Large orders of automorphisms of smooth curves in $\mathbb P^1\times \mathbb P^1$

TL;DR

This paper analyzes automorphisms of smooth curves of bidegree $(a,b)$ on the surface ${\mathbb P}^1\times{\mathbb P}^1$ for $a,b\ge 3$. It embeds the curve automorphisms into the ambient group ${\rm Aut}({\mathbb P}^1\times{\mathbb P}^1)$, derives sharp divisibility bounds for the possible orders ${\rm ord}(f)$, and gives sufficient conditions under which the quotient by an automorphism is ${\mathbb P}^1$. The authors identify families of curves achieving large automorphism orders and, in several maximal-order cases, present explicit plane-model equations parametrized by a moduli parameter. The results illuminate the interplay between symmetry and geometry for curves on a ruled surface, extending understanding beyond plane curves by furnishing concrete classifications and constructions on ${\mathbb P}^1\times{\mathbb P}^1$.

Abstract

For $a,b\geq 3$, we calculate the orders of automorphisms of smooth curves with bidegree $(a,b)$ in the product $\pp$ of the projective line $\mathbb P^1$. We identify smooth curves in $\pp$ which have automorphisms with the largest orders. In addition, we study the relationship between symmetry and geometric structure of curves. We provide a sufficient condition for the quotient space by an automorphism to be $\mathbb P^1$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Lemma 2.6
• proof