Playlist - Algebraic geometry (Borcherds)

The first lecture describes a birational map from a circle to a line.

This lecture discusses two examples of cubic curves: a nodal cubic, and an elliptic curve.

This lecture gives more examples and applications of algebraic geometry, including Bezout's theorem, Pauppus's theorem, and Pascal's theorem.

This lecture continues with more examples from algebraic geometry, in particular Kakeya sets over finite fields and the 27 lines on a cubic surface.

This lecture covers the definition of affine space and its Zariski topology.

This lecture covers Noetherian rings, Noetherian spaces, and irreducible sets.

This lecture describes the weak Nullstellensatz, giving the maximal ideals of polynomial rings over algebraically closed fields.

This lecture covers the proof of the strong Nullstellensatz using the Rabinowitsch trick, and gives some examples.

This lecture describes the Lasker-Noether theorem expressing an ideal as an intersection of primary ideals.

This lecture covers the proof of the Lasker-Noether theorem.

This lecture describes taking a quotient of an algebraic set by a group.

This lecture covers the proof of Hilbert's finiteness theorem for rings of invariants. (This is not the same as Hilbert's finiteness theorem for ideals, though the two theorems are related.)

This lecture covers three examples of quotients by groups: a cyclic quotient singularity, the parameter space of cyclohexane, and the moduli space of elliptic curves.

This lecture covers the dimension of a topological space, algebraic set, or ring.

This lecture introduces projective space and describes the synthetic and analytic approaches to projective geometry.

This lecture covers Desargues's theorem and duality of projective space.

This lecture covers the relation between affine and projective varieties, with some examples such as a cubic curve and the twisted cubic.

This lecture covers products of affine and projective varieties and the Segre embedding.

This lecture covers two examples of projective varieties: the Veronese surface in 5-dimensional projective space, and the variety of all lines in 3-dimensional space.

This lecture is about Grassmannians and some of their applications.

This lecture covers projective space bundles, with Hirzebruch surfaces and scrolls as examples. It also includes a brief discussion of abstract varieties.

This lecture describes toric varieties as examples of abstract varieties. For more about these see the book "Introduction to toric varieties" by Fulton.

This lecture gives a quick review of category theory as background for the definition of morphisms of algebraic varieties.

This lecture covers regular functions on affine and quasiprojective varieties.

This lecture covers the definition of a morphism of varieties and compares algebraic varieties with other types of locally ringed spaces.

This lecture covers the relation between morphisms of affine algebraic sets and homomorphisms of commutative rings. As examples it describes some homomorphisms of commutative rings coming from the products of affine algebraic groups.

This lecture describes two examples: the twisted cubic is isomorphic to a projective line, and the affine plane without the origin is not isomorphic to any affine algebraic set.

This lecture describes how to use the Segre embedding to show that the categorical product of two projective varieties exists and is projective.

This lecture describes the automorphisms of affine and projective space, and gives a brief discussion of the Jacobian conjecture.

This lecture covers the Ax-Grothendieck theorem, which states that an injective regular map between varieties is surjective. The proof uses a strange technique: first prove the result in characteristic greater than 0, then prove it in characteristic 0 using the fact that the first-order theory of algebraically closed fields of characteristic 0 is complete.

This lecture covers the definition of rational functions and rational maps, and gives an example of a cubic curve that is not birational to the affine line.

This lecture covers the relation between elliptic functions and cubic curves, and uses this to show that cubic curves are not rational.

This lecture gives two rather informal and incomplete arguments for why non-singular cubic surfaces are rational.

This lecture covers blowing up a point of affine space, and gives some examples of using this to resolve singularities of plane curves.

This lecture continues the discussion of blowing up in the previous video, with examples, of blowing up the real affine plane, blowing up an ideal, and regularizing a rational map from ℙ¹ × ℙ¹ to ℙ².

This lecture describes an example of a birational map called the Atiyah flop.

This lecture defines singular points and tangents spaces, and shows that the set of non-singular points of a variety is open and dense.

This lecture covers the Zariski tangent space, and describes some other ways of viewing tangent spaces.

This lecture discusses the Du Val singularites, and sketches how to desingularize the E₈ Du Val singularity.

This lecture gives some examples of resoutions of singularities, and describes an application of resolution to a problem about analytic continuation of integrals.

This lecture reviews completions of rings and Hensel's lemma, and gives an example of two analytically isomorphic singularities.

This lecture introduces elimination theory and reviews resultants.

This lecture discusses proper maps and shows that projective varieties are complete.

This lecture gives an informal survey of complex curves of small genus.

This lecture discusses Hurwitz curves and sketches a proof of Hurwitz's bound for the symmetry group of a complex curve.

This lecture gives examples of complex curves of genus 2 and 3 with the largest possible symmetry groups.

This lecture sketches a proof that singularities of plane curves in characteristic 0 can be resolved by repeated blowups, using a method essentially due to Issac Newton.

This lecture describes how to use Newton's rotating ruler to expand algebraic functions as power series. One application is that the field of complex Puiseux series is algebraically closed.

This lecture gives a review of the Hilbert polynomial of a graded module over a graded ring, and classifies integer-valued polynomials.

This lecture defines the degree of a projective variety and gives a few examples.

This lecture is about Bezout's theorem and its variations, which say that under some conditions the degree of an intersection of algebraic sets is the product of their degrees.