Instructor: Nathan Chen Office hours: Mon/Wed 4:30-5:30pm in Math 609

• Lectures on Mon/Wed 2:40-3:55pm in Math 307

Email: nathanchen@math.columbia.edu

TA: Jacob Daum TA office hours: Thursday 12-2pm.

Prerequisites: This course will be quite difficult. Some prerequisites include complex analysis (complex function theory and contour integrals), abstract algebra 1 and 2, and point-set topology. Familiarity with smooth manifold theory will be helpful.

Topics include algebraic curves/Riemann surfaces, holomorphic and meromorphic differentials, singularities of plane curves, normalization, divisors and intersection numbers, branched covers, ramification and Riemann-Hurwitz, Riemann-Roch, divisors/line bundles, embeddings of low genus curves, Abel's theorem, Jacobi-inversion theorem, Clifford's theorem.

Textbook: Introduction to Algebraic Curves by Phillip Griffiths.

The syllabus is here.

First class: January 17, 2024. Last class: April 29, 2024. Notes.

Lecture 1 (1/17): Homogeneous polynomials, projective plane curves. Section 1.1.

Lecture 2 (1/22): Riemann surfaces, examples, Euler characteristic, holomorphic functions. Sections 1.2, 1.3.

Lecture 3 (1/24): Meromorphic/holomorphic functions and differentials, zeros/poles, contour integrals and Stokes' theorem. Section 1.4.

Lecture 4 (1/29): Residue theorem, zeros/poles of differentials. Section 1.5.

Lecture 5 (1/31): Index, winding number, Poincaré-Hopf, complex manifolds. Sections 1.6, 1.7.

Lecture 6 (2/5): Affine/projective algebraic varieties, smooth points. Sections 1.8, 1.9.

Lecture 7 (2/7): Tangent space, singular points, multiplicity, plane singularities. Section 2.1.

Lecture 8 (2/12): Connectedness of irreducible plane curves. Section 2.2.

Lecture 9 (2/14): Local structure of plane curves, normalization, Weierstrauss polynomial. Sections 2.3-2.6.

Lecture 10 (2/19): Intersection multiplicities and Bezout's theorem. Section 2.7.

Lecture 11 (2/21): Ramification divisors and Riemann-Hurwitz. Section 2.8.

Lecture 12 (2/26): Genus formula for plane curve with nodes. Section 2.9.

Lecture 13 (2/28): Linear equivalence of divisors, vector space of sections of a divisor.

Lecture 14 (3/4): Canonical divisor, holomorphic 1-forms.

Lecture 15 (3/6): Midterm Exam

SPRING BREAK (3/11 - 3/15).

Lecture 16 (3/18): Riemann-Roch inequalities

Lecture 16 (3/20): Riemann-Roch inequalities

Lecture 17 (3/25): Riemann-Roch, constructing holomorphic differentials

Lecture 17 (3/27): Genus 1 curves, complete linear series

Lecture 18 (4/1): Complete linear series, embedding, canonical linear series

Lecture 19 (4/3): Revisiting hyperelliptic curves, genus 2 and genus 3 curves.

Lecture 20 (4/8): Genus 4 curves, hypersurfaces containing curves in projective space

Here is the pdf for the homework. This will be updated every week and is due at 11:59pm ET.

• Problem set 1 posted. Due Wednesday, January 31.

• Problem set 2 posted. Due Thursday, February 8.

• Problem set 3 posted. Due Thursday, February 15.

• Problem set 4 posted. Due Thursday, February 22.

• Problem set 5 posted. Due Saturday, March 2.

• Problem set 6 posted. Due Thursday, March 21.

• Problem set 7 posted. Due Thursday, April 4.

• Problem set 8 posted. Due Thursday, April 11.

• Problem set 9 posted. Due Tuesday, April 23.