Minimal algebraic space filling curves on the product of projective lines

TL;DR

The paper advances the explicit construction of smooth algebraic space filling curves on the product of projective lines over finite fields by showing that there exist many minimal-degree curves of bidegree $(q+1,q+1)$ given by $F=f(Y_0,Y_1)(X_0^qX_1-X_0X_1^q)+g(X_0,X_1)(Y_0^qY_1-Y_0Y_1^q)$ with $f$ lacking $\mathbb{F}_q$-points; it proves a refined smoothness criterion and demonstrates, through substantial computations for small primes and a general existence argument, that smoothness is prevalent, including symmetric examples $C_{f,f}$ for odd $q$; the results substantially extend the known landscape beyond prior non-constructive proofs. The work combines explicit algebraic criteria with computational evidence to show that obstruction to smooth space filling curves is mild in this setting. Overall, the paper provides both constructive families and statistical support for abundant minimal-degree smooth space filling curves in $\mathbb{P}^1 \times \mathbb{P}^1$ over finite fields.

Abstract

We investigate minimal degree smooth algebraic space filling curves on the product of projective lines. We prove that there are plenty of examples in an explicit sense, extending the existence result of Homma and Kim.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Lemma 3.1