Math 145: Undergraduate Algebraic Geometry

Winter 2017

Tuesdays and Thursdays 9-10:20 in 381-U

In this class, you will be introduced to some of the central ideas in algebraic geometry. Because the field is a synthesis of ideas from many different parts of mathematics, it usually requires a lot of background and experience. My intent is to try to aim this class at people with a strong background in algebra and a willingness to develop geometric intuition, but to also have it accessible to those who have taken Math 120 and are willing to work hard and learn new things on the fly.

I want to get across some of the main ideas while doing lots of calculations. As far as possible, I want the class to be able to understand proofs completely, while also seeing enjoyable consequences. This means that the course will have "episodes" of different topics, and I will change plans on the fly as it becomes clear what the audience needs in terms of background.

You needn't be a student in the class in order to participate. Periodic email to the participants will be sent out through canvas. If you would like to be involved, please let me know and I will add you to the mailing list. Course links:

  • canvas (mailing list and weekly check-in)
  • piazza (for discussion)
  • the current version of the notes (in overleaf), livetexed by Nitya Mani, and with individual classes edited by individual students.

    Instructor: Ravi Vakil (vakil@math, office 383-Q, office hours Wednesdays 9:15-11:15 am and Fridays 2:30-3:30 pm).

    Course assistant: Laurent Cote (lcote@math, office 381-L, office hours Wednesdays 3:30-4:15 pm and Thursdays 7-8:15 pm.).

    Prerequisites: Comfort with rings and modules. At the very least, a strong background from Math 120. Background in commutative algebra, number theory, complex analysis (in particular Riemann surfaces), differential geometry, and algebraic topology will help. But I realize that many people in the class will have seen none of these things.)

    References: There will be no textbook for the course, but there are a number of good references. Relevant to this course:

  • Miles Reid's "Undergraduate Algebraic Geometry",
  • Bill Fulton's "Algebraic Curves" (freely and legally available here),
  • Frances Kirwan's "Complex Algebraic Curves".

  • Other good stuff: Brendan Hassett's "Introduction to Algebraic Geometry", Cox-Little-O'Shea's "Ideals Varieties and Algorithms", ... A number of these books (Reid, Kirwan, and Cox-Little-O'Shea) are freely electronically available through the library (to those at Stanford).

    Syllabus: The syllabus will evolve depending on the class, but it will roughly cover the following material, in roughly the following order.
  • linear geometry (degree on)
  • hypersurfaces take 1
  • degree 2: conics, pythagorean triples, quadrics
  • degree 3: elliptic curves
  • algebraic sets: the maps V and I; the Zariski topology; Noetherian rings; irreducible components; Hilbert's Nullstellensatz; morphisms(=maps) of algebraic sets
  • affine algebraic varieties; morphisms of affine algebraic varieties
  • algebraic varieties: definitions; projective varieties; morphisms; products, Haussdorffness
  • rational functions and rational maps
  • images of morphisms; elimination theory; fibers of morphisms
  • calculus (derivatives and differentials), smoothness, dimension theory
  • algebraic curves

    What you should do in this class (including grading)

    You should be active in class, keeping me honest, and asking me questions (no matter how silly you think they are).

    You should be editing and reading the notes (in sharelatex), and for one of the classes you will be responsible for the notes, and making them as useful and readable as possible. This means figuring out things (by asking me, or discussing with others, or reading). Remember, you are not allowed to ever complain again about a mathematics text, until you make your day's notes a work of art.

    You should be testing your understanding by doing problems on the problem set, and discussing with friends, going to office hours, and discussing on piazza.

    You will write something short exploring a related topic (the "term paper").

    The final grade will be:

  • 50% problem sets (including online check-ins)
  • 30% participation (online participation includes editing of notes)
  • 20% one topic written up (likely to be a page's worth, but in the notes or latexed)

    Problem sets

    Problem sets will come out on the weekend, and be due in Laurent Cote's mailbox the next Friday at 4 pm. (He may actually pick them up a little later, but makes no promises.) You are encouraged to discuss the problems with each other (in person, or on piazza) but must credit people (and other sources) for ideas when writing up solutions, and you must write up solutions individually and independently. The lowest homework score will be dropped. One homework can be late, but with a 25 per cent penalty; late sets can be handed in up until the end of week 9 (Friday 4 pm in Laurent's mailbox).

  • Problem set 1 (due Friday January 20) is here. There is also an "online check-in" component on canvas.
  • The revised version of problem set 2 (due Friday January 27) is here. There is also an "online check-in" component on canvas.
  • Problem set 3 (due Friday February 3) is here. There is also an "online check-in" component on canvas.
  • The revised version of problem set 4 (due Friday February 10) is here. There is also an "online check-in" component on canvas.
  • Problem set 5 (due Friday February 17) is here. There is also an "online check-in" component on canvas.
  • Problem set 6 (due Friday February 24) is here. There is also an "online check-in" component on canvas.
  • Problem set 7 (due Friday March 3) is here. There is also an "online check-in" component on canvas.

