Prerequisite: Some algebra, some topology, some geometry but lots of enthusiasm

Algebraic Geometry R. Hartshorne, Graduate Texts in Mathematics (GTM) 52, Springer Verlag 1977.

Principles of Algebraic Geometry P. Griffiths and J. Harris, Wiley Interscience 1978.

Algebraic Geometry: A first course. J. Harris, GTM 133, Springer 1992.

The Geometry of Schemes. J. Harris and D. Eisenbud, GTM 197, Springer 2000.

Course Description:

Classical Algebraic Curves, Riemann-Roch theorem, Periods, Abel-Jacobi theorem, Jacobians. (Since this is biased towards complex analysis, we shall use modular curves as examples.)

Modern (1960's) Affine varieties, Schemes, Sheaves, Cohomology, Serre duality. (bias towards geometry aka complex manifolds). Projective space, divisors, ampleness, linear systems, GAGA.

Arithmetic Number theory as a special case of algebraic geometry. Curves over finite fields, Riemann hypothesis, Weil conjectures. The curve Spec Z and its branched covers; Galois groups as fundamental groups.

Moduli problems (1970's). Moduli spaces of curves, representability of functors, vector bundles on curves.\ms The hope is that (by the end, if not earlier, of the course,) you will see that schemes are actually very geometric objects, that number theory is actually geometry in disguise. Since there is no one textbook that covers all these topics, I have decided to put several relevant books on reserve at the library (the decision has not yet been - and alas perhaps might never be! - put into action); the above listed recommended books will suffice for most purposes. There are also lots of course notes/exercises/introductions online which I urge you to check out on your own.

Format of the Course: Many exercises will be given out in class. A reasonable number (approximately 15 to 20) of them need to handed in during the semester. There will also be a seminar report component that we will discuss in detail in class.

Grading: Based on class participation and homeworks (+ seminar report).

The problem at issue can be stated as follows:

for configurations of the most general type which simultaneously pertain to arithmetic and to the theory of functions - that is, which depend on any given algebraic numbers and algebraic functions of whatsoever parameters -, try to attain, in the most general case, the same degree of achievement and completion as characterizes in some measure or other - the simplest results brought to light.

There opens up here before us a vast expanse in the domain of pure theory. The universal regularity and harmony which rules here confers upon it a supreme degree of fascination and beauty. It must be observed, besides, that this domain is still far from practical applications, though in future the situation may change.
 
  "Ich beschra"nke mich darauf, noch einmal das allgemeinste Problem welches hier vorliegt, im Anschluss an Kroneckers Festschrift von 1881 zu charakterisieren. Es handelt sich nicht nur um die reinen Zahlko"rper oder Ko"rper, die von einem Parameter Z abha"ngen, oder um die Analogisierung dieser Ko"rper, sondern es handelt sich schliesslich darum, fu"r Gebilde, die gleichzeitig arithmetisch und funktiontheoretisch sind, also von gegebenen algebraischen Zahlen und gegebenen algebraischen Funktionen irgendwelcher Parameter algebraisch abha"ngen, das selbe zu liesten, was mehr oder weniger vollsta"nding in den einfachsten Fa"llen gelungen ist. Es bietet sich da ein ungeheurer Ausblick auf ein rein theoretisches Gebeit, welches durch seine allgemeinen Gesetzma"ssigkeiten den gro"ssten a"stetischen Reiz ausu"bt, aber, wie wir nicht unterlassen du"rfen hier zu bemerken, alen praktischen Anwendungen zuna"chst ganz fern liegt."

Felix Klein in Die Entwicklung der Mathematik im 19tne Jahrhundert, Chelsea Publishing Co. 1956. Chapter 7, page 334.
 

The most satisfactory realization of this dream is Grothendieck's theory of schemes.
A postscript version of this page PS

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