Algebraic Geometry I (18.725)

The introduction of algebraic geometry from Wiki:
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include linescirclesparabolasellipseshyperbolascubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysistopology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

This is the first part of a year long course. We will continue on 2020 Spring.

Classroom: 2-135

Office hour: Chenyang Xu Thr 14:00-16:00, 2-468,

Prerequisite: 18.901 (basic topology) and 18.705 (commutative algebra)

Time: Tu/Th 11-12:30

Email: cyxu [ at ] math.mit.edu

Materials:

Textbook:  [AG] “Algebraic Geometry” by Hartshorne

Main reference: [RSAG] “The rising sea: Foundation of Algebraic Geometry” by Ravi Vakil

Remark: While Hartshorne’s book is classical, it’s also known to be dense. So we will add related materials from [RSAG] and other places to make the class more comprehensive.

Score: 40% Problem set 30% Midterm (in class) 30% Final (take home)

Tentative Schedule:

Sep 5-24: algebraic varieties ([AG] Chapter 1)

Sep 26- Oct 22: Scheme theory ([AG] Chapter 2, Section 1-4)

Nov. 7 Midterm  (Nonstandard office hour for midterm: Nov 4, 4-6pm)

2pm Dec. 3 - 10am Dec. 5  Final (See the link, takehome) (Nonstandard office hour for midterm: Dec. 2, 4-6pm)

Problem sets:

No.1: [AG] Chapter I: 1.1, 1.4, 1.7, 1.8, 1.11(*), 2.3, 2.4, 2.5, 2.8, 2.14, 2.15, 2.17 (a-c)(*). Due on Sep. 17.

No.2: [AG] Chapter I: 3.5, 3.10, 3.14, 3.15, 3.17, 4.2, 4.3, 4.4, 4.6, 4.10, 5.1, 5.2, 6.6. Due on Oct. 1.

No.3: [AG] Chapter II: 1.2, 1.16, 1.18, 1.21, 2.3, 2.11, 2.17, 2.18, 3.10, 3.11, 3.13, 3.18, 3.19, 3.22(*). Due on Oct. 17.

No.4: [AG] Chapter II: 4.2, 4.3. 4.4, 4.5, 4.8, 4.9, 4.11, 5.1, 5.3, 5.9, 5.12, 5.14, 5.18. Due on Oct. 29.

No.5: [AG] Chapter II: 5.10, 6.1, 6.3, 6.5, 6.9, 6.10.  Due on Nov. 21.

No.6: [AG] Chapter II: 7.1, 7.2, 7.3, 7.5, 7.10, 7.11. Due on Dec. 3.

(*) means extra points