Irreducible plane curves with the Albanese dimension 2

Abstract

Let B B be a plane curve given by an equation F ( X 0 , X 1 , X 2 ) = 0 F(X_{0}, X_{1}, X_{2}) = 0 , and let B a B_{a} be the affine plane curve given by f ( x , y ) = F ( 1 , x , y ) = 0 f(x, y) = F(1,x, y) = 0 . Let S n S_{n} denote a cyclic covering of P 2 {\mathbf {P}}^{2} determined by z n = f ( x , y ) z^{n} = f(x, y) . The number max n ∈ N ( dim ⁡ ℑ ( S n → Alb ⁡ ( S n ) ) ) \max _{ n \in {\mathbf {N}}} \left ( \operatorname {dim} \Im (S_{n} \to \operatorname {Alb} (S_{n})) \right ) is called the Albanese dimension of B a B_{a} . In this article, we shall give examples of B a B_{a} with the Albanese dimension 2.