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Hilbert-Mumford criterion for genus 3 curves

发布于 2026-07-11T13:45:11Z · 公开内容更新于 2026-08-17T13:52:57Z

Hilbert-Mumford criterion for genus 3 curves 1 Hilbert-Mumford criterion We recall the Hilbert-Mumford criterion following [ Hoskins2023GIT ] and [ MFK1994GIT ]

Hilbert-Mumford criterion for genus 3 curves

1 Hilbert-Mumford criterion

We recall the Hilbert-Mumford criterion following [HOS23] and [MFK94].

Definition 1.1.

Let $𝐺$ be a reductive group acting linearly on a projective scheme $𝑋 \subseteq ℙ 𝑛$ . Assume $𝑅 (𝑋) = ⨁ 𝑟 \geq 0 𝐻 0 (𝑋, O (𝑟))$ .

(i)

We say $𝑥 \in 𝑋$ is semistable if there exists a $𝐺$ -invariant homogeneous function $𝑓 \in 𝑅 (𝑋) 𝐺 𝑟$ for some $𝑟 > 0$ such that $𝑓 (𝑥) \neq 0$ . We write $𝑋 𝑠 𝑠$ for the open set in $𝑋$ of semistable points.
(ii)

We say $𝑥 \in 𝑋$ is stable if its orbit is closed in $𝑋 𝑠 𝑠$ and its stabiliser is zero dimensional. We write $𝑋 𝑠$ for the open set in $𝑋$ of stable points.

Lemma 1.2.

Let $𝐺$ be a geometrically reductive group acting on an affine scheme $𝑋$ . If $𝑊 1$ and $𝑊 2$ are disjoint $𝐺$ -invariant closed subsets of $𝑋$ , then there is an invariant function $𝑓 \in O (𝑋) 𝐺$ which separates these sets, i.e.

𝑓 (𝑊 1) = 0 a n d 𝑓 (𝑊 2) = 1 .

For simplicity, we assume we have a linear representation $𝐺 \to 𝐺 𝐿 (𝑉)$ of a reductive group $𝐺$ and consider the associated linear action on $𝑋 = ℙ (𝑉)$ . For a closed subscheme $𝑌 \subseteq 𝑋$ with a linear $𝐺$ -action, we have $𝑌 𝑠 𝑠 = 𝑌 \times 𝑋 𝑋 𝑠 𝑠$ and so it suffices to understand semistability on the ambient projective space.

Proposition 1.3 (Topological Hilbert-Mumford criterion).

For a linear action of reductive group $𝐺$ on $ℙ (𝑉)$ , the following statements hold for $𝑥 = [𝑣] \in ℙ (𝑉)$ .

(a)

$𝑥$ is semistable $⟺$ $0 \notin ―――― 𝐺 \cdot 𝑣$ .
(b)

$𝑥$ is stable $⟺$ $d i m 𝐺 𝑣 = 0$ and $𝐺 \cdot 𝑣$ is closed in $𝑉$ .

Proof.

We will just give the proof of the first statement. By definition $𝑥 = [𝑣]$ is semistable $⟺$ there is a $𝐺$ -invariant homogeneous polynomial $𝑓 \in 𝑅 (𝑋) 𝐺$ which is non-zero at $𝑥$ . Since $𝑓$ is $𝐺$ -invariant it is constant on orbit closures, and so $𝑓$ separates the closed schemes $―――― 𝐺 \cdot 𝑣$ and $0$ , which shows these closed subschemes are disjoint. Conversely if the closed $𝐺$ -invariant schemes $―――― 𝐺 \cdot 𝑣$ and $0$ in $𝑉$ are disjoint, then as $𝐺$ is geometrically reductive, there exists a $𝐺$ -invariant polynomial $𝑓 \in O (𝑉) 𝐺$ separating these subsets

𝑓 (―――― 𝐺 \cdot 𝑣) = 1 w i t h 𝑓 (0) = 0

by Lemma 1.2. By considering the decomposition of $𝑓 = \sum 𝑖 𝑓 𝑖$ into ( $𝐺$ -invariant) homogeneous pieces, we see there is a $𝐺$ -invariant homogeneous piece $𝑓 𝑖$ which is non-vanishing at $𝑥$ . ∎

Definition 1.4.

For a linear action of a torus $𝑇 = 𝔾 𝑛 𝑚$ on $ℙ (𝑉)$ , consider the associated weight decomposition

𝑉 = ⨁ 𝜒 \in 𝑋 * (𝑇) 𝑉 𝜒,

where $𝑋 * (𝑇) : = H o m (𝑇, 𝔾 𝑚) = H o m (𝔾 𝑛 𝑚, 𝔾 𝑚) = ℤ 𝑛$ . We refer to the support of this decomposition as the $𝑇$ -weights on $ℙ (𝑉)$ . For $𝑥 = [𝑣] \in ℙ (𝑉)$ , we write $𝑣 = \sum 𝑣 𝜒$ and define the $𝑇$ -weight set of this point to be

𝑤 𝑡 𝑇 (𝑥) = 𝑤 𝑡 𝑇 (𝑣) = {𝜒 \in 𝑋 * (𝑇) ≅ ℤ 𝑛 : 𝑣 𝜒 \neq 0} .

