Reduction type of smooth quartics

Reynald Lercier; Qing Liu; Elisa Lorenzo Garc\'ia; Christophe; Ritzenthaler

arXiv:1803.05816·math.NT·October 20, 2021

Reduction type of smooth quartics

Reynald Lercier, Qing Liu, Elisa Lorenzo Garc\'ia, Christophe, Ritzenthaler

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TL;DR

This paper characterizes the reduction types of smooth quartic curves over discrete valuation fields using algebraic invariants and geometric invariant theory, providing explicit criteria and models, especially for Picard curves.

Contribution

It introduces a new characterization of reduction types of smooth quartics via algebraic invariants and models, extending geometric invariant theory results over arbitrary rings.

Findings

01

Characterization of reduction types in terms of algebraic invariants.

02

Existence of homogeneous systems of parameters over discrete valuation rings.

03

Explicit criteria for reduction types of Picard curves.

Abstract

Let $C / K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\overset{ˉ}{K}$ , in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2, 3, 5$ or $7$ . On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of (Seshadri 1977). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$ . We exhibit explicit ones for ternary quartic forms under the action of $SL_{3, R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give…

Tables4

Table 1. Table 1. Singular quartics in characteristic p ⩾ 5 𝑝 5 p\geqslant 5 and 0 0

Case	Condition		$F (x_{1}, x_{2}, x_{3})$
0	$Π$ of order $⩾ 3$		$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{1} x_{2}^{3} + a_{22} x_{1}^{2} x_{2}^{2} + a_{31} x_{1}^{3} x_{2} + a_{40} x_{1}^{4} + \\ (a_{03} x_{2}^{3} + a_{12} x_{1} x_{2}^{2} + a_{21} x_{1}^{2} x_{2} + a_{30} x_{1}^{3}) x_{3} \end{matrix}$
1.1	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} = 0 \end{matrix}$	$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{2}^{3} x_{1} + a_{22} x_{2}^{2} x_{1}^{2} + a_{31} x_{2} x_{1}^{3} + a_{40} x_{1}^{4} + \\ (a_{12} x_{2}^{2} x_{1} + a_{21} x_{2} x_{1}^{2} + a_{30} x_{1}^{3}) x_{3} + x_{1}^{2} x_{3}^{2} \end{matrix}$
1.2	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \end{matrix}$	$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{2}^{3} x_{1} + a_{22} x_{2}^{2} x_{1}^{2} + a_{31} x_{2} x_{1}^{3} + a_{40} x_{1}^{4} + \\ x_{2}^{3} x_{3} + x_{1}^{2} x_{3}^{2} \end{matrix}$
2.1	$Π$ of order 2 & $x_{1} = 0$ , $x_{2} = 0$ are both tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{2}^{3} x_{1} + a_{22} x_{2}^{2} x_{1}^{2} + a_{31} x_{2} x_{1}^{3} + a_{40} x_{1}^{4} + \\ (x_{2}^{3} + x_{1}^{3}) x_{3} + x_{2} x_{1} x_{3}^{2} \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{2}^{3} x_{1} + a_{22} x_{2}^{2} x_{1}^{2} + a_{31} x_{2} x_{1}^{3} + a_{40} x_{1}^{4} + \\ x_{1}^{3} x_{3} + x_{2} x_{1} x_{3}^{2} \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} = 0 \end{matrix}$	$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{2}^{3} x_{1} + a_{22} x_{2}^{2} x_{1}^{2} + a_{31} x_{2} x_{1}^{3} + a_{40} x_{1}^{4} + \\ x_{2} x_{1} x_{3}^{2} \end{matrix}$

Table 2. Table 2. Singular quartics in characteristic p = 2 𝑝 2 p=2

Case	Condition		$F (x_{1}, x_{2}, x_{3})$
0	$Π$ of order $⩾ 3$		$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{1} x_{2}^{3} + a_{22} x_{1}^{2} x_{2}^{2} + a_{31} x_{1}^{3} x_{2} + a_{40} x_{1}^{4} + \\ (a_{03} x_{2}^{3} + a_{12} x_{1} x_{2}^{2} + a_{21} x_{1}^{2} x_{2} + a_{30} x_{1}^{3}) x_{3} \end{matrix}$
1.1	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} = 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{13} x_{2}^{3} + a_{22} x_{1} x_{2}^{2} + a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} + \\ (a_{21} x_{1} x_{2} + a_{30} x_{1}^{2}) x_{3} + x_{1} x_{3}^{2}) \end{matrix}$
1.2	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \end{matrix}$	$\begin{matrix} x_{2} (a_{31} x_{1}^{3} + x_{2}^{2} x_{3} + x_{1} x_{3}^{2}) \end{matrix}$
2.1	$Π$ of order 2 & $x_{1} = 0$ , $x_{2} = 0$ are both tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} x_{3} (x_{2}^{3} + a_{12} x_{1} x_{2}^{2} + (a_{12} + 1) x_{1}^{2} x_{2} + x_{1}^{3} + x_{1} x_{2} x_{3}) \\ or x_{3} (x_{2}^{3} + x_{1} x_{2}^{2} + x_{1}^{2} x_{2} + x_{1}^{3} + x_{1} x_{2} x_{3}) \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{13} x_{2}^{3} + a_{22} x_{1} x_{2}^{2} + a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} \\ + (a_{21} x_{1} x_{2} + x_{1}^{2}) x_{3} + x_{2} x_{3}^{2}) \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} = 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{13} x_{2}^{3} + a_{22} x_{1} x_{2}^{2} + a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} \\ + a_{21} x_{1} x_{2} x_{3} + x_{2} x_{3}^{2}) \end{matrix}$

Table 3. Table 3. Singular quartics in characteristic p = 3 𝑝 3 p=3

Case	Condition		$F (x_{1}, x_{2}, x_{3})$
0	$Π$ of order $⩾ 3$		$\begin{matrix} a_{04} x_{2}^{4} + a_{13} x_{1} x_{2}^{3} + a_{22} x_{1}^{2} x_{2}^{2} + a_{31} x_{1}^{3} x_{2} + a_{40} x_{1}^{4} + \\ (a_{03} x_{2}^{3} + a_{12} x_{1} x_{2}^{2} + a_{21} x_{1}^{2} x_{2} + a_{30} x_{1}^{3}) x_{3} \end{matrix}$
1.1	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} = 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{13} x_{2}^{3} + a_{22} x_{1} x_{2}^{2} + a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} + \\ (a_{21} x_{1} x_{2} + a_{30} x_{1}^{2}) x_{3} + x_{1} x_{3}^{2}) \end{matrix}$
1.2	$Π$ of order 2 & $x_{1} = 0$ is the only tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \end{matrix}$	$\begin{matrix} x_{3} (x_{2}^{3} + a_{30} x_{1}^{3} + x_{1}^{2} x_{3}) \end{matrix}$
2.1	$Π$ of order 2 & $x_{1} = 0$ , $x_{2} = 0$ are both tangent at $Π$	$\begin{matrix} a_{03} \neq 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} x_{3} (x_{2}^{3} + x_{1}^{3} + x_{1} x_{2} x_{3}) \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} \neq 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} + x_{1}^{2} x_{3} + x_{2} x_{3}^{2}) \end{matrix}$
2.2		$\begin{matrix} a_{03} = 0 \\ & a_{30} = 0 \end{matrix}$	$\begin{matrix} x_{1} (a_{31} x_{1}^{2} x_{2} + a_{40} x_{1}^{3} + x_{2} x_{3}^{2}) \\ or x_{2} (a_{04} x_{2}^{3} + a_{13} x_{1} x_{2}^{2} + x_{1} x_{3}^{2}) \end{matrix}$

