Math 145: Algebraic Geometry

This class will Tu-Th at 12 Noon in 380F.

• Instructor: Daniel Bump (bump at math dot stanford dot edu)
• Office hours (Bump): Tu-Th 10:30-11:55 AM or Wed 12-1. (Other times by appointment)
• Course Assistant: Shirui Liu. Office Hours Th 9-10:20 or Fri 10-11 in 381-F.

Canvas and Gradescope

• Canvas link.
• Gradescope for this class.

Exams

• Midterm Date: Wednesday, October 28 at 7 PM (week 6)
• Final Exam Date: December 11, 12:15-3:15 in 380X

Text

We will use Fulton's book Algebraic Curves. Although the book is out of print, Fulton made a pdf file of a revised version available on his web page. Obtain the book here:

• Algebraic Curves by William Fulton.

Notes

• An important difference between Dummit and Foote and Fulton is that in Fulton, a ring homomorphism $f : R \longrightarrow S$ is supposed to take $1_R$ to $1_S$. Dummit and Foote do not make this assumption. As a special case, if $R$ is a subring of $S$, we could take $f$ to be the inclusion map and call $R$ a subring of $S$. Dummit and Foote do not assume that a subring contains $1_R$. So for Dummit and Foote, an ideal is a subring. Fulton does not make this assumption. Fulton's convention is more usual. For example, Lang's Algebra also assumes that $f (1_R) = 1_S$ and does not call an ideal a subring.
• If $k$ is a field, a $k$-algebra is a ring $R$ containing $k$ in its center. (In Fulton, or commutative algebra in general rings are commutative, the requirement that $k\subseteq Z(R)$ is automatic.) A $k$-algebra homomorphism $R\to S$ is defined to be a ring homomorphism that is the identity on $k$. With this in mind, on page 17 Fulton says "All rings and fields will contain $k$ as a subring. By a homomorphism $\varphi:R\to S$ of such rings we will mean a ring homomorphism such that $\varphi(\lambda)=\lambda$ for $\lambda\in k$." A more standard way of saying the same thing is that he is working in the category of commutative $k$-algebras, and homomorphisms are $k$-algebra homomorphisms.
• The following notation is introduced in (B) on page 13 of Fulton. If $R$ is a ring and $v_1,\cdots,v_n$ are elements of a ring $S$ containing $R$ then $R[v_1,\cdots,v_n]$ is the smallest subring of $S$ containing $R$, $v_1,\cdots,v_n$. If $R=K$ is a field it is correct to call $K[v_1,\cdots,v_n]$ the $K$-algebra generated by $v_1,\cdots,v_n$. The square brackets notation does not imply that $R[v_1,\cdots,v_n]$ is a polynomial ring. This is however consistent with the notation where $R[X_1,\cdots,X_n]$ is the polynomial ring on $X_1,\cdots,X_n$, since that is a special case of the same notation.
• If $R=F$ is a field we also denote by $F(v_1,\cdots,v_n)$ the smallest field generated by $v_1,\cdots,v_n$. The square and round bracket notations are widely used, for example in Lang's Algebra and other literature, though perhaps not Dummit and Foote.
• Fulton uses the term quotient field to mean field of fractions. I recommend not using the term quotient field this way. If $R$ is a ring and $I$ is an ideal, $R/I$ is a quotient ring which might be a field. I think the preferred terminology is field of fractions for the smallest field containing $R$, if $R$ is an integral domain.

Homeworks

Homeworks will usually be due on Wednesdays on Gradescope.

Wednesday, September 30, 2026	Homework 1	1.3, 1.16, 1.18, 1.20, 1.21	TeX file	My solutions
Wednesday, October 7, 2026	Homework 2	1.22, 1.25, 1.28, 1.34, 1.38	TeX file	My solutions
Wednesday, October 14, 2026	Homework 3	1.33(a), 1.44, 1.46, 2.2, 2.12	TeX file	My solutions
Wednesday, October 21, 2026	Homework 4	2.24, 2.34, 2.41, 2.48, 4.1, 3.2	TeX file	My solutions
Thursday, October 29, 2026	Homework 5	2.3, 2.27, 2.28, 3.1, 4.4, 4.5	TeX file	My solutions
Wednesday, November 4, 2026	Homework 6	2.50, 2.51, 3.1, 3.10 (curve only), 3.13	TeX file
Wednesday, November 11, 2026	Homework 7	2.43, 3.5, 3.6, 3.19, 5.1	TeX file