On the Prym map of degree 4 cyclic covers of genus 2 curves

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus 2 curves. We show that its non-empty fibers, apart from two exceptional fibers, are isomorphic to the intersection of an elliptic normal curve in P³ with an affine space A³ ⊂ P³. Moreover, we obtain a new description of the space of degree 4 étale cyclic covers of genus 2 curves R₂⁴ and find equations of elliptic and hyperelliptic curves arising from such covers.