    Short writing assignment

    You will also write a short mathematical exposition for others in the class, so they can learn about something in more detail. The length should be at least a page, but not much longer. It will be due no earlier than the 9th week, but I would like to see a draft earlier. I hope to get almost everyone set up with a topic by some time in the 6th week of quarter (the week of Feb. 13-17). For background, you can use any sources. You might want to start with the references mentioned here, as well as google and wikipedia. Sample possible topics:

  • Weierstrass P-function
  • rational points on elliptic curves
  • a full glossary for the notes (including links to definitions in the notes, or to other sources)
  • proof of Nullstellensatz
  • rational points on cubic curves: finding lots of them
  • prove enough of Bezout for elliptic curves
  • complex varieties
  • 27 lines on a cubic surface (2 people working together or sequentially?)
  • the Riemann-Hurwitz formula
  • affine schemes (not just reduced ones)
  • intersection multiplicities of curves in the plane (following Fulton)

    Class summaries

  • Class 1, Tues. Jan. 10 (notes editor: Kate Stowell). Welcome. What is algebraic geometry? Geometry, algebra, diophantine equations, complex analysis, topology. Themes and punchlines. Linear algebra as linear geometry.
  • Class 2, Thurs. Jan. 12 (Brett Harvey). A touch of category theory. Linear maps between vector spaces. Projective n-space, and coordinates. Key words: affine space over a field, GL(n), projective n-space over k, projective coordinates, hyperplane at infinity, projective line, projective plane, projective transform = change of projective coordinates, PGL(n+1), the group PGL(2) = linear fractional transformations (a group that is 3-transitive).
  • Class 3, Tues. Jan. 17 (Will McCloskey). Cross ratio. Degree d form, quadratic form, cubic form, homogeneous form. Homogenization, dehomogenization. Hypersurface, quadric hypersurface, cubic hypersurface. Conic curve, cubic curve. Dimension and smoothness of hypersurfaces. Degree 2: diagonizing quadratic forms, rank, full rank, primitive pythagorean triples, conics over the complex numbers are spheres,
  • Class 4, Thurs. Jan. 19 (Lingxiao Li). Smoothness and dimension of hypersurfaces. Tangent lines. Newton's method for power series, and solving diophantine equations modulo 15. Derivatives over arbitrary fields. Cubic curves. The group law on cubic curves.
  • Class 5, Tues. Jan. 24 (Tommy Fan). More on the group law on cubics. Proof of associativity.
  • Class 6, Thurs. Jan. 26 (Ethan Sussman). Pappus' and Pascal's theorem. Topological spaces. The Zariski topology on affine space. The "vanishing set" map from polynomials to subsets of affine space. Properties of the vanishing map V(.). Noetherian rings.
  • Class 7, Tues. Jan. 31 (Chen Lu). The Zariski topology on affine space. It is a topology. Algebraic sets = closed subsets. Polynomial maps are continuous. Hilbert basis theorem. Irreducibility, and irreducible component. Algebraic sets have finitely many irreducible components. The radical of an ideal, and radical ideals. The map I(.).
  • Class 8, Thurs. Feb. 2 (Vineet Gupta). Hilbert's Nullstellensatz: three equivalent statements. Closed subsets correspond to radical ideals.
  • Class 9, Tues. Feb. 7 (Ian Torres). For today, we work over an algebraically closed field. Functions on affine algebraic sets, maps of affine algebraic sets. Affine varieties over an algebraically closed field = (basically, with arrows reversed) finitely generated nilpotent-free k-algebras.
  • Class 10, Thurs. Feb. 9 (Jiren Zhu). Affine varieties over an algebraically closed field.
  • Class 11, Tues. Feb. 14 (Tudor Ciobanu). Affine varieties over a general field. Toward varieties in general: the notion of a sheaf. Examples: the sheaf of differentiable functions, the sheaf of continuous functions. Restriction maps, stalk, germ. Identity axiom gluability axiom.
  • Class 12, Thurs. Feb. 16 (Brian Morris). More on sheaves.
  • Class 13, Tues. Feb. 21 (Leslie Tu).
  • Class 14, Thurs. Feb. 23 (Nikolas Castro).
  • Class 15, Tues. Feb. 28 (Nina Zubrillina).
  • Class 16, Thurs. Mar. 2 (Quinn Greicius).
  • Class 17, Tues. Mar. 7 (Anthony Chen).
  • Class 18, Thurs. Mar. 9.
  • Class 19, Tues. Mar. 14.
  • Class 20, Thurs. Mar. 16.

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