Definition 1.5.

If a morphism $𝑓 : 𝔾 𝑚 \to 𝑆$ , with $𝑆$ separated, extends to $̃ 𝑓 : 𝔸 1 \to 𝑆$ , then this extension is unique and we write $l i m 𝑡 \to 0 𝑓 (𝑡) : = ̃ 𝑓 (0)$ . Similarly if $𝑓$ extends to $ℙ 1$ , we write $l i m 𝑡 \to \infty 𝑓 (𝑡) : = ̃ 𝑓 (\infty)$ .

Proposition 1.6 (Hilbert-Mumford for $𝔾 𝑚$ actions).

For a linear action of $𝔾 𝑚$ on $ℙ (𝑉)$ and $𝑥 \in ℙ (𝑉)$ , the following statements hold:

(a)

$𝑥$ is $𝔾 𝑚$ -semistable $⟺$ $0 \in 𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝔾 𝑚 (𝑥))$ .
(b)

$𝑥$ is $𝔾 𝑚$ -stable $⟺$ $0 \in 𝐼 𝑛 𝑡 (𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝔾 𝑚 (𝑥)))$ .

Proof.

We again just prove the statement for semistability. By Proposition 1.3, we have that $𝑥 = [𝑣] \in ℙ (𝑉)$ is $𝔾 𝑚$ -semistable $⟺$ $0 \notin ―――― 𝔾 𝑚 \cdot 𝑣$ .

Since $0 \notin 𝔾 𝑚 \cdot 𝑣$ , if $0 \in ―――― 𝔾 𝑚 \cdot 𝑣$ , then $0∈――――𝔾𝑚⋅𝑣\𝔾𝑚⋅𝑣$ . Any point in the boundary of this orbit closure is either

l i m 𝑡 \to 0 𝑡 \cdot 𝑣 o r l i m 𝑡 \to \infty 𝑡 \cdot 𝑣 = l i m 𝑡 \to 0 𝑡 - 1 \cdot 𝑣 .

Moreover, we have $l i m 𝑡 \to 0 𝑡 \cdot 𝑣 = 0$ $⟺$ $𝑤 𝑡 𝔾 𝑚 (𝑣) \subseteq ℤ > 0$ and similarly $l i m 𝑡 \to \infty 𝑡 \cdot 𝑣 = 0$ $⟺$ $𝑤 𝑡 𝔾 𝑚 (𝑣) \subseteq ℤ < 0$ . Hence $𝑥 = [𝑣] \in ℙ (𝑉)$ is $𝔾 𝑚$ -semistable $⟺$ there exists $𝑟 0 \in 𝑤 𝑡 𝔾 𝑚$ with $𝑟 0 \leq 0$ and there exists $𝑟 \infty \in 𝑤 𝑡 𝔾 𝑚$ with $𝑟 \infty \geq 0$ , or equivalently $0 \in 𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝔾 𝑚 (𝑥))$ . ∎

Example 1.7.

Let $𝑋 = ℙ 𝑛 = ℙ (𝑉)$ , where $𝑉 = 𝑘 𝑛 + 1$ has basis $𝑒 0, 𝑒 1, \dots, 𝑒 𝑛$ . Consider the linear action of $𝔾 𝑚$ on $𝑉$ given by

𝑡 \cdot 𝑒 0 = 𝑡 - 1 𝑒 0, 𝑡 \cdot 𝑒 𝑖 = 𝑡 𝑒 𝑖, 𝑖 = 1, \dots, 𝑛 .

Equivalently, the induced action on $ℙ 𝑛$ is

𝑡 \cdot [𝑥 0 : 𝑥 1 : \dots : 𝑥 𝑛] = [𝑡 - 1 𝑥 0 : 𝑡 𝑥 1 : \dots : 𝑡 𝑥 𝑛] .

Since $𝑋 * (𝔾 𝑚) ≅ ℤ$ , where $𝑚 \in ℤ$ corresponds to the character $𝑡 \mapsto 𝑡 𝑚$ , the weight decomposition of $𝑉$ is

𝑉 = 𝑉 - 1 \oplus 𝑉 1,

where

𝑉 - 1 = 𝑘 𝑒 0, 𝑉 1 = ⟨ 𝑒 1, \dots, 𝑒 𝑛 ⟩ .

All other weight spaces are zero.

Let

𝑥 = [𝑣] = [𝑥 0 : 𝑥 1 : \dots : 𝑥 𝑛] \in ℙ (𝑉),

where

𝑣 = 𝑥 0 𝑒 0 + 𝑥 1 𝑒 1 + \dots + 𝑥 𝑛 𝑒 𝑛 .

With respect to the decomposition $𝑉 = 𝑉 - 1 \oplus 𝑉 1$ , we have

𝑣 = 𝑣 - 1 + 𝑣 1,

where

𝑣 - 1 = 𝑥 0 𝑒 0 \in 𝑉 - 1, 𝑣 1 = 𝑥 1 𝑒 1 + \dots + 𝑥 𝑛 𝑒 𝑛 \in 𝑉 1 .