Table 4. Table 4. CM-curves from [ KLL + 18 ] with the reduction type for the primes dividing the discriminant

Curve	Pot. non-hyp.	Pot. hyp.	No pot. good	Unknown
$X_{1}$	$13$	$2^{†}$ , $37$ , $15187$	$5$ , $7$
$X_{2}$		$2^{†}$ , $701$	$3$ , $7$
$X_{3}$	$31$	$2^{†}$ , $233$ , $356399$	$3$ , $5$	$7$
$X_{5}$	$13$	$2^{†}$ , $37$ , $127$	$3$	$7$
$X_{6}$	$19$	$2^{†}$ , $127$ , $211$ , $20707$	$3$ , $17$	$7$
$X_{7}$	$73$	$2^{†}$ , $71$ , $17665559$	$3$ , $83$	$7$
$X_{8}$	$19$	$2^{†}$ , $499$	$7$
$X_{9}$	$13$	$79$ , $233$ , $857$	$5$	$2$ , $7$
$X_{10}$		$41$ , $71$		$2$ , $7$
$X_{11}$	$31$	$23$ , $47$ , $27527$		$2$ , $7$
$X_{12}$		$5711$ , $73064203493$	$11$	$7$
$X_{13}$	$43$	$547$ , $11827$ , $189169$	$2$ , $11$
$X_{14}$	$19$	$101$ , $107$ , $8378707$	$11$
$X_{15}$		$19$
$X_{16}$		$19$ , $37$ , $79$ , $13373064392147$	$3$
$X_{17}$		$19$ , $1229$ , $3913841117$	$2$	$3$
$X_{18}$	$13$	$19$ , $101$ , $251$ , $7468843725186901$	$2$
$X_{19}$		$11$ , $43$	$2$
$X_{20}$		$67$ , $1439$ , $2739021126001$

Equations213

F = (x_{2} + x_{3}) x_{1}^{3} - (2 x_{2}^{2} + x_{3} x_{2}) x_{1}^{2} + (x_{2}^{3} - x_{3} x_{2}^{2} + 2 x_{3}^{2} x_{2} - x_{3}^{3}) x_{1} - 2 x_{3}^{2} x_{2}^{2} + 3 x_{3}^{3} x_{2} .

F = (x_{2} + x_{3}) x_{1}^{3} - (2 x_{2}^{2} + x_{3} x_{2}) x_{1}^{2} + (x_{2}^{3} - x_{3} x_{2}^{2} + 2 x_{3}^{2} x_{2} - x_{3}^{3}) x_{1} - 2 x_{3}^{2} x_{2}^{2} + 3 x_{3}^{3} x_{2} .

{y^{2} + G = 0, π^{s} y - Q = 0.

{y^{2} + G = 0, π^{s} y - Q = 0.

v_{x} (y) = v (y) / e - min {v (y) / e, v (x_{i}) / d_{i} : i = 0, \dots, n} \in Q_{\geq 0} \cup {\infty},

v_{x} (y) = v (y) / e - min {v (y) / e, v (x_{i}) / d_{i} : i = 0, \dots, n} \in Q_{\geq 0} \cup {\infty},

E_{+} \subseteq (I_{0}, \dots, I_{n})

E_{+} \subseteq (I_{0}, \dots, I_{n})

v_{DO} (I_{3} (F)) = 0, v_{DO} (I_{27} (F)) > 0 and v_{ι} (I_{3} (F)^{5} I_{27} (F)) = 0.

v_{DO} (I_{3} (F)) = 0, v_{DO} (I_{27} (F)) > 0 and v_{ι} (I_{3} (F)^{5} I_{27} (F)) = 0.

f : C \to P (H^{0} (C, ω_{C / O})) ≃ P_{O}^{2} .

f : C \to P (H^{0} (C, ω_{C / O})) ≃ P_{O}^{2} .

Z = V_{+} (F) \subset P_{O}^{2} .

Z = V_{+} (F) \subset P_{O}^{2} .

Z_{red} = V_{+} (q) \subset P_{k}^{2} .

Z_{red} = V_{+} (q) \subset P_{k}^{2} .

\overline{F} = q^{2} .

\overline{F} = q^{2} .

F = Q^{2} + π^{r} G,

F = Q^{2} + π^{r} G,

B = O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + π^{ε} T), T \in O [x_{1}, x_{2}, x_{3}] + O [x_{1}, x_{2}, x_{3}] y

B = O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + π^{ε} T), T \in O [x_{1}, x_{2}, x_{3}] + O [x_{1}, x_{2}, x_{3}] y

f : C \to P (H^{0} (C, ω_{C / O})) ≃ P_{O}^{2} .

f : C \to P (H^{0} (C, ω_{C / O})) ≃ P_{O}^{2} .

W = Proj (k [x_{1}, x_{2}, x_{3}, y] / (q, y^{2} + g))

W = Proj (k [x_{1}, x_{2}, x_{3}, y] / (q, y^{2} + g))

B_{k} = A \oplus A y \subset Frac (A) \oplus Frac (A) y = Frac (A) [y] / (y^{2} + g)

B_{k} = A \oplus A y \subset Frac (A) \oplus Frac (A) y = Frac (A) [y] / (y^{2} + g)

B_{k} = k [x_{1}, x_{2}, x_{3}, y] / (q, y^{2} + h y + g)

B_{k} = k [x_{1}, x_{2}, x_{3}, y] / (q, y^{2} + h y + g)

(\partial q / \partial x_{1} \hfill y_{0} \partial h / \partial x_{1} + \partial g / \partial x_{1} \partial q / \partial x_{2} \hfill y_{0} \partial h / \partial x_{2} + \partial g / \partial x_{2} \partial q / \partial x_{3} y_{0} \partial h / \partial x_{3} + \partial g / \partial x_{3}),

(\partial q / \partial x_{1} \hfill y_{0} \partial h / \partial x_{1} + \partial g / \partial x_{1} \partial q / \partial x_{2} \hfill y_{0} \partial h / \partial x_{2} + \partial g / \partial x_{2} \partial q / \partial x_{3} y_{0} \partial h / \partial x_{3} + \partial g / \partial x_{3}),

Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + G))

Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + G))

W = Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + π^{ε} G)) .

W = Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + π^{ε} G)) .

Q, H, G \in O [x_{1}, x_{2}, x_{3}]

Q, H, G \in O [x_{1}, x_{2}, x_{3}]

Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + H y + G))

Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{s} y, y^{2} + H y + G))

F = Q^{2} + π^{s} Q H + π^{r} P

F = Q^{2} + π^{s} Q H + π^{r} P

F = Q_{1}^{2} + π^{s_{1}} Q_{1} H_{1} + π^{r + 1} P_{1}

F = Q_{1}^{2} + π^{s_{1}} Q_{1} H_{1} + π^{r + 1} P_{1}

W = Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{ℓ} y, y^{2} + π^{s - ℓ} H y + π^{ε} P)) .

W = Proj (O [x_{1}, x_{2}, x_{3}, y] / (Q - π^{ℓ} y, y^{2} + π^{s - ℓ} H y + π^{ε} P)) .

P \in (π, Q_{0}, y) \cap O [x_{1}, x_{2}, x_{3}] = (π, Q) O [x_{1}, x_{2}, x_{3}] .

P \in (π, Q_{0}, y) \cap O [x_{1}, x_{2}, x_{3}] = (π, Q) O [x_{1}, x_{2}, x_{3}] .

F = Q^{2} + (π^{s} H + π^{r} H_{0}) Q + π^{r + 1} P_{1} .

F = Q^{2} + (π^{s} H + π^{r} H_{0}) Q + π^{r + 1} P_{1} .

F = Q_{1}^{2} + (π^{s} H + π^{r} H_{0} - 2 π^{ℓ} R) Q_{1} + π^{r + 1} P_{1} .

F = Q_{1}^{2} + (π^{s} H + π^{r} H_{0} - 2 π^{ℓ} R) Q_{1} + π^{r + 1} P_{1} .

u F = Q^{2} + π^{s} Q H + π^{2 s} G

u F = Q^{2} + π^{s} Q H + π^{2 s} G

F = Q^{2} + π^{s} Q H + π^{2 s} G .

F = Q^{2} + π^{s} Q H + π^{2 s} G .

F = Q_{n}^{2} + π^{s_{n}} Q_{n} H_{n} + π^{r + n} P_{n}

F = Q_{n}^{2} + π^{s_{n}} Q_{n} H_{n} + π^{r + n} P_{n}

Q_{n + 1} - Q_{n} \in π^{[(r + n) /2]} O [x_{1}, x_{2}, x_{3}], s_{n} \geq min {s_{1}, r, v (2) + r /2} for all n \geq 1 .

Q_{n + 1} - Q_{n} \in π^{[(r + n) /2]} O [x_{1}, x_{2}, x_{3}], s_{n} \geq min {s_{1}, r, v (2) + r /2} for all n \geq 1 .

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Full text

Reduction type of smooth plane quartics

Reynald Lercier

Reynald Lercier, DGA & Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.

[email protected]

,

Qing Liu

Qing Liu, Institut de mathématiques de Bordeaux, CNRS - UMR 5251, Université de Bordeaux, F-33405 Talence, France.

[email protected]

,

Elisa Lorenzo García

Elisa Lorenzo García, Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.

[email protected]

and

Christophe Ritzenthaler

Christophe Ritzenthaler, Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.

[email protected]

(Date: March 1, 2024)

Abstract.

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$ , in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$ . On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of [Ses77]. For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$ . We exhibit explicit ones for ternary quartic forms under the action of $\operatorname{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

Key words and phrases:

smooth plane quartic, reduction, hyperelliptic, invariants, valuation

2010 Mathematics Subject Classification:

11G20, 14Q05, 14D10, 14D20, 14H25

The authors would like to thank Jeroen Sijsling for his careful reading of an earlier version of the paper, Matthieu Romagny for useful references, Bianca Viray and Armand Brumer for inspiring discussions, and the referee for his/her comments improving the presentation of the manuscript. The second named author is partly supported by Xiamen University and the NSFC grant no. 11031004.

1. Introduction and main results

Let $K$ be a discrete valuation field with valuation $v$ , valuation ring $\mathcal{O}$ and a uniformizer $\pi$ . Let $k=\mathcal{O}/\langle\pi\rangle$ be the residue field of characteristic $p\geq 0$ . When $F$ is an integral polynomial, i.e. with coefficients in $\mathcal{O}$ , we denote by $\bar{F}$ its reduction modulo $\pi$ . A smooth projective geometrically connected curve of genus $3$ is either a plane quartic or a hyperelliptic curve depending on whether the canonical divisor is very ample or not. Let $C/K$ be a smooth plane quartic.

•

We say that $C$ has good (resp. good quartic, resp. good hyperelliptic) reduction over $K$ if $C$ has a projective smooth model $\mathcal{C}$ over $\mathcal{O}$ (resp. whose special fiber is a smooth plane quartic, resp. a hyperelliptic curve).

•

We say that $C$ has potentially good (resp. potentially good quartic, resp. potentially good hyperelliptic) reduction if for some finite separable extension $(K^{\prime},v^{\prime})$ of $(K,v)$ , $C_{K^{\prime}}/K^{\prime}$ has good (resp. good quartic, resp. good hyperelliptic) reduction over $K^{\prime}$ .

•

We say that $C$ has bad reduction over $K$ if it does not have good reduction over $K$ .

Note that the type of the potential reduction does not depend on the choice of $(K^{\prime},v^{\prime})$ and is given by the type of reduction of the stable model of $C$ over some finite separable extension of $K$ . Hence, in the following, the expression “after a possible finite extension of $K$ ”, or potentially, means that we are allowed to take a finite extension of $K$ and still call $K,\mathcal{O},v,\pi$ the corresponding notions. Let us point out already here that when using the adjective (semi)-stable alone, we always mean in the Deligne-Mumford sense of (semi)-stability. Otherwise, while talking later about Geometric Invariant Theory (GIT) stability, we will always write GIT-(semi-)stable.

Given such a curve $C/K$ , we want to determine the reduction type of $C$ among the above possibilities. In the first two cases, we also look for an explicit equation of the special fiber. In the first case, it will be again a smooth plane quartic over $k$ , whereas in the second case it will be isomorphic over $\bar{k}$ to $y^{2}=f(x,z)$ where $f$ is a binary octic with no multiple roots.

*Example 1.1**.*

Let us consider the plane smooth quartic $C/\mathbb{Q}:F=0$ where

[TABLE]

This curve is an equation of $X_{ns}(13)\simeq X_{s}(13)\simeq X_{0}^{+}(169)$ which is studied in [BDM*+*19, Cor.6.8]. By using a general result from [Edi90], they prove that this curve has good quartic reduction away from $13$ and potentially good hyperelliptic reduction at 13 after a ramified extension of degree 84. By using the characterizations we are giving in this article and our implementations, we can automatically check that the curve has potentially good hyperelliptic reduction at $13$ and that the special fiber is the hyperelliptic curve of affine equation $y^{2}=x^{7}-1$ (see Example 5.13).

