Prym Varieties and Fourfold Covers

Contents 1. Introduction 2 2. Prym varieties for covers of curves 3 3. Galois covers 8 4. Degree two covers 10 5. Covers of degree three 13 6. Covers of degree four 15 6.1. The cyclic case 15 6.2. The Klein case 17 7. The dihedral case 22 7.1. The bigonal construction 34 8. The alternating case 37 8.1. The trigonal construction for the case A 4 43 9. The symmetric case 44 9.1. The classical case of the trigonal construction 55 9.2. Completion of the principally polarized P (S/R) case 57 9.3. The general case 58 10. Other applications 63 10.1. Examples 63 10.2. Jacobians of curves isogenous to a product of Jacobians 64 10.3. Rigid Jacobians with S 4 actions 66 10.4. One dimensional families of curves with S 4 action 67 10.5. Prym varieties of genus seven double covers of genus three curves, isogeno Appendix 70 References 77