Rational maps from products of curves to surfaces with $p_g = q = 0$

Abstract: We study dominant rational maps from a product of two curves to surfaces with $p_{g} = q = 0$. Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.