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Minimal Weierstrass models and regular models of hyperelliptic curves

Qing Liu

Abstract

Let $C$ be a hyperelliptic curve of genus $g\ge 2$ over a discrete valuation field $K$ with perfect residue field. We study the minimal Weierstrass models of $C$. When there is more than one such model, we find interesting properties on the minimal regular model and the canonical model of $C$. For curves of genus $2$, we characterize the existence of the stable reduction in terms of the minimal Weierstrass models. When there is more than one such model, we can compute the Euler factor of $\mathrm{Jac}(C)$ and a volume form of the Néron model of $\mathrm{Jac}(C)$, using two specific minimal Weierstrass models.

Minimal Weierstrass models and regular models of hyperelliptic curves

Abstract

Let be a hyperelliptic curve of genus over a discrete valuation field with perfect residue field. We study the minimal Weierstrass models of . When there is more than one such model, we find interesting properties on the minimal regular model and the canonical model of . For curves of genus , we characterize the existence of the stable reduction in terms of the minimal Weierstrass models. When there is more than one such model, we can compute the Euler factor of and a volume form of the Néron model of , using two specific minimal Weierstrass models.
Paper Structure (20 sections, 57 theorems, 123 equations, 6 figures)

This paper contains 20 sections, 57 theorems, 123 equations, 6 figures.

Table of Contents

  1. Minimal Weierstrass models
  2. Weierstrass models
  3. Chains of Weierstrass models
  4. Minimal Weierstrass models and multiplicities
  5. Extremal and inner minimal Weierstrass models
  6. Number of minimal Weierstrass models
  7. Minimal Weierstrass models and base changes
  8. Algorithm for determining all minimal Weierstrass models
  9. Applications to the regular models and the stable reduction
  10. Blowing-up Weierstrass models and regularity
  11. Extended Tate's algorithm
  12. Minimal regular models
  13. Canonical models
  14. Rigid analytic structure
  15. Stable reduction
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose $g$ is even and that $C$ has more than one minimal Weierstrass models. Then there exists an even $n\ge 2$ and a Weierstrass equation of $C$ such that the minimal Weierstrass models of $C$ are given by the equations (for even $0\le i\le n$): with $x_i=x/\pi^{i}$ and $y_i=y/\pi^{(g+1)i/2}$.

Figures (6)

  • Figure 1: $\Omega$, $\Omega'$ are the respective strict transforms of $Z_k$ and $Z'_k$.
  • Figure 2: The closed fiber of $(\widetilde{Z\vee Z'})_k$.
  • Figure 3: The closed fiber of $W_0\vee \cdots \vee W_n$.
  • Figure 4: Type IV$^*$ for $(W, p_0)$.
  • Figure 5: The closed fiber of the minimal regular model $\mathcal{C}$ of $C$.
  • ...and 1 more figures

Theorems & Definitions (126)

  • Theorem 1: Corollary \ref{['cor:equa_term']}, see also Theorem \ref{['chain-mwm']}
  • Theorem 2: Theorem \ref{['thm:bound_n']}
  • Theorem 3: Theorem \ref{['regular-even']}
  • Corollary 4: Corollary \ref{['cor:gl']}
  • Proposition 5: Proposition \ref{['prop:positive']}
  • Theorem 6: See Theorem \ref{['K1-K2']} and also Corollary \ref{['cor:g2_summary']}
  • Theorem 7: See Theorem \ref{['thm:stableg2']}
  • Theorem 8: Proposition \ref{['prop:volume']} and Theorem \ref{['thm:g2-Euler']}
  • Lemma 1.1
  • proof
  • ...and 116 more