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

TL;DR

This paper determines the maximum orders of automorphisms of smooth curves with given bidegree in the product of two projective lines and explores how symmetry relates to the curves' geometric structure.

Contribution

It computes the automorphism orders for smooth curves in a0a0a0a0 and identifies those with the largest automorphism orders, also examining symmetry-structure relationships.

Findings

Calculated automorphism orders for curves with bidegree (a,b) in a0a0a0a0

Identified curves with maximal automorphism orders

Provided conditions for the quotient space to be a0a0a0a0

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 $\mathbb P^1$.