable Hilbert point $x\in\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$ where

[TABLE]

satisfying $\sum_{i=0}^{r}a_{i}=0$. If $b=\max_{0\leq i\leq r}a_{i}$ and $a=\min_{0\leq i\leq r}a_{i}$, then

[TABLE]

Proof.

By Lemma 4.2, we can assume that

[TABLE]

and

[TABLE]

Without loss of generality, $\lambda\in\Lambda_{x,d}$. It it doesn’t, we may choose a permutation matrix $q$ satisfying $q\star\lambda\in\Lambda_{x,d}$ and let $q\star\lambda$ be another representative of $[\lambda]$. It is possible because $x\in E_{d,[\lambda],\delta}$, by Theorem 2.1. By definition, $\Delta_{x,d}$ contains

[TABLE]

Since $\Delta_{x,d}$ is the convex hull of $\Xi_{x,d}$, Lemma 4.1 and our assumption on $x$ imply that

[TABLE]

if we consider the maximum value of the degree of $x_{0}$ in each monomial $m\in\Xi_{x,d}$. Now suppose

[TABLE]

then for all $0\leq j\leq r$ satisfying $a=a_{j}$ and for all $(b_{0},\ldots,b_{r})\in\Delta_{x,d}$,

[TABLE]

by (3) so that

[TABLE]

This means that $n_{e,H_{p.x}}>n_{H_{x}}=n_{H_{p.x}}$, for the transposition matrix $p$ which permutes [math] and $j$. This is a contradiction. ∎

If $\lambda=p\star(-r,1,\ldots,1)\in\Gamma(T)$ for some permutation matrix $p\in G$, then we see that

[TABLE]

so that $n_{H_{x}}$ must be a fixed value by (6). We also see that

[TABLE]

This implies that every unstable hypersurface of degree $d\geq r+1$ is singular. This is also a weaker version of [8, Proposition 4.2., Chapter 3]. (6) has been derived by the existence of certain coordinate but smoothness of $x\in\textup{Hilb}^{P}(\mathbb{P}^{r}_{k})$ requires a general property of each state polytope in $\{\Delta_{g.x,d}|g\in G,|\Delta_{g.x,d}|=\max_{h\in G}|\Delta_{h.x,d}|\}$. However, we can check that (6) is sharp. If $r=3$, $d=4$ and $\phi_{d}(x)=[x_{0}^{4}]\in\mathbb{P}(k[x_{0},x_{1},x_{2},x_{3}]_{4})$ then $\lambda=(3,-1,-1,-1)\in\Lambda_{x,d}$ so that

[TABLE]