Hence the $𝔾 𝑚$ -weight set of $𝑥$ is

w t 𝔾 𝑚 (𝑥) = {- 1 : 𝑥 0 \neq 0} \cup {1 : (𝑥 1, \dots, 𝑥 𝑛) \neq 0} .

By the Hilbert–Mumford criterion for $𝔾 𝑚$ -actions,

𝑥 \in 𝑋 𝑠 𝑠 ⟺ 0 \in c o n v (w t 𝔾 𝑚 (𝑥)),

and

𝑥 \in 𝑋 𝑠 ⟺ 0 \in I n t c o n v (w t 𝔾 𝑚 (𝑥)) .

Therefore $𝑥$ is semistable if and only if both weights $- 1$ and $1$ occur in $w t 𝔾 𝑚 (𝑥)$ . Equivalently,

𝑥 0 \neq 0 a n d (𝑥 1, \dots, 𝑥 𝑛) \neq 0 .

In this case

w t 𝔾 𝑚 (𝑥) = {- 1, 1},

and

0 \in I n t ([- 1, 1]) .

Thus the semistable and stable loci coincide:

𝑋 𝑠 𝑠 = 𝑋 𝑠 = {[𝑥 0 : \dots : 𝑥 𝑛] \in ℙ 𝑛 : 𝑥 0 \neq 0, (𝑥 1, \dots, 𝑥 𝑛) \neq 0} .

Since $𝐷 + (𝑥 0) ≅ 𝔸 𝑛$ via

[𝑥 0 : 𝑥 1 : \dots : 𝑥 𝑛] ⟼ (𝑥 1 𝑥 0, \dots, 𝑥 𝑛 𝑥 0),

we get

𝑋 𝑠 𝑠 = 𝑋 𝑠 ≅ 𝔸 𝑛 ∖ {0} .

Finally, the invariant section ring is

𝑅 (ℙ 𝑛, O (1)) 𝔾 𝑚 = 𝑘 [𝑥 0 𝑥 1, \dots, 𝑥 0 𝑥 𝑛] .

Indeed, a monomial

𝑥 𝑎 0 0 𝑥 𝑎 1 1 \dots 𝑥 𝑎 𝑛 𝑛

has $𝔾 𝑚$ -weight

- 𝑎 0 + 𝑎 1 + \dots + 𝑎 𝑛,

so it is invariant if and only if

𝑎 0 = 𝑎 1 + \dots + 𝑎 𝑛 .

Hence every invariant monomial is a monomial in

𝑥 0 𝑥 1, \dots, 𝑥 0 𝑥 𝑛 .

Therefore

ℙ 𝑛 / / 𝔾 𝑚 = P r o j 𝑘 [𝑥 0 𝑥 1, \dots, 𝑥 0 𝑥 𝑛] ≅ ℙ 𝑛 - 1 .

The quotient map on the semistable locus is

𝑋 𝑠 𝑠 ⟶ ℙ 𝑛 - 1, [𝑥 0 : 𝑥 1 : \dots : 𝑥 𝑛] ⟼ [𝑥 1 : \dots : 𝑥 𝑛] .

The Hilbert-Mumford criterion will ultimately be a numerical criterion that phrases semistability in terms of the weights of $1$ -parameter subgroups (1-PS), which are non-trivial group homomorphisms $𝜆 : 𝔾 𝑚 \to 𝐺$ .

Definition 1.8.

For a linear action of a reductive group $𝐺$ on $ℙ (𝑉)$ , we define the Hilbert-Mumford weight of $𝑥 = [𝑣]$ at a $1$ -parameter subgroup $𝜆 : 𝔾 𝑚 \to 𝐺$ to be

𝜇 (𝑥, 𝜆) : = - m i n 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) = - m i n {𝜒 \in H o m (𝜆 (𝔾 𝑚), 𝔾 𝑚) ≅ ℤ : 𝑣 𝜒 \neq 0} .

Remark 1.9.

𝜇 (𝑥, 𝜆 - 1) = - m i n 𝑤 𝑡 𝜆 - 1 (𝔾 𝑚) (𝑥) = m a x 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) .

Proposition 1.10.

(1)

$𝜇 (𝑥, 𝜆)$ is the unique integer $𝜇$ such that $l i m 𝑡 \to 0 𝑡 𝜇 𝜆 (𝑡) \cdot 𝑣$ exists and is non-zero.
(2)

$𝜇 (𝑥, 𝜆) = 𝜇 (𝑥 0, 𝜆)$ where $𝑥 0 = l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑥$ (and this limit exists as $𝑋$ is projective).
(3)

$𝜇 (𝑥, 𝜆) \leq 0 ⟺ l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣$ exists, with equality $⟺$ $l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 \neq 0$ .
(4)

$𝜇 (𝑔 \cdot 𝑥, 𝑔 𝜆 𝑔 - 1) = 𝜇 (𝑥, 𝜆)$ for all $𝑔 \in 𝐺$ .
(5)

$𝜇 (𝑥, 𝜆 𝑛) = 𝑛 𝜇 (𝑥, 𝜆)$ for a positive integer $𝑛$ .

Proof.