The most complicated task will be to distinguish good hyperelliptic reduction from bad reduction. By identifying a plane quartic with its $15$ coefficients up to a multiplicative constant, we can view the set $\bm{X}$ of smooth plane quartics as a subset of $\mathbb{P}\,^{14}$ . Since isomorphisms between smooth plane quartics are linear, $\bm{X}/\operatorname{SL}_{3}(\bar{K})$ gives a description of the locus $\bm{M}_{3}^{\textrm{qrt}}$ of plane quartics inside the moduli space $\bm{M}_{3}$ of all curves of genus $3$ . Considering the locus of hyperelliptic curves as “a boundary” of the locus of $\bm{M}_{3}^{\textrm{qrt}}$ inside $\bm{M}_{3}$ , the challenge is to understand whether a family of plane smooth quartics has limit in this boundary. Similar situations have been studied to describe the boundary of the compactification of $\bm{M}_{3}$ and there is a rich literature on this subject [Art09, Gla79, Has99, HL10, Kan00, Kon00, Mum77, Yuk86]. Surprisingly, the mentioned problem has never been fully addressed before (still, see [HLP00, Prop.14 and Prop.15] in the case of Ciani quartics). Known results about the relations between plane quartics and hyperelliptic genus 3 curves are basically contained in the following proposition [Cle80, p.156], [Har87], [HM98, p.134].

Proposition 1.2.

Suppose $p\neq 2$ . Let $s>0$ be an integer, $G\in\mathcal{O}[x_{1},x_{2},x_{3}]$ be a primitive (i.e. the gcd of its coefficients is 1) quartic form and $Q\in\mathcal{O}[x_{1},x_{2},x_{3}]$ be a primitive quadratic form. Let us assume that $\bar{Q}$ is irreducible and that $\bar{Q}=0$ intersects $\bar{G}=0$ transversely in $8$ distinct $\overline{k}$ -points. Let us denote $\mathcal{C}/\mathcal{O}$ the subscheme of the weighted projective space $\mathbb{P}\,^{(1,1,1,2)}$ defined by

[TABLE]

Then the generic fiber is isomorphic to the plane smooth quartic $C/K:Q^{2}+\pi^{2s}G=0$ which has good hyperelliptic reduction. The special fiber of $\mathcal{C}$ is isomorphic to the double cover of $\bar{Q}=0$ ramified over the 8 distinct intersection $\overline{k}$ -points of $\bar{Q}=0$ with $\bar{G}=0$ .

This motivates the following definitions.

Definition 1.3.

Let $C/K$ be a plane smooth quartic. When $p\neq 2$ , we say that $C$ admits a toggle model if there exist an even integer $s>0$ , a primitive quartic form $G\in\mathcal{O}[x_{1},x_{2},x_{3}]$ and a primitive quadric $Q\in\mathcal{O}[x_{1},x_{2},x_{3}]$ with $\bar{Q}$ irreducible such that $Q^{2}+\pi^{2s}G=0$ is $K$ -isomorphic to $C$ . If, moreover, $\bar{Q}=0$ intersects $\bar{G}=0$ transversely in $8$ distinct $\overline{k}$ -points, the model $Q^{2}+\pi^{2s}G=0$ is a good toggle model of $C$ . We refer to Theorem 2.9 for the definition when $p=2$ .

When $C/K$ admits a plane quartic model over $\mathcal{O}$ which special fiber is smooth then obviously this plane model is a stable model of $C$ . When $C$ admits a good toggle model, it has good hyperelliptic reduction and (1.1) is a stable (actually smooth) model of $C$ . The converse holds as well, this is the main result of Section 2.

Theorem 1.4 (See Theorems 2.8 and

2.9).

Let $K$ be a discrete valuation field with residue field of characteristic $p\geq 0$ . Let $C/K$ be a smooth plane quartic. Then $C$ admits good hyperelliptic reduction over $K$ if and only if $C$ has a good toggle model over $K$ .

By using this result, the methods of [Kol97] and [ES20] and some ingredients from invariant theory, the question of the reduction type (good quartic, good hyperelliptic or bad) over $K$ of a smooth quartic may be answered (see remark 5.2).

We turn now to the determination of the type when one allows extensions of $K$ . In the spirit of the work of [Liu93] on genus 2, we aim at getting a characterization of the potential reduction type in terms of the valuations of certain invariants of $C/K$ under the action of $\operatorname{SL}_{3}(\bar{K})$ . Let us start by recalling some results on the homogeneous algebra of invariants of ternary quartic forms.

In [Dix87], Dixmier worked out a list of 7 homogeneous polynomial invariants for the equivalence of ternary quartic forms under the action of $\operatorname{SL}_{3}(\mathbb{C})$ , denoted $I_{3},I_{6},I_{9},I_{12},I_{15},I_{18}$ and $I_{27}$ . They form a Homogeneous System Of Parameters (HSOP) for the algebra of invariants $\mathcal{A}$ over $\mathbb{C}$ , which means that the radical of the ideal they generate is equal to the irrelevant ideal of the homogeneous algebra $\mathcal{A}$ (see Definition 3.7 and remark 3.9). Later, this list is completed by Ohno ([Ohn07, Theorem 4.1], see also [Els15]) into a list of 13 homogeneous generators of $\mathcal{A}$ as $\mathbb{C}$ -algebra. Theses invariants are defined over $\mathbb{Z}[\frac{1}{2\cdot 3}]$ , and are now called the Dixmier-Ohno invariants. We denote $\operatorname{\underline{DO}}$ the list of these invariants, $\operatorname{\underline{DO}}(F)\in K^{13}$ their values at a ternary quartic form $F$ and when at least one of these values is not zero, we denote $\operatorname{DO}(F)$ the corresponding point in the weighted projective space with weights $3,6,9,9,12,12,15,15,18,18,21,21,27$ . They determine the isomorphism classes of smooth quartic curves over $\mathbb{C}$ in the sense that two smooth quartics $C_{i}:\,F_{i}=0$ are isomorphic over $\mathbb{C}$ if and only if $\operatorname{DO}(F_{1})=\operatorname{DO}(F_{2})$ .

