Some geometric-arithmetic aspects of separated variable curves

The paper shows certain geometric-arithmetic aspects with some new observations in consideration of variables separated curves, e.g. relationships with generalized Chebyshev polynomials, Chebyshev pencils, local variant at infinity of Stothers-Mason abc-theorem, Stothers- Langevin pairs, Pell-Abel conics (or polynomial Pell equations), Belyi maps, etc. Several case studies and open problems are discussed.