Let $𝜆 : 𝔾 𝑚 \to 𝐺$ be a one-parameter subgroup and let $𝑥 = [𝑣] \in ℙ (𝑉)$ . Decompose $𝑉$ into $𝜆$ -weight spaces:

𝑉 = ⨁ 𝑟 \in ℤ 𝑉 𝑟, 𝜆 (𝑡) \cdot 𝑤 = 𝑡 𝑟 𝑤 f o r 𝑤 \in 𝑉 𝑟 .

Write

𝑣 = \sum 𝑟 \in ℤ 𝑣 𝑟, 𝑣 𝑟 \in 𝑉 𝑟,

and set

w t 𝜆 (𝑣) : = {𝑟 \in ℤ : 𝑣 𝑟 \neq 0} .

Let

𝑟 0 : = m i n w t 𝜆 (𝑣) .

We first prove (1). For any integer $𝜇$ , we have

𝑡 𝜇 𝜆 (𝑡) \cdot 𝑣 = \sum 𝑟 𝑡 𝜇 + 𝑟 𝑣 𝑟 .

This has a limit in $𝑉$ as $𝑡 \to 0$ if and only if

𝜇 + 𝑟 \geq 0 f o r a l l 𝑟 \in w t 𝜆 (𝑣) .

The limit is nonzero if and only if equality holds for at least one $𝑟 \in w t 𝜆 (𝑣)$ . Hence the unique such integer is

𝜇 = - 𝑟 0 .

Therefore

𝜇 (𝑥, 𝜆) = - 𝑟 0 .

For (2), consider

𝜆 (𝑡) \cdot 𝑥 = [\sum 𝑟 𝑡 𝑟 𝑣 𝑟] .

Multiplying the representative by $𝑡 - 𝑟 0$ , we obtain

𝜆 (𝑡) \cdot 𝑥 = [\sum 𝑟 𝑡 𝑟 - 𝑟 0 𝑣 𝑟] .

Letting $𝑡 \to 0$ , all terms with $𝑟 > 𝑟 0$ vanish, while the term $𝑣 𝑟 0$ remains. Thus

𝑥 0 : = l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑥 = [𝑣 𝑟 0] .

Since $𝑣 𝑟 0$ has pure $𝜆$ -weight $𝑟 0$ , we get

𝜇 (𝑥 0, 𝜆) = - 𝑟 0 = 𝜇 (𝑥, 𝜆) .

For (3), the ordinary vector limit

l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 = l i m 𝑡 \to 0 \sum 𝑟 𝑡 𝑟 𝑣 𝑟

exists if and only if no negative power of $𝑡$ occurs, that is, if and only if

𝑟 0 \geq 0 .

Since $𝜇 (𝑥, 𝜆) = - 𝑟 0$ , this is equivalent to

𝜇 (𝑥, 𝜆) \leq 0 .

Moreover, the limit is nonzero if and only if the lowest occurring weight is $0$ , namely $𝑟 0 = 0$ . Equivalently,

l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 \neq 0 ⟺ 𝜇 (𝑥, 𝜆) = 0 .

Thus

𝜇 (𝑥, 𝜆) \leq 0 ⟺ l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 e x i s t s,

with equality if and only if this limit is nonzero.

For (4), let $𝑔 \in 𝐺$ . Then $𝑔 \cdot 𝑥 = [𝑔 \cdot 𝑣]$ , and the conjugate one-parameter subgroup is

𝑔 𝜆 𝑔 - 1 : 𝔾 𝑚 \to 𝐺, 𝑡 \mapsto 𝑔 𝜆 (𝑡) 𝑔 - 1 .

For any integer $𝜇$ ,

𝑡 𝜇 (𝑔 𝜆 𝑔 - 1) (𝑡) \cdot (𝑔 𝑣) = 𝑔 (𝑡 𝜇 𝜆 (𝑡) \cdot 𝑣) .

Since $𝑔 : 𝑉 \to 𝑉$ is a linear isomorphism, the left-hand side has a nonzero limit if and only if

𝑡 𝜇 𝜆 (𝑡) \cdot 𝑣

has a nonzero limit. By the uniqueness in (1), we get

𝜇 (𝑔 \cdot 𝑥, 𝑔 𝜆 𝑔 - 1) = 𝜇 (𝑥, 𝜆) .

Finally, for (5), let $𝑛$ be a positive integer and define

𝜆 𝑛 (𝑡) : = 𝜆 (𝑡 𝑛) .

If $𝑣 𝑟 \in 𝑉 𝑟$ for the $𝜆$ -action, then

𝜆 𝑛 (𝑡) \cdot 𝑣 𝑟 = 𝜆 (𝑡 𝑛) \cdot 𝑣 𝑟 = 𝑡 𝑛 𝑟 𝑣 𝑟 .

Thus the occurring weights for $𝜆 𝑛$ are

{𝑛 𝑟 : 𝑟 \in w t 𝜆 (𝑣)} .

Hence

𝜇 (𝑥, 𝜆 𝑛) = - m i n {𝑛 𝑟 : 𝑟 \in w t 𝜆 (𝑣)} = - 𝑛 𝑟 0 = 𝑛 𝜇 (𝑥, 𝜆) .