This result over $\mathbb{C}$ is clearly not sufficient for our purposes as we want to deal with fields of arbitrary characteristic and with reduction properties. Fortunately, Seshadri in [Ses77] has generalized the Geometric Invariant Theory (GIT) of Mumford over an arbitrary ring $R$ . For the convenience of the reader, we include in Section 3.1 some general results on this topic and proofs we were unable to find in the literature. We also prove, under some assumptions on $R$ 111the assumptions are satisfied for instance for local rings and rings of integers of global fields., that there exists a HSOP over $R$ , i.e. a finite list of invariants defined over $R$ being a HSOP after base change to all residue fields of $R$ . It is a bit of a surprise as one knows that there exist invariants over fields of small characteristics which do not come from reducing invariants in characteristic [math] (see [Bas15] for binary forms and remark 4.6 for ternary quartics). In principle, one should be able to exhibit a HSOP over $\mathbb{Z}$ , but in practice the degrees may be too large to be practical. This is why in Theorem 4.1, we determine explicit HSOP over $\mathcal{O}$ for ternary quartic forms which depend only on the characteristic of the residue field. This is achieved in all characteristics but $3$ , for which we only have a conjectural HSOP. (see Theorem 4.3). Note also that even for some large characteristics, like $523$ , the Dixmier invariants alone may not be a HSOP unlike in characteristic [math]. With the normalization of [GKZ94, p.426], [Dem12], the discriminant $D_{27}$ of a ternary quartic is a degree $27$ polynomial with coefficients in $\mathbb{Z}$ and always belongs to our HSOP (up to a multiplicative unit). It is related to the Dixmier invariant $I_{27}$ by $2^{40}\cdot I_{27}=D_{27}$ . A plane quartic $F=0$ over any field is smooth if and only if $D_{27}(F)\neq 0$ .

As we want to deal with valuations of the invariants, we need to introduce the following notation before stating our results.

Definition 1.5.

Let $\underline{d}=(d_{0},\ldots,d_{n})\in\mathbb{Z}_{>0}^{n+1}$ . Given an element $\underline{x}=(x_{0},\ldots,x_{n})\in K^{n+1}\setminus\{0\}$ , we denote by $x=(x_{0}:\ldots:x_{n})$ the corresponding point in the weighted projective space $\mathbb{P}\,^{\underline{d}}(K)$ . After a possible finite extension of $K$ , one can always find a representative $(X_{0},\ldots,X_{n})\in\mathcal{O}^{n+1}$ of $x$ such that one of the $X_{i}$ ’s has valuation [math], so it represents a section in $\mathbb{P}\,^{\underline{d}}(\mathcal{O})$ . We call such a representative a minimal representative and denote it $\underline{x}^{\min}$ .

A minimal representative is not uniquely defined but two minimal representatives differ by the action of a unit and their valuations are coordinate-wise equal.

Definition 1.6.

If $y\in K$ and $e\in\mathbb{Z}_{>0}$ , we call the normalized valuation of degree $e$ of $y$ with respect to $\underline{x}$ the (possibly infinite) number

[TABLE]

(By convention $v(0)=\infty$ ). We have $v_{x}(y)=0$ if and only if $y\neq 0$ and $x_{i}^{e}y^{-d_{i}}\in\mathcal{O}$ for all $i\leq n$ .

Notice that if $(X_{0},\ldots,X_{n},Y)$ is a minimal representative of the point $(x_{0}:\ldots:x_{n}:y)\in\mathbb{P}\,^{\underline{d},e}$ , then $v_{x}(y)=v(Y)/e$ .

In the context of invariants, we will use the following definition and notation.

Definition 1.7.

Let $\underline{I}=(I_{0},\ldots,I_{n})$ be a list of homogeneous polynomials of degree $\underline{d}$ in $K[T_{0},\ldots,T_{m}]$ . For $F\in K^{m+1}\setminus\{0\}$ , if $\underline{I}(F)=(I_{0}(F),\ldots,I_{n}(F))$ is not the zero vector, and if $J$ is a homogeneous polynomials of degree $e$ we denote by $v_{I}(J(F))$ the normalized valuation of degree $e$ of $J(F)$ with respect to $\underline{I}(F)$ . It only depends on the point of $\mathbb{P}\,^{\underline{d}}(K)$ defined by $F$ . Finally, if $F_{0}\in\mathcal{O}^{m+1}$ and $\underline{I}(F_{0})=\underline{I}(F_{0})^{\min}$ we say that $F_{0}$ is a minimal form.

*Remark 1.8**.*

Let $J,I_{0},\dots,I_{n}\in E\subseteq\mathcal{O}[T_{0},\dots,T_{m}]$ be homogeneous elements of some graded sub- $\mathcal{O}$ -algebra and suppose that

[TABLE]

that is, $\{I_{0},\dots,I_{n}\}$ is a homogeneous system of radical generators (HSORG) of $E$ , see Definition 3.7. In particular, HSOP are HSORG with further condition on the cardinality. The condition $v_{I}(J(F))=0$ is equivalent to $F\in D_{+}(J)$ , the principal open subset of $\operatorname{Proj}E$ associated to $J$ . Here $F$ is viewed as a point of $(\operatorname{Proj}E)(K)$ . In particular, the condition $v_{I}(J(F))=0$ does not depend on the choice of the system $I_{0},\dots,I_{n}$ as long as the latter is a HSORG of $E$ .

Considering ternary quartic forms $F$ , a natural question, and a key argument for the sequel, is to know if, after a possible finite extension of $K$ , there exists a minimal form $\operatorname{GL}_{3}(K)$ -equivalent to $F$ . We show in Proposition 3.13 that this is the case when $F=0$ is GIT-stable over $K$ and the list $\underline{I}$ is a HSORG over $\mathcal{O}$ . Using this result, we obtain the following theorem.

Theorem 1.9 (See Theorem 3.15).

Let $K$ be a discrete valuation field with valuation $v$ , valuation ring $\mathcal{O}$ and residue field $k$ of characteristic $p\geq 0$ . Let $\underline{I}$ be a list of invariants giving rise to a HSORG for ternary quartic forms under the action of $\operatorname{SL}_{3,\mathcal{O}}$ . Then a smooth plane quartic $C/K:F=0$ has potentially good quartic reduction if and only if $v_{I}(D_{27}(F))=0$ (in other words, for any invariant $I_{i}$ of degree $d_{i}$ in the list $\underline{I}$ , we have $I_{i}(F)^{27}/D_{27}(F)^{d_{i}}\in\mathcal{O}$ ).

This theorem is proved in Section 3.3. Once worked out explicit HSOP in Section 4.1, it gives a practical criterion in all characteristics. Indeed, notice, that though we use the existence of a minimal form, it is not necessary to compute it to check whether $v_{I}(D_{27}(F))=0$ . Similarly, we do not need the minimal form to compute an equation of the special fiber when the curve has potentially good quartic reduction, at least generically and if $p>7$ (see remark 4.4). Some illustrations are given in Example 4.5 and in Theorems 4.7, 4.8 and 4.9 with the characterization of potentially good quartic reduction for Picard curves in terms of low degree polynomials in their coefficients when $\operatorname{char}(K)\neq 3$ .

From Section 5.1 and beyond, we restrict to a discrete valuation ring $\mathcal{O}$ with residue field $k$ of characteristic $p=0$ or $p>7$ (see a discussion about this restriction in remark 5.15). We focus there on distinguishing between curves which have potentially good hyperelliptic reduction or have not potentially good reduction. In Proposition 5.3, we first prove that $C/K:F=0$ potentially admits a (not necessarily good) toggle model if and only if any minimal representative of $\operatorname{DO}(F)$ reduces modulo $\pi$ to $\operatorname{DO}(Q_{0}^{2})$ where $Q_{0}=x_{2}^{2}-4x_{1}x_{3}$ is a fixed non-degenerate quadric (the choice of this particular one will become clearer later).

