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.

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