This proves all the assertions. ∎

For a linear $𝔾 𝑚$ -action on $ℙ (𝑉)$ , let $𝜆 (𝑡) = 𝑡$ . Write

𝑉 = ⨁ 𝑟 \in ℤ 𝑉 𝑟, 𝑣 = \sum 𝑟 𝑣 𝑟, w t 𝔾 𝑚 (𝑣) = {𝑟 : 𝑣 𝑟 \neq 0} .

Then

𝜇 (𝑥, 𝜆) = - m i n w t 𝔾 𝑚 (𝑣),

whereas

𝜇 (𝑥, 𝜆 - 1) = m a x w t 𝔾 𝑚 (𝑣) .

Indeed, the weights for $𝜆 - 1$ are the negatives of the weights for $𝜆$ .

Thus

𝜇 (𝑥, 𝜆) \geq 0 ⟺ m i n w t 𝔾 𝑚 (𝑣) \leq 0,

and

𝜇 (𝑥, 𝜆 - 1) \geq 0 ⟺ m a x w t 𝔾 𝑚 (𝑣) \geq 0 .

Consequently,

0 \in c o n v (w t 𝔾 𝑚 (𝑥)) ⟺ 𝜇 (𝑥, 𝜆) \geq 0 a n d 𝜇 (𝑥, 𝜆 - 1) \geq 0 .

Similarly,

0 \in I n t c o n v (w t 𝔾 𝑚 (𝑥)) ⟺ 𝜇 (𝑥, 𝜆) > 0 a n d 𝜇 (𝑥, 𝜆 - 1) > 0 .

It is important to distinguish these numerical inequalities from the ordinary vector limit $l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣$ . Since

𝜆 (𝑡) \cdot 𝑣 = \sum 𝑟 𝑡 𝑟 𝑣 𝑟,

this ordinary limit exists if and only if

m i n w t 𝔾 𝑚 (𝑣) \geq 0,

or equivalently

𝜇 (𝑥, 𝜆) \leq 0 .

Moreover, the limit exists and is nonzero if and only if the minimum weight is $0$ , that is,

l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 e x i s t s a n d i s n o n z e r o ⟺ 𝜇 (𝑥, 𝜆) = 0 .

Hence the condition $𝜇 (𝑥, 𝜆) \geq 0$ should not be interpreted as the existence of a nonzero ordinary limit of $𝜆 (𝑡) \cdot 𝑣$ ; rather, it is the condition that some occurring $𝜆$ -weight of $𝑣$ is non-positive.

Theorem 1.11 (Hilbert-Mumford criterion).

For a reductive group $𝐺$ acting linearly on a projective scheme $𝑋 \subseteq ℙ 𝑛$ , the following statements hold for $𝑥 \in 𝑋$ .

(a)

$𝑥$ is semistable $⟺$ $𝜇 (𝑥, 𝜆) \geq 0$ for all $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ ,
(b)

$𝑥$ is stable $⟺$ $𝜇 (𝑥, 𝜆) > 0$ for all nontrivial $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ .

Remark 1.12.

$𝜇 (𝑥, 𝜆) \geq 0$ for all $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ $⟺$ $𝜇 (𝑥, 𝜆) = - m i n 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) \geq 0$ and $𝜇 (𝑥, 𝜆 - 1) = m a x 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) \geq 0$ for all $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ $⟺$ $0 \in 𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥))$ for all $𝜆 : 𝔾 𝑚 \to 𝐺$ .

$𝜇 (𝑥, 𝜆) > 0$ for all nontrivial $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ $⟺$ $𝜇 (𝑥, 𝜆) = - m i n 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) > 0$ and $𝜇 (𝑥, 𝜆 - 1) = m a x 𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥) > 0$ for all nontrivial $1$ -PS $𝜆 : 𝔾 𝑚 \to 𝐺$ $⟺$ $0 \in 𝐼 𝑛 𝑡 (𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝜆 (𝔾 𝑚) (𝑥)))$ for all nontrivial $𝜆 : 𝔾 𝑚 \to 𝐺$ .

Remark 1.13.

Let $𝑋$ be a projective $𝐺$ -scheme and let $L$ be an ample $𝐺$ -linearised line bundle. Since $L$ is ample, for $𝑛 ≫ 0$ the line bundle $L \otimes 𝑛$ is very ample. The $𝐺$ -linearisation on $L$ induces a $𝐺$ -linearisation on $L \otimes 𝑛$ , hence a linear representation of $𝐺$ on

𝐻 0 (𝑋, L \otimes 𝑛) .

Thus $L \otimes 𝑛$ defines a $𝐺$ -equivariant closed embedding

𝑖 𝑛 : 𝑋 ↪ ℙ (𝐻 0 (𝑋, L \otimes 𝑛) \lor),

and

L \otimes 𝑛 ≅ 𝑖 * 𝑛 O (1) .

Moreover,

𝜇 L \otimes 𝑛 (𝑥, 𝜆) = 𝑛 𝜇 L (𝑥, 𝜆),

so replacing $L$ by a positive tensor power does not change the Hilbert–Mumford inequalities defining semistability or stability. Hence, after replacing $L$ by a sufficiently high tensor power, the general projective GIT problem reduces to the linear action on a projective space.