It remains to characterize in terms of the invariants when a toggle model $Q_{0}^{2}+\pi^{2s}G=0$ is good, i.e. when the intersection of $\bar{Q}_{0}=0$ with $\bar{G}=0$ is transversal. This is the content of Section 5.2. In order to do so, we introduce a well-known $\operatorname{SL}_{2}(\bar{K})$ -equivariant linear morphism $b_{8}$ from the space of ternary quartic forms $F$ to the space of binary octic forms. The action on the ternary quartics is given through a morphism $h:\operatorname{SL}_{2}(\bar{K})\to\operatorname{SL}_{3}(\bar{K})$ whose image denoted $\operatorname{SO}_{3}(\bar{K})$ is the stabilizer of $Q_{0}$ . Starting from a good toggle model $F=Q_{0}^{2}+\pi^{2s}G$ of $C/K$ , a quick computation shows that $b_{8}(F)=\pi^{2s}b_{8}(G)$ and that $y^{2}=\overline{b_{8}(G)}$ is an affine equation of the special fiber of the stable model. The algebra of invariants of binary octics under the action of $\operatorname{SL}_{2}(\bar{K})$ is generated by the classical Shioda invariants $j_{2},\ldots,j_{10}$ , see [Shi67]. The invariants $j_{2},\ldots,j_{7}$ still form a HSOP over $\mathcal{O}$ , see Corollary 3.17 and the discussion before. Evaluating the Dixmier-Ohno invariants on a toggle model $Q_{0}^{2}+\pi^{2s}G=0$ and the Shioda invariants $j_{2},\ldots,j_{7}$ on $b_{8}(G)$ gives algebraic relations between the two sets and considering their $\pi$ -expansion reveals a list $\iota=(\iota_{6},\iota_{9},\iota_{12},\iota_{15},\iota_{18},\iota_{21})$ of six explicit invariants for ternary quartic forms, see equation (5.6). Moreover we get a similar congruence (5.4) between the discriminant of binary octics, $D_{14}$ , and $I_{3}^{5}\,I_{27}$ . For a good toggle model, one has $v_{\iota}(I_{3}^{5}(F)\,I_{27}(F))=v(D_{14}(b_{8}(G)))=0$ . Actually, we have the following equivalence.

Theorem 1.10.

Let $K$ be a discrete valuation field with valuation $v$ , valuation ring $\mathcal{O}$ and a uniformizer $\pi$ . Let $k=\mathcal{O}/\langle\pi\rangle$ be the residue field of characteristic $p\neq 2,3,5$ and $7$ . Let $C/K$ be a smooth plane quartic defined by $F=0$ . Then $C$ has potentially good hyperelliptic reduction if and only if

[TABLE]

Under these conditions, one can also obtain an explicit equation for the special fiber.

To prove that the conditions of Theorem 1.10 are also sufficient, one first observes that they imply the existence of a toggle model $F=Q_{0}^{2}+\pi^{2s}G$ for $C$ . Then, a crucial step is to prove that we can choose it such that $\overline{b_{8}(G)}$ is GIT-semi-stable. In order to do this, we rigidify a toggle model in the following way: there is, up to a multiplicative constant, a unique contravariant $\rho$ of order 2 and degree 4, which can be seen as a quadratic form whose coefficients are degree $4$ polynomials in the coefficients of $F$ (see [Dix87] and [LRS18]). After a possible finite extension of $K$ and completion of $K$ , one can always find a toggle model $F$ such that $\rho(F)=\rho(Q_{0}^{2})$ . This extra-information allows us to control simultaneously the integrality of the coefficients of the toggle model and of the form $b_{8}(G)$ we are looking for (see Lemma 5.12).

Let us point out that the conditions in Theorem 1.10 may give the wrong idea that the study of the valuations of $I_{3}$ and $I_{27}$ would be enough to characterize the reduction. But this is not true since we need to consider them inside the broader lists of invariants $\operatorname{DO}$ and $\iota$ , and normalize them accordingly to the valuations of all the invariants of these lists. Notice that a similar phenomena occurs for Siegel modular forms. In [LR19], we show that the pull-back of the Siegel modular form $\chi_{28}$ to the space of quartics is $-2^{171}\cdot 3^{3}\cdot I_{27}^{3}I_{3}$ . Now, over $\mathbb{C}$ , the non-vanishing of this modular form together with the vanishing of the modular form $\chi_{18}$ (which is known to be related to $I_{27}^{2}$ ) characterizes the locus of hyperelliptic Jacobians among abelian varieties (see [Tsu86, p.847]). A natural guess is to expect that they would be the key forms to find the reduction type of a quartic. However, their expressions in terms of invariants seem to lead to the absurd statement that the vanishing of $\chi_{18}$ always implies the vanishing of $\chi_{28}$ . Within this article, one sees that this is because these forms would have to be considered inside broader lists to become significant.

One of the motivations of this work was, in the continuation of [BCL*+*15], [KLL*+*16] and [KLL*+*18], to understand the primes dividing the discriminant $I_{27}$ of a smooth plane quartic with complex multiplication over a number field only in terms of the CM-field. We hope that the present work can be useful for this task. Another direction is to refine the study of the bad reduction type in the spirit of [BW17], [DDMM19] and to describe more precisely the degenerated cases. We only have a quick peek at this question in remarks 2.3 and 2.10 and set aside an exhaustive study for a forthcoming article.

2. Hyperelliptic reduction in terms of toggle models

In this section $K$ is a discrete valuation field with valuation $v$ , valuation ring $\mathcal{O}$ and a uniformizer $\pi$ . Let $k=\mathcal{O}/\langle\pi\rangle$ be the residue field of characteristic $p\geq 0$ . Before proving Theorem 1.4, we need some preliminary results.

2.1. Some general facts

In this section we prove the existence of suitable models when $C$ has good hyperelliptic reduction over $K$ (see Theorems 2.8 and 2.9).

We start with some geometric observations. Suppose $C$ has good reduction over $K$ . Let ${\mathcal{C}}$ be a smooth projective model over ${\mathcal{O}}$ . Let $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ be the dualizing sheaf ( $=\Omega^{1}_{{\mathcal{C}}/{\mathcal{O}}}$ as ${\mathcal{C}}$ is smooth). Then $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is a base-point free invertible sheaf and induces a morphism

[TABLE]

On the generic fiber, $f$ is a closed immersion because $C$ is a plane quartic. Let ${\mathcal{Z}}=f({\mathcal{C}})$ be the image of $f$ endowed with the reduced subscheme structure. Then $f:{\mathcal{C}}\to{\mathcal{Z}}$ is a birational finite morphism. As ${\mathcal{C}}$ is normal, $f$ is just the normalization morphism.

