# Algebraic Geometry

## Part II

Welcome to the course webpage for algebraic geometry, taught in the 2020 Lent term at Cambridge. This course is intended for Part II students in pure mathematics. Instructor: Dhruv Ranganathan (E 1.01). Meeting time: MWF 12-1PM in MR4

Overview: Algebraic geometry is an old and amazingly interdisciplinary and active subject, borrowing ideas from topology, differential geometry, number theory, and analysis. In this course the goal is to become acquainted with the basics, affine and projective varieties, and algebraic curves and some of the fundamental theory that governs their geometry.

Resources: We will not follow any textbook word-for-word, but excellent resources are the wonderful-and/or-irritating Undergraduate Algebraic Geometry by Reid and the first chapter in Hartshorne's Algebraic Geometry. I also like Algebraic Curves by Fulton which is free.

Lecture Notes: Complete lecture notes can be found here. I should warn you that (1) I often make game-time decisions about how I want to explain things, what examples I want to do, ordering of material, etc. I do this even after I've written my notes already. As a result, these notes are not exactly what I said. (2) The notes are really written for myself. So I sometimes write things like "definition of isomorphism" without actually writing it in my notes, just as a reminder to define it. So these are rather loose. Also (3), I didn't give the first 4 lectures, my notes will not match up with reality for those. But hopefully it's still helpful for you.

Lecture schedule:

• January 17: Lecture cancelled due to Dhruv being unwell.
• January 20: Mark lectured.
• January 22: Mark lectured.
• January 24: Lecture cancelled due to Dhruv being unwell.
• January 27: Mark lectured.
• January 29: Mark lectured.

Until this point, the material covered included definitions of affine varieties, the relationship between ideals and varieties, coordinate rings, irreducibility and irreducible decompositions, statement of Hilbert's Nullstellensatz.

• January 31: Projective space, basic motivations, and homogeneous polynomials.
• February 3: More on projective space, homogeneous ideals, projective varieties and their function fields.
• February 5: Examples of projective varieties, plane curves and intersections, the twisted cubic.
• February 7: The surface P1xP1. Homogenization for polynomials and ideals, computing projective closures.
• February 10: Morphisms and rational maps. Birationality. Examples of morphisms.
• February 12: Products of quasi-projective varieties. Blowups. Smoothness and tangent spaces for hypersurfaces.
• February 14: Tangent spaces, smoothness, and dimension. Differentials and birational invariance and dimension.
• February 17: Transcendence degree and birationality to hypersurfaces. A little more dimension theory.
• February 19: Algebraic curves, local rings, smoothness, local parameters.
• February 21: Orders of vanishing, morphisms from algebraic curves, degree and ramification.
• February 24: Organize the function theory! Divisors on algebraic curves, hyperplane sections.
• February 26: More on divisors and differentials on algebraic curves I.
• February 28: Differentials on algebraic curves -- formalism . The canonical class and (finally) the genus!
• March 2: The canonical divisor for a plane curve. The Riemann-Roch theorem and its consequences.
• March 4: Group law on the elliptic curve, the group law on plane cubics, the Abel-Jacobi map. Aside on enumerative geometry
• March 6: The Riemann-Hurwitz formula for covers. Applications: Lüroth's theorem, FLT for polynomials.
• March 9: Canonical projective morphisms, ampleness and very ampleness, criteria for embeddings.
• March 11: A non-hyperelliptic curve is embedded by its canonical class. High degree embeddings. Also see this.
• March 13 - Non-Examinable: One structure (schemes), one space (the moduli space of curves), and one formula (χ(Mg) = ζ(1-2g)/(2-2g)).

Divisor Calculations: With divisor theory calculations, it can really help to see a worked example. Here are a couple that I dug up that are on the internet. (1) Calculating zeroes and poles. (2) Another basic example of a divisor on a curve. (3) Divisors associated to hyperplane sections.

Examples sheets: There were four example sheets and one supplemental sheet. They appear here.

Disclaimer: Algebraic geometry is a beautiful subject known for its technicality, but the reason for that technicality is that the range of geometric phenomena that appear in the subject is extremely large. Consequently, if you're serious about the subject, you should spend this course trying to see as much of this range as you possibly can -- you should explore and store examples. We'll see a very large number in lectures too of course. The examples in Harris's Algebraic Geometry: A First Course are wonderful and crucial, and I recommend learning a few examples a week just to build your geometric database.

Check Engine I (February 24): At this point in the course, including Example Sheet III, here are what I consider to be the key "non-vocabulary" aspects of what we've done: 1. There is a unique conic through 5 points in P2. 2. An elliptic curve has no non-constant rational maps from A1. 3. The Segre and Veronese varieties, including the quadric surface and the twisted cubic. 4. There are two families of lines, or "rulings", on a smooth quadric surface. 5. There are finitely many lines on smooth cubic surfaces (we've only seen shadows of this result at this point). 6. A general surface in P3 of degree 4 or larger has no lines on it (Example Sheet III). 7. The nodal cubic and cuspidal cubic are the first examples of singular irreducible curves. They are both birational to P1. 8. There exist varieties that are neither affine nor projective. 9. The image of a variety under a morphism need not be a variety. 10. Every variety is birational to a hypersurface.

Check Engine II: At the end of the section on curve theory, the crux of what we've studied is the following: 1. Maps from curves to projective space are essentially equivalent to vector spaces L(D) of rational functions with poles bounded by D. 2. There are two easy way to produce divisors: the divisor associated to a rational function; the divisor associated to an intersection with a hyperplane (always taken with multiplicity). 3. The vector space of meromorphic differentials is a theory developed in parallel with the field of meromorphic (i.e. rational) functions. Its role is to give us a new source of divisors. 4. Riemann-Roch connects k(X) to ΩX -- a non-zero differential in ΩX gives us a canonical divisor and Riemann-Roch is a "duality" statement relating L(D) and L(K-D). 5. Consequences of Riemann-Roch are plentiful: the degree of the canonical, the degree-genus formula, the Riemann-Hurwitz theorem. 6. The vector space L(K) of holomorphic (i.e. regular) differentials gives a map from any curve to projective space of dimension g-1. This map is an embedding if and only if the curve is non-hyperelliptic. 7. High degree (larger than 2g) divisors always give embeddings