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Aims Theaimofthisbookistoprovideaguidetoarichandfascinatings- ject: algebraic curves, and how they vary in families. The revolution that the ?eld of algebraic geometry has undergone with the introd- tion of schemes, together with new ideas, techniques and viewpoints introduced by Mumford and others, have made it possible for us to understandthebehaviorofcurvesinwaysthatsimplywerenotpos- ble a half-century ago. This in turn has led, over the last few decades, to a burst of activity in the area, resolving long-standing problems and generating new and unforeseen results and questions. We hope to acquaint you both with these results and with the ideas that have made them possible. The book isn’t intended to be a de?nitive reference: the subject is developing too rapidly for that to be a feasible goal, even if we had the expertise necessary for the task. Our preference has been to - cus on examples and applications rather than on foundations. When discussing techniques we’ve chosen to sacri?ce proofs of some, even basic,results—particularlywherewecanprovideagoodreference— inordertoshowhowthemethodsareusedtostudymoduliofcurves. Likewise, we often prove results in special cases which we feel bring out the important ideas with a minimum of technical complication.

512.77 --- 511.33 --- Algebraic curves. Algebraic surfaces. Three-dimensional algebraic varieties --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Curves, Algebraic. --- Moduli theory. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 512.77 Algebraic curves. Algebraic surfaces. Three-dimensional algebraic varieties --- Moduli theory --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Curves, Algebraic --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebraic curves --- Algebraic varieties --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Geometry, algebraic. --- Algebraic geometry --- Geometry