Theorem 1.14.

Let $𝐺$ be a reductive group acting on an affine space $𝑉$ . If $𝑣 \in 𝑉$ and $0 \in ―――― 𝐺 \cdot 𝑣$ , then there is a $1$ -PS $𝜆$ of $𝐺$ such that $l i m 𝑡 \to 0 𝜆 (𝑡) \cdot 𝑣 = 0$ .

Proposition 1.15.

For a reductive group $𝐺$ acting on a projective scheme $𝑋 \subseteq ℙ 𝑛$ linearly, fix a maximal torus $𝑇 < 𝐺$ . For $𝑥 \in 𝑋$ , the following statements hold.

(a)

$𝑥$ is $𝐺$ -(semi)stable $⟺$ $𝑔 \cdot 𝑥$ is $𝑇$ -(semi)stable for all $𝑔 \in 𝐺$ .
(b)

$𝑥$ is $𝑇$ -semistable $⟺$ $0 \in 𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝑇 (𝑥))$ .
(c)

$𝑥$ is $𝑇$ -stable $⟺$ $0 \in 𝐼 𝑛 𝑡 (𝑐 𝑜 𝑛 𝑣 (𝑤 𝑡 𝑇 (𝑥)))$ .

Proposition 1.16 (Semistability for binary forms).

Consider the action of $𝑆 𝐿 2$ on the space of degree $𝑑$ binary forms $ℙ 𝑑 = ℙ (𝑘 [𝑥, 𝑦] 𝑑)$ . For $𝑝 𝐹 \in ℙ 𝑑$ corresponding to $𝐹 (𝑥, 𝑦) \in 𝑘 [𝑥, 𝑦] 𝑑$ , then

(a)

$𝐹$ is semistable $⟺$ all roots of $𝐹$ have multiplicity $\leq 𝑑 / 2$ .
(b)

$𝐹$ is stable $⟺$ all roots of $𝐹$ have multiplicity $< 𝑑 / 2$ .

2 Hilbert–Mumford criterion for genus 3 curves

The discussion in this section is inspired by [ART09]. Throughout this section, let $𝑘$ be an algebraically closed field of characteristic $0$ .

2.1 Non-hyperelliptic curves of genus 3: plane quartics

Let $𝐶$ be a smooth projective curve of genus $3$ . By [VAK25] 19.7.3, $𝐶$ is not hyperelliptic $⟺$ the canonical map describes $𝐶$ as a degree $4$ curve in $ℙ 2$ .

Let $𝑊 = 𝑘 3$ , with coordinates $𝑋, 𝑌, 𝑍$ , and set

𝑋 4 : = ℙ (S y m 4 𝑊 \lor) .

A point $[𝐹] \in 𝑋 4$ represents the plane quartic

𝐶 𝐹 = {𝐹 = 0} \subset ℙ (𝑊) .

The group $P G L (𝑊)$ acts on $𝑋 4$ by change of coordinates. Hence the moduli problem of non-hyperelliptic genus $3$ curves is described by the quotient of the open locus

𝑈 4 : = {[𝐹] \in 𝑋 4 : 𝐶 𝐹 i s s m o o t h}

by $P G L 3$ .

Let

𝜆 : 𝔾 𝑚 \to S L (𝑊)

be a one-parameter subgroup. After a change of coordinates, we may assume

𝜆 (𝑡) = d i a g (𝑡 𝑎, 𝑡 𝑏, 𝑡 𝑐), 𝑎 \geq 𝑏 \geq 𝑐, 𝑎 + 𝑏 + 𝑐 = 0 .

Writing

𝐹 = \sum 𝑖 + 𝑗 + 𝑘 = 4 𝑐 𝑖 𝑗 𝑘 𝑋 𝑖 𝑌 𝑗 𝑍 𝑘,

the monomial $𝑋 𝑖 𝑌 𝑗 𝑍 𝑘$ has $𝜆$ -weight

𝑎 𝑖 + 𝑏 𝑗 + 𝑐 𝑘 .

Thus

w t 𝜆 (𝐹) = {𝑎 𝑖 + 𝑏 𝑗 + 𝑐 𝑘 : 𝑐 𝑖 𝑗 𝑘 \neq 0} .

Proposition 2.1.

Every smooth plane quartic is $P G L 3$ -stable for the natural $P G L 3$ -linearisation on

ℙ (S y m 4 𝑊 \lor) .

Proof.

By the Hilbert–Mumford criterion, it suffices to show that for every non-trivial one-parameter subgroup $𝜆$ , the occurring weights of $𝐹$ contain both a negative and a positive weight.

Write

𝜆 (𝑡) = d i a g (𝑡 𝑎, 𝑡 𝑏, 𝑡 𝑐), 𝑎 \geq 𝑏 \geq 𝑐, 𝑎 + 𝑏 + 𝑐 = 0 .

Since $𝜆$ is non-trivial, we have

𝑎 > 0, 𝑐 < 0 .

Suppose first that every monomial occurring in $𝐹$ has non-negative $𝜆$ -weight. Consider the point

𝑃 = [0 : 0 : 1] .

