Lecture class: Algebraic Geometry 2 (Summer 2026)

This is the continuation of the lecture course Algebraic Geometry held in the winter term 2025/26.

We will continue to study the language of schemes. A key question that motivates much of what we will do (there are also plenty of other motivations) is the following: We want to understand better which schemes can be embedded as closed subschemes in projective space $\mathbb P^n_k$ (over a field $k$, or maybe over a more general base). If a scheme $X$ can be embedded into projective space, we would like to understand in which different ways it can be embedded into projective space. These are at the same time very classical questions and questions that are of great importance in the study of schemes, and hence also, in certain forms, the subject of current research.

Some of the topics we will discuss

Proper schemes and morphisms

Similarly as separatedness is the algebro-geometric version of the Hausdorff property, properness is the algebro-geometric version of compactness; we will see that – as expected, cf. the Riemann sphere – projective space and all of its closed subschemes are proper. (The converse is, interestingly, not true.))

Divisors and line bundles

Assume we have a closed subscheme $X \subseteq \mathbb P^n_k$. We can intersect $X$ with the “coordinate hyperplanes” $V_+(X_i)$ to obtain subschemes of $X$ of a very special form, so-called divisors. We will discuss several perspectives on the notion of divisor. It is an interesting question whether, given a divisor, it arises in the way described above.

$\mathscr O_X$-Modules

Similarly to the notion of module over a ring, there is a notion of sheaves of modules over a sheaf of rings which we can apply in particular to the structure sheaf of a scheme or of any ringed space. Such $\mathscr O_X$-modules contain a lot of interesting information about the underlying space $X$. There is a close connection to the notion of divisor which we will discuss in the course.

Cohomology of sheaves

We know that a surjective sheaf morphism might not induce surjections on the sets of sections over a fixed open of the underlying space (for simplicity take the whole space $X$). So the functor $\Gamma(X, -)$ taking global sections (on the category of sheaves of abelian groups say, where we have a notion of exact sequences) is not exact. Studying in which way exactness fails leads, by the general formalism of “derived functors”, to the notion of cohomology groups $H^i(X, \mathscr F)$ on $X$ with coefficients in a sheaf $\mathscr F$ of abelian groups. (Often $\mathscr F$ will be a $\mathscr O_X$-module). This is an extremely powerful tool to “algebraize” geometric information. After this “controled loss of information” often things are easier to work with.

The Theorem of Riemann-Roch

This famous theorem is a specific example where cohomology of $\mathscr O_X$-modules attached to divisors on a smooth projective curve can be used very profitably in order to study the geometry of the curve. Among many other things it yields a very clean description of the group structure on an elliptic curve, giving us the associativity (which we had to leave open in Part 1) basically for free.

Date/time: Tue, 2-4pm (S-U-3.02), Wed, 10am-12pm (S-U-3.01). First lecture on Tuesday, April 14.

Notes on the lecture course: … will come shortly. At html you find the lecture notes for a similar class which I taught in the summer 2023.

References to the literature: See Algebraic Geometry

Exercise group

Date/time: to be discussed during the first lecture. The exercise group will start in the week of April 13.