## Math 763 Introduction to Algebraic Geometry A first course Lecturer: Andrei Caldararu

### Fall 2018

About the course: This is an introduction to the basic ideas and methods of algebraic geometry. It will introduce the main objects of study of the subject, affine and projective varieties, and then we will concentrate on curves, divisors on curves, etc. A secret goal will be to get to state and prove Riemann-Roch for curves. We will try to emphasize examples over the theory.

We will use a fairly algebraic approach, hence a good control of commutative ring theory (at a minimum at the level of 741/742) is needed.

Official syllabus:Available here.

Approximate syllabus:

• Affine and projective varieties.
• Morphisms and rational maps.
• Local properties: smoothness and dimension. Tangent space.
• Degree of projective varieties and Bezout's Theorem.
• Divisors.
• Low-dimensional varieties: curves and surfaces. Blow-ups.
• The Riemann-Roch Theorem.

Lectures: There will be two 75 minute lectures per week:

TR 11:00-12:15 in B129 Van Vleck

Text: The textbook for the course is

Algebraic Geometry by Robin Hartshorne, GTM 52 (1977), Springer-Verlag

However, I shall draw inspiration from a few other texts, among them Joe Harris' "Algebraic Geometry" and Shafarevich's books.

You can also consult Milne's notes Milne's notes http://www.jmilne.org/math/CourseNotes/ag.html

Office hours: Tuesdays 1:30-2:30 in Van Vleck 605 and by appointment.

Grading: The term grade will be primarily based on homework. I may decide to assign some final reading and exposition projects. This will be discussed later in the semester.

Homework: Homework will be due a week after it is assigned, during lecture. All claims that you make in your homework MUST BE PROVED in order to receive credit.

Homework assignments:

Homework assignment 1, due September 25: Hartshorne §1.1: 1, 2, 5, 8, 9, 11

Homework assignment 2, due October 2: Hartshorne §1.2: 4, 6, 11, 13, 14, 15, 16

Homework assignment 3, due October 11: Hartshorne §1.3: 1, 2, 5, 6, 7, 15, 21

Homework assignment 4, due October 25: Hartshorne §1.3: 17; §1.4: 1, 3, 5, 6, 7

Homework assignment 5, due November 15: Hartshorne §1.4: 9; §1.5: 1, 3, 4, 7, 9, 10

Also do the exercise assigned in class: blow up the singular point on z^2 - yx^2 +4y^{n+1} = 0 and further singular points on the resulting blown-up surfaces until the resulting surface is no longer singular, and draw the dual graph of the exceptional locus.

Homework assignment 6, due December 6: Hartshorne §1.6: §1.6: 1, 2, 3, 4, 6

Homework assignment 7, due December 13: Solve the following two problems:

a) Assume that an elliptic curve (i.e., a smooth projective curve of genus 1) is embedded by a complete linear system in P^3. Also assume that the resulting curve in P^3 is a complete intersection of two hypersurfaces. Use the Riemann-Roch theorem to find the degrees of these two hypersurfaces.

b) For a curve C of genus g>=2, it is a known fact (that we won't prove) that the divisor 2*K_C gives an embedding into a projective space P^N. Determine N as a function of g.