Algebraic geometry

Lectures: 2 sessions / week, 1.5 hours / session

There are no prerequisite courses, but 18.705 Commutative Algebra must be taken concurrently. In addition, students should understand the basic notions of commutative algebra, such as localization, Noetherian property and prime ideals. Familiarity with the following topics is helpful though not strictly necessary: Basic notions of category theory: Yoneda Lemma, (co)limits, (co)products; calculus on manifolds, including vector fields, differential forms and de Rham cohomology; beginning graduate topology: (co)homology, fundamental groups, topological vector bundles; complex analysis: Compact Riemann surfaces.

This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.

The theory will be presented mostly following:

Kempf, G. Algebraic Varieties. Cambridge University Press, 1993. ISBN: 9780521426138 [Preview with Google Books], complemented with examples from other sources, especially Shafarevich, Igor R. Basic Algebraic Geometry 1: Varieties in Projective Space. Springer, 2013. ISBN: 9783642379550 and Hartshorne, Robin. Algebraic Geometry. Springer, 1997. ISBN: 9780387902449. [Preview with Google Books]

Time permitting we will introduce schemes following:

Hartshorne, Robin. Algebraic Geometry. Springer, 1997. ISBN: 9780387902449 [Preview with Google Books] and / or cover some intersection theory following Shafarevich, Igor R. Chapter 4 in Basic Algebraic Geometry 1: Varieties in Projective Space. Springer, 2013. ISBN: 9783642379550.

Homework will be assigned weekly. There are 11 problem sets. The last problem set, Problem Set 11, is for you to solve on your own – it will not be graded or collected. Collaborations are allowed but you must write up your solution yourself. Acknowledge your collaborators and sources (other than textbook) consulted on your paper.

Final grade is based on homeworks, 10% each, no final exam will be given. You have the option to write an expository paper for extra credit.

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