The monomials

𝑍 4, 𝑋 𝑍 3, 𝑌 𝑍 3

have weights

4 𝑐, 𝑎 + 3 𝑐, 𝑏 + 3 𝑐 .

These are all negative. Indeed,

4 𝑐 < 0 .

Moreover, since $𝑎 = - 𝑏 - 𝑐$ and $𝑏 \geq 𝑐$ , we have

𝑎 \leq - 2 𝑐,

and hence

𝑎 + 3 𝑐 \leq 𝑐 < 0 .

Also

𝑏 + 3 𝑐 = - 𝑎 + 2 𝑐 < 0 .

Therefore none of the monomials

𝑍 4, 𝑋 𝑍 3, 𝑌 𝑍 3

occurs in $𝐹$ .

On the affine chart $𝑍 \neq 0$ , with coordinates

𝑢 = 𝑋 𝑍, 𝑣 = 𝑌 𝑍,

the polynomial $𝐹 / 𝑍 4$ has zero constant term and zero linear terms at $(𝑢, 𝑣) = (0, 0)$ . Hence

𝐹 (𝑃) = 𝐹 𝑋 (𝑃) = 𝐹 𝑌 (𝑃) = 0 .

By Euler’s identity

𝑋 𝐹 𝑋 + 𝑌 𝐹 𝑌 + 𝑍 𝐹 𝑍 = 4 𝐹,

we also get

𝐹 𝑍 (𝑃) = 0 .

Thus $𝑃$ is a singular point of $𝐶 𝐹$ , contradicting smoothness. Hence some occurring monomial has negative $𝜆$ -weight.

Applying the same argument to $𝜆 - 1$ , some occurring monomial has positive $𝜆$ -weight. Therefore

m i n w t 𝜆 (𝐹) < 0 < m a x w t 𝜆 (𝐹) .

Equivalently,

0 \in I n t c o n v (w t 𝜆 (𝐹))

for every non-trivial $𝜆$ . Hence $[𝐹]$ is $P G L 3$ stable. ∎

Consequently, every smooth plane quartic defines a $P G L 3$ -stable point of

𝑋 4 = ℙ (S y m 4 𝑊 \lor) .

Equivalently,

𝑈 4 \subset 𝑋 𝑠, P G L 3 4 .

Thus the geometric quotient

𝑈 4 / P G L 3

parametrizes smooth plane quartics. Since the canonical embedding of a non-hyperelliptic curve of genus $3$ realizes it as a smooth plane quartic, and since isomorphisms of curves preserve the canonical linear system, this quotient parametrizes non-hyperelliptic smooth curves of genus $3$ .

2.2 Hyperelliptic curves of genus 3: binary octics

A hyperelliptic curve of genus $3$ is a double cover

𝐶 ⟶ ℙ 1

branched over a divisor of degree

2 𝑔 + 2 = 8 .

After choosing homogeneous coordinates $𝑋, 𝑌$ on $ℙ 1$ , such a curve can be written as

𝐶 𝑓 : 𝑧 2 = 𝑓 (𝑋, 𝑌),

where

𝑓 \in S y m 8 𝑘 2

is a binary octic. The curve $𝐶 𝑓$ is smooth if and only if $𝑓$ has eight distinct roots on $ℙ 1$ .

Thus the parameter space is

𝑋 8 : = ℙ (S y m 8 𝑘 2),

with the natural action of

P G L 2 .

The hyperelliptic locus $H 3$ is described by the quotient of the square-free binary octics by $P G L 2$ .

Proposition 2.2.

Let $𝑓$ be a binary octic. Then $[𝑓] \in ℙ (S y m 8 𝑘 2)$ is $P G L 2$ semistable if and only if

𝑚 𝑝 (𝑓) \leq 4 f o r a l l 𝑝 \in ℙ 1 .

It is $P G L 2$ stable if and only if

𝑚 𝑝 (𝑓) \leq 3 f o r a l l 𝑝 \in ℙ 1 .

Proof.

A binary octic is a binary form of degree $8$ . After choosing homogeneous coordinates $𝑋, 𝑌$ on $ℙ 1$ , we have an identification

S y m 8 𝑘 2 ≅ 𝑘 [𝑋, 𝑌] 8 .

Thus

ℙ (S y m 8 𝑘 2) ≅ ℙ (𝑘 [𝑋, 𝑌] 8) .

The stability and semistability conditions may be checked using the $𝑆 𝐿 2$ -action. Indeed, the center ${\pm 𝐼} \subset 𝑆 𝐿 2$ acts trivially on $ℙ (S y m 8 𝑘 2)$ , since $- 𝐼$ acts on a degree $8$ binary form by multiplication by $(- 1) 8 = 1$ . Hence the induced $𝑆 𝐿 2$ -stability and $P G L 2$ -stability conditions agree.

For each point $𝑝 \in ℙ 1$ , let $𝑚 𝑝 (𝑓)$ denote the multiplicity of $𝑝$ as a root of the homogeneous binary form $𝑓$ . Applying Proposition 1.16 with $𝑑 = 8$ , we obtain

[𝑓] i s s e m i s t a b l e ⟺ e v e r y r o o t o f 𝑓 h a s m u l t i p l i c i t y \leq 82 .