*Fact 2.1**.*

If we have a finite birational morphism ${{\mathcal{W}}}\to{\mathcal{Z}}$ , and if ${{\mathcal{W}}}$ is normal, then ${\mathcal{W}}\to{\mathcal{Z}}$ is the normalization morphism ([Liu02, Proposition 4.1.22]), hence ${{\mathcal{W}}}$ is isomorphic to ${\mathcal{C}}$ .

If $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is very ample, then $f$ is a closed immersion and ${\mathcal{C}}$ is a smooth quartic in $\mathbb{P}\,^{2}_{\mathcal{O}}$ . Suppose from now on that $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is not very ample. Then $f_{k}:{\mathcal{C}}_{k}\to\mathbb{P}\,^{2}_{k}$ is a finite generically étale morphism of degree $2$ into its image $({\mathcal{Z}}_{k})_{\mathrm{red}}$ which is a smooth conic over $k$ . The scheme ${\mathcal{Z}}$ is defined by an irreducible quartic form $F\in{\mathcal{O}}[x_{1},x_{2},x_{3}]$ as

[TABLE]

Let $q\in k[x_{1},x_{2},x_{3}]$ be a nondegenerate quadratic form defining $({\mathcal{Z}})_{\mathrm{red}}$ :

[TABLE]

In the sequel, $q$ will be fixed once for all. As $V_{+}(\overline{F})={\mathcal{Z}}_{k}$ , the reduction $\overline{F}$ is equal to a multiple of $q^{2}$ . Scaling $F$ by a unit of ${\mathcal{O}}$ , we can suppose that

[TABLE]

Given such an $F$ , equation (2.3) implies that there exist a positive integer $r>0$ , a quadratic form $Q\in{\mathcal{O}}[x_{1},x_{2},x_{3}]$ and a quartic form $G\in{\mathcal{O}}[x_{1},x_{2},x_{3}]$ such that

[TABLE]

with $\overline{Q}=q$ and $\overline{G}\neq 0$ . Note that if $\hat{{\mathcal{O}}}$ is the completion of ${\mathcal{O}}$ , then $F$ is not a square in $\hat{{\mathcal{O}}}[x_{1},x_{2},x_{3}]$ because $C$ is geometrically reduced. So the biggest possible power $\pi^{r}$ dividing $F-Q^{2}$ in ${\mathcal{O}}[x_{1},x_{2},x_{3}]$ for all liftings $Q$ of $q$ exists.

We endow the polynomial ring ${\mathcal{O}}[x_{1},x_{2},x_{3},y]$ with the grading $\deg x_{i}=1$ and $\deg y=2$ . The following lemma will be repeatedly used:

Lemma 2.2.

Let $B$ be a graded ${\mathcal{O}}$ -algebra of the following form

[TABLE]

with $s\geq 0$ , $\varepsilon\in\{0,1\}$ and $2s+\varepsilon>0$ . Then the following properties hold:

(a)

$B$ * is flat over ${\mathcal{O}}$ .* 2. (b)

Suppose that $B\otimes K$ is normal. If $B\otimes k$ is reduced when $\varepsilon=0$ and $T\notin(\pi,Q,y)$ when $\varepsilon=1$ , then $B$ is normal. 3. (c)

If furthermore, $F\in(y^{2}+\pi^{\varepsilon}T)K^{*}$ in $K[x_{1},x_{2},x_{3},y]/(Q-\pi^{s}y)$ , then $\operatorname{Proj}B\simeq{\mathcal{C}}$ .

Proof.

(a) The flatness is equivalent to $B$ being $\pi$ -torsion-free. The latter follows from a straightforward computation.

(b) As $\dim B\otimes K=\dim B\otimes k=2$ , $\operatorname{Spec}B$ is equidimensional. It is then easy to see that $B$ is a local complete intersection ([Liu02], Exercise 6.3.4(c)). By Serre’s criterion ([Liu02], Corollary 8.2.24), it is then enough to check that $B_{\mathfrak{p}}$ is normal for height $1$ prime ideals $\mathfrak{p}$ of $B$ . This follows from the hypothesis on $B\otimes K$ if $\pi\notin\mathfrak{p}$ .

Suppose that $\pi\in\mathfrak{p}$ . Then $\mathfrak{p}$ is a minimal prime ideal over $\pi B$ . Let us show that $B_{\mathfrak{p}}$ is a discrete valuation ring, or equivalently, that $\mathfrak{p}B_{\mathfrak{p}}$ is principal. If $\varepsilon=0$ , as $B\otimes k$ is reduced, $\mathfrak{p}B_{\mathfrak{p}}$ is generated by $\pi$ . If $\varepsilon=1$ , then $\mathfrak{p}=(\pi,y)$ is the unique prime ideal over $\pi B$ . This prime ideal is the image in $B$ of the prime ideal $(\pi,Q,y)$ of ${\mathcal{O}}[x_{1},x_{2},x_{3},y]$ . By hypothesis, $T\notin\mathfrak{p}$ in $B$ , therefore $\mathfrak{p}B_{\mathfrak{p}}$ is generated by $y$ .

(c) The canonical map ${\mathcal{O}}[x_{1},x_{2},x_{3}]\to B$ is finite. The hypothesis on $F$ easily implies that $F$ belongs to the ideal $(Q-\pi^{s}y,y^{2}+\pi^{\varepsilon}T)$ in ${\mathcal{O}}[x_{1},x_{2},x_{3},y]$ . So we get a canonical map $\phi:{\mathcal{O}}[x_{1},x_{2},x_{3}]/(F)\to B$ which is an isomorphism after tensoring with $K$ , thus $\phi$ is finite and injective and the corresponding morphism $\operatorname{Proj}B\to{\mathcal{Z}}$ is finite and birational. We conclude by Fact 2.1. ∎

*Remark 2.3**.*

When $C$ has irreducible stable (not necessarily smooth) reduction over $K$ , the situation is very similar to the good reduction case. Let ${\mathcal{C}}$ be the stable model of $C$ over ${\mathcal{O}}$ . Then $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is still base-point free and induces a morphism

[TABLE]

The reduced image ${\mathcal{Z}}=f({\mathcal{C}})$ is a quartic and $f:{\mathcal{C}}\to{\mathcal{Z}}$ is a birational finite morphism, equal to the normalization morphism. If $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is very ample, then ${\mathcal{C}}\simeq{\mathcal{Z}}$ is a stable quartic in $\mathbb{P}\,^{2}_{\mathcal{O}}$ . If $\omega_{{\mathcal{C}}/{\mathcal{O}}}$ is not very ample, then one can show that $({\mathcal{Z}}_{k})_{\mathrm{red}}$ is a smooth conic over $k$ and $f_{k}:{\mathcal{C}}_{k}\to({\mathcal{Z}}_{k})_{\mathrm{red}}$ is a finite generically étale morphism of degree $2$ . The reduction is a singular hyperelliptic curve.

