@MISC{Vojta98chapter0, author = {Paul Vojta}, title = {CHAPTER 0 PRELIMINARY MATERIAL}, year = {1998} }

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Abstract

This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic preliminary notation, both mathematical and logistical. Section 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Section 3 gives a few more details on the field of definition of a variety. Section 4 does the same as Section 2 for number theory. The remaining sections of this chapter give slightly longer descriptions of some topics in algebraic geometry that will be needed: Kodaira’s lemma in Section 5, and descent in Section 6. §1. General notation The symbols Z, Q, R, and C stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol N signifies the natural numbers, which in this book start at zero: N = {0, 1, 2, 3,...}. When it is necessary to refer to the positive integers, we use subscripts: Z>0 = {1, 2, 3,...}. Similarly, R≥0 stands for the set of nonnegative real numbers, etc.