Since $82 = 4$ , this is equivalent to

𝑚 𝑝 (𝑓) \leq 4 f o r a l l 𝑝 \in ℙ 1 .

This proves the semistability statement.

Similarly, Proposition 1.16 gives

[𝑓] i s s t a b l e ⟺ e v e r y r o o t o f 𝑓 h a s m u l t i p l i c i t y < 82 .

Equivalently,

𝑚 𝑝 (𝑓) < 4 f o r a l l 𝑝 \in ℙ 1 .

Since $𝑚 𝑝 (𝑓)$ is an integer, this is equivalent to

𝑚 𝑝 (𝑓) \leq 3 f o r a l l 𝑝 \in ℙ 1 .

This proves the proposition. ∎

In particular, every square-free binary octic defines a $P G L 2$ -stable point of

𝑋 8 = ℙ (S y m 8 𝑘 2) .

Equivalently,

𝑈 s f 8 \subset 𝑋 𝑠, P G L 2 8,

where

𝑈 s f 8 : = {[𝑓] \in 𝑋 8 : 𝑓 i s s q u a r e - f r e e} .

Thus the geometric quotient

𝑈 s f 8 / P G L 2

parametrizes reduced effective divisors of degree $8$ on $ℙ 1$ up to projective change of coordinates. Equivalently, it parametrizes smooth hyperelliptic curves of genus $3$ .

2.3 Summary

In this section, we applied the Hilbert–Mumford criterion to smooth curves of genus $3$ . The main point is that genus $3$ curves naturally split into two types: non-hyperelliptic curves, which are represented by plane quartics, and hyperelliptic curves, which are represented by binary octics.

For the non-hyperelliptic case, let $𝑊 = 𝑘 3$ and

𝑋 4 : = ℙ (S y m 4 𝑊 \lor) .

A point $[𝐹] \in 𝑋 4$ defines a plane quartic

𝐶 𝐹 = {𝐹 = 0} \subset ℙ (𝑊) .

Let

𝑈 4 : = {[𝐹] \in 𝑋 4 : 𝐶 𝐹 i s s m o o t h} .

We proved that every point of $𝑈 4$ is $P G L 3$ -stable. Hence

𝑈 4 \subset 𝑋 𝑠, P G L 3 4,

and the quotient

𝑈 4 / P G L 3

parametrizes non-hyperelliptic smooth curves of genus $3$ .

For the hyperelliptic case, a smooth curve of genus $3$ is a double cover of $ℙ 1$ branched over a reduced divisor of degree $8$ . Thus it is represented by a square-free binary octic. Let

𝑋 8 : = ℙ (S y m 8 𝑘 2), 𝑈 s f 8 : = {[𝑓] \in 𝑋 8 : 𝑓 i s s q u a r e - f r e e} .

Using the Hilbert–Mumford criterion for binary forms, we proved that $[𝑓] \in 𝑋 8$ is $P G L 2$ -stable if and only if

𝑚 𝑝 (𝑓) \leq 3 f o r a l l 𝑝 \in ℙ 1 .

Since every square-free binary octic has only simple roots, we have

𝑈 s f 8 \subset 𝑋 𝑠, P G L 2 8 .

Therefore

𝑈 s f 8 / P G L 2

parametrizes smooth hyperelliptic curves of genus $3$ .

Therefore, in both the non-hyperelliptic and hyperelliptic cases, smooth curves of genus $3$ give stable points for the relevant projective parameter spaces.

References

[ART09] M. Artebani (2009) A compactification of $M 3$ via K3 surfaces. Nagoya Mathematical Journal 196, pp. 1–26. Cited by: §2.
[HOS23] V. Hoskins (2023) Moduli spaces and geometric invariant theory: old and new perspectives. External Links: 2302.14499, Link Cited by: §1.
[MFK94] D. Mumford, J. Fogarty, and F. Kirwan (1994) Geometric invariant theory. 3 edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 34, Springer-Verlag, Berlin. Cited by: §1.
[VAK25] R. Vakil (2025) The rising sea: foundations of algebraic geometry. Princeton University Press. External Links: ISBN 9780691268668 Cited by: §2.1.

Hilbert-Mumford criterion for genus 3 curves

1 Hilbert-Mumford criterion

Definition 1.1.

Lemma 1.2.

Proposition 1.3 (Topological Hilbert-Mumford criterion).

Proof.

Definition 1.4.

Definition 1.5.

Proposition 1.6 (Hilbert-Mumford for 𝔾𝑚 actions).

Proof.

Example 1.7.

Definition 1.8.

Remark 1.9.

Proposition 1.10.

Proof.

Theorem 1.11 (Hilbert-Mumford criterion).

Remark 1.12.

Remark 1.13.

Theorem 1.14.

Proposition 1.15.

Proposition 1.16 (Semistability for binary forms).

2 Hilbert–Mumford criterion for genus 3 curves

2.1 Non-hyperelliptic curves of genus 3: plane quartics

Proposition 2.1.

Proof.

2.2 Hyperelliptic curves of genus 3: binary octics

Proposition 2.2.

Proof.

2.3 Summary

References

Proposition 1.6 (Hilbert-Mumford for $𝔾 𝑚$ actions).