Next we examine when the special fiber of $\operatorname{Proj}B$ is reduced or semi-stable (in the Deligne-Mumford sense). A point in a scheme over $k$ is said to be separable over $k$ if its residue field is a separable extension of $k$ .

Lemma 2.4.

Suppose $\mathrm{char}(k)\neq 2$ . Let $g\in k[x_{1},x_{2},x_{3}]$ be a quartic form and let

[TABLE]

with $\deg x_{i}=1$ and $\deg y=2$ . Then the following properties are true:

(a)

$W$ * is reduced if and only if $g\not\equiv 0\mod q$ ;* 2. (b)

$W$ * is semi-stable (respectively smooth) if and only if the curves $q=0$ and $g=0$ in $\mathbb{P}\,^{2}_{k}$ intersect at separable points over $k$ , with multiplicity at most $2$ (respectively equal to $1$ ).*

Proof.

(a) The condition is clearly necessary. Suppose that the condition is satisfied. Let us prove that $B_{k}:=k[x_{1},x_{2},x_{3},y]/(q,y^{2}+g)$ is reduced. Let $A=k[x_{1},x_{2},x_{3}]/(q)$ . This is an integrally closed domain and we have $B_{k}=A\oplus Ay$ . By hypothesis $g\neq 0$ in $A$ . So $\operatorname{Frac}(A)[y]/(y^{2}+g)$ is separable over $\operatorname{Frac}(A)$ . Hence,

[TABLE]

is reduced.

(b) The natural projection $\phi:W\to V_{+}(q)$ is finite and étale away from the subscheme $D:=V_{+}(q,g)$ . Let $w_{0}\in W$ be a point lying over some point of $D$ . Then $\phi$ is ramified at $w_{0}$ . Let us show that $w_{0}$ is a separable point over $k$ . When $W$ is smooth at $w_{0}$ , this is proved for instance in [LL99], Lemma 3.3. The case when $W$ is singular at $w_{0}$ is proved in [Liu02], Proposition 10.3.7(b).

For the rest of (b), we can suppose that $k$ is algebraically closed. A local equation of $W$ at a point lying over an intersection point $z\in D$ is $y^{2}=x^{r}a$ where $r$ is the intersection number at $z$ and $a\in{\mathcal{O}}_{V_{+}(q),z}^{*}$ . The condition for $W$ to be semi-stable or smooth is then clear. ∎

Lemma 2.5.

Let $\mathrm{char}(k)=2$ . Let $h,g\in k[x_{1},x_{2},x_{3}]$ be respectively a quadratic and a quartic forms. Let

[TABLE]

and let $W=\operatorname{Proj}B_{k}$ with $\deg x_{i}=1$ and $\deg y=2$ . Then the following properties are true:

(a)

$W$ * is reduced if and only if $h$ is prime to $q$ or, $h$ is divisible by $q$ and $g$ is not a square mod $q$ ;* 2. (b)

if $h$ is divisible by $q$ , then $W$ is not semi-stable; 3. (c)

if the curves $V_{+}(q)$ and $V_{+}(h)$ intersect transversely at separable points of $\mathbb{P}\,^{2}_{k}$ , then $W$ is semi-stable.

Proof.

(a) The condition is clearly necessary. Let us show the converse. Let $A$ be defined as in the proof of Lemma 2.4(a). If $h$ is prime to $q$ , then the proof is similar to that of Lemma 2.4(a). Suppose now that $h\equiv 0\pmod{q}$ . If $b=a_{0}+a_{1}y\neq 0$ is nilpotent in $B_{k}$ , then so is $b^{2}=a_{0}^{2}+a_{1}^{2}g\in A$ . Therefore $a_{0}^{2}+a_{1}^{2}g=0$ in $A$ . As the latter is integrally closed, $a_{0}/a_{1}\in A$ and $g$ is a square in $A$ , contradiction.

(b) Denote by $\phi:W\to Z:=V_{+}(q)$ the natural projection. If $h$ is divisible by $q$ , then $\phi$ is purely inseparable, so the singularities of $W$ are all cuspidal. On the other hand, $W$ is not smooth as its arithmetic genus is $3$ while its geometric genus is [math]. So $W$ is not semi-stable.

(c) We can suppose $k$ is separably closed. The above cover $W\to V_{+}(q)$ is étale away from $D$ . So $W\setminus\phi^{-1}(D)$ is smooth. The local equation of $W$ at a point lying over $z\in D$ is $y^{2}+tu(t)y+g(t)$ where $t$ is a local parameter of $Z\simeq\mathbb{P}\,^{1}_{k}$ at $z$ , with $u(0)\neq 0$ . After a translation of $y$ by some constant in $\overline{k}$ , the local equation has the shape $y^{2}+tu(t)y+t^{r}f(t)$ with $r\geq 1$ and $f(0)\neq 0$ . If $r=1$ , the point is smooth, otherwise it is a node. ∎

*Remark 2.6**.*

The condition in (c) is not necessary. We can have inseparable non-transverse intersection points with $W$ smooth.

*Remark 2.7**.*

The smoothness condition is less easy to express than in the $\mathrm{char}(k)\neq 2$ case. Keep the notation of the proof of Part (b). Let $w_{0}\in W$ . We know that $w_{0}$ is a smooth point if $w_{0}\notin\phi^{-1}(D)$ . Suppose that $\phi(w_{0})=z_{0}\in D$ . The Jacobian criterion implies that $W$ is smooth at $w_{0}$ if and only if the matrix

[TABLE]

where $y_{0}\in\bar{k}$ is a square root of $g (z_{0})$

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Reduction type of smooth plane quartics

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2010 Mathematics Subject Classification:

1. Introduction and main results

Example 1.1*.*

Proposition 1.2**.**

Definition 1.3**.**

Theorem 1.4** **(See Theorems 2.8 and

Definition 1.5**.**

Definition 1.6**.**

Definition 1.7**.**

Remark 1.8*.*

Theorem 1.9** (See Theorem 3.15).**

Theorem 1.10**.**

2. Hyperelliptic reduction in terms of toggle models

2.1. Some general facts

Fact 2.1*.*

Lemma 2.2**.**

Proof.

Remark 2.3*.*

Lemma 2.4**.**

Proof.

Lemma 2.5**.**

Proof.

Remark 2.6*.*

Remark 2.7*.*

*Example 1.1**.*

Proposition 1.2.

Definition 1.3.

Theorem 1.4 (See Theorems 2.8 and

Definition 1.5.

Definition 1.6.

Definition 1.7.

*Remark 1.8**.*

Theorem 1.9 (See Theorem 3.15).

Theorem 1.10.

*Fact 2.1**.*

Lemma 2.2.

*Remark 2.3**.*

Lemma 2.4.

Lemma 2.5.

*Remark 2.6**.*

*Remark 2.7**.*