[Paper Review] The rationality of the moduli spaces of trigonal curves

Shouhei Ma|arXiv (Cornell University)|Jul 1, 2012

Algebraic Geometry and Number Theory6 references4 citations

TL;DR

This paper proves that the moduli space of trigonal curves of genus $g = 4N$ is rational by leveraging invariant theory for $\mathrm{SL}_2 \times \mathrm{SL}_2$, constructing a dominant rational map via a carefully chosen bi-transvectant to reduce the rationality of the quotient $\mathbb{P}V_{3,b}/(\mathrm{SL}_2 \times \mathrm{SL}_2)$ to stable rationality of a Grassmannian, completing the rationality classification for all $g \geq 5$. The result confirms that $\mathcal{T}_g$ is rational for all $g \geq 5$, resolving the final case in the classification of trigonal moduli spaces.

ABSTRACT

The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.

Motivation & Objective

• To complete the classification of rationality for the moduli space $\mathcal{T}_g$ of trigonal curves of genus $g \geq 5$, which was previously known for $g \equiv 2 \pmod{4}$ and odd $g$, but not for $g \equiv 0 \pmod{4}$.
• To establish the rationality of $\mathbb{P}V_{3,b}/(\mathrm{SL}_2 \times \mathrm{SL}_2)$ for all odd $b \geq 5$, which parametrizes trigonal curves of genus $g = 4N$ via canonical embeddings in $\mathbb{P}^1 \times \mathbb{P}^1$.
• To develop a computational method using $\mathrm{SL}_2 \times \mathrm{SL}_2$-bilinear maps (bi-transvectants) to construct a dominant rational map from $V_{3,b}$ to a Grassmannian, enabling reduction of the rationality problem to stable rationality of the Grassmannian.
• To verify non-degeneracy of the induced rational map $v \mapsto \operatorname{Ker}(T(v, \cdot))$ for carefully chosen bi-transvectants, ensuring the bundle structure is well-defined and dominant.

Proposed method

• The paper uses the double bundle method: a bi-transvectant $T: V_{3,b} \times V_{a',b'} \to V_{a'',b''}$ is constructed such that $\dim V_{a',b'} > \dim V_{a'',b''}$, defining a rational map $v \mapsto \operatorname{Ker}(T(v, \cdot))$ from $V_{3,b}$ to the Grassmannian $G(c, V_{a',b'})$ with $c = \dim V_{a',b'} - \dim V_{a'',b''}$.
• The choice of bi-transvectant depends on $b \mod 5$, with explicit constructions tailored to each residue class to ensure $c$ is small and $a', b', c$ are odd, which is essential for handling the $-1$ scalar action in the group action.
• Non-degeneracy of the kernel map is verified by analyzing the induced linear maps on monomial bases using the Clebsch-Gordan formula and explicit transvectant formulas, particularly for $r = e$ and $r = e-1$.
• The rationality of $\mathbb{P}V_{3,b}/(\mathrm{SL}_2 \times \mathrm{SL}_2)$ is reduced to the stable rationality of $G(c, V_{a',b'})/(\mathrm{SL}_2 \times \mathrm{SL}_2)$, which is established via standard techniques in invariant theory.
• The proof handles the case $b \equiv 0 \pmod{5}$, $b \equiv 1 \pmod{5}$, $b \equiv 2 \pmod{5}$, $b \equiv 3 \pmod{5}$, and $b \equiv 4 \pmod{5}$ separately, with detailed verification for each, including the special case $b = 7$.
• For $b = 7$, a specific vector $v \in V_{3,7}$ and $w \in V_{3,3}$ are constructed such that $w$ spans the kernel of $T(v, \cdot)$, and surjectivity of $T(\cdot, w)$ is confirmed by checking image containment on monomial subspaces.

Experimental results

Research questions

• RQ1 Is the moduli space $\mathcal{T}_g$ of trigonal curves of genus $g \equiv 0 \pmod{4}$ rational?
• RQ2 Can the rationality of $\mathbb{P}V_{3,b}/(\mathrm{SL}_2 \times \mathrm{SL}_2)$ be established for all odd $b \geq 5$ using invariant-theoretic methods?
• RQ3 Does there exist a dominant rational map $V_{3,b} \dashrightarrow G(c, V_{a',b'})$ induced by a bi-transvectant with $c$ small and $a', b', c$ odd, enabling reduction to stable rationality?
• RQ4 For $b \equiv 4 \pmod{5}$, does the kernel of $T(v, \cdot)$ on $V_{1,3n+4}$ have dimension 1 for a suitable $v \in V_{3,5n+4}$, ensuring the map is well-defined and dominant?
• RQ5 Is the map $T(\cdot, w): V_{3,7} \to V_{2,4}$ surjective for the chosen $w \in V_{3,3}$, confirming the final case in the proof?

Key findings

• The moduli space $\mathcal{T}_g$ of trigonal curves of genus $g$ is rational for all $g \geq 5$, completing the classification of rationality for trigonal moduli spaces.
• For all odd $b \geq 5$, the quotient $\mathbb{P}V_{3,b}/(\mathrm{SL}_2 \times \mathrm{SL}_2)$ is rational, which corresponds to the moduli space of trigonal curves of genus $g = 4N$ with $b = 2N+1$.
• A dominant rational map $v \mapsto \operatorname{Ker}(T(v, \cdot))$ is constructed from $V_{3,b}$ to a Grassmannian $G(c, V_{a',b'})$ via a carefully chosen bi-transvectant, with $c$ small and $a', b', c$ odd, ensuring compatibility with the $-1$ scalar action.
• Non-degeneracy of the kernel map is verified for each residue class of $b \mod 5$ by analyzing monomial bases and using the non-degeneracy of transvectants $T^{(r)}$ when the degrees are coprime.
• For $b = 7$, the kernel of $T(v, \cdot)$ on $V_{3,3}$ is shown to be 1-dimensional, and the map $T(\cdot, w): V_{3,7} \to V_{2,4}$ is proven surjective by checking image containment on $x^2$, $xy$, and $y^2$ components.
• The stable rationality of the Grassmannian $G(c, V_{a',b'})$ under $\mathrm{SL}_2 \times \mathrm{SL}_2$ action is established as a key intermediate step, reducing the rationality of the original quotient to a known stable rationality result.