A Scrapbook of Complex Curve Theory
This is very nice book by C.H. Clemens and would make an indispensable tool for anyone willing to enter the realm of algebraic geometry. Although not written in the tight mathematical form Definition-Theorem-Proof, it covers and explains motivation for the introduction of algebraic language to problems in geometry using many suitable examples. The first chapter recalls all notions of classical geometry related to conic curves - projective space, linear system of conics, cross ratio, hyperbolic space and rational points on quadrics. The second chapter describes cubic (i.e., elliptic) curves. A unifying picture emerges by comparison of elliptic curves over complex numbers (i.e., the variation of Hodge structure) and elliptic curves over finite fields (i.e., generating function for the number of rational points over successive extensions of finite fields). The third and fourth chapters offer an introduction to elliptic, modular and theta function theory, moduli spaces and basic facts of cohomology theory - Jacobians, duality in sheaf cohomology, Abel-Jacobi map, etc. The last two chapters contain topics from geometry of higher genus Riemannian surfaces, e.g. Prym varieties, Schottky groups and hyperelliptic curves of higher genus.