Topics in Geometry 2, Spring 2006

For a general description of this course, see the description in the `studiegids', and for the schedule (time and location) see schedules in the `studiegids',

This page gives a progress report on what has been done (course and problem session), and gives a planning of what is to follow. Homework exercises that are to be graded should be given to Johan one week after the problem session in which they have been assigned.

In the beginning of the course we are still working from the book of Miles Reid: Undergraduate Algebraic Geometry.

1. Febuary 6.
In the lecture, a systematic summary of sections 1 to 5 was given. Varieties (affine, projective, quasi-affine and quasi-projective) were defined as in the book. Morphisms were defined as continuous maps that transform regular functions into regular functions. To be explicit, if V and W are varieties, then a morphism from V to W is a map f from V to W that is continuous for the Zariski topologies on V and W, such that for every open subset U of W and for any regular function g on U the function gf is regular on the inverse image of U under f. Then rational maps were defined and it was shown that our definition of morphism is equivalent to the one in the book.

Exercises: 5.2 (graded), 5.4 and 5.8 (char. not .2)

2. February 13.
Tangent spaces and singular and non-singular points. Sections 6.1 to 6.8 were treated, except 6.7 and the very end of 6.8. The dimension of V is the minimal dimension of its tangent spaces.

Exercises: 6.3 (graded), 6.4 and 6.5. It is not required to prove irreducibility of all closed sets in these exercises.

3. February 20.
Section 6.8 was finished, as well as a stronger statement than Cor. 6.7: the tangent space of V at P depends only on the local ring of V at P. Section 6.7, exercise 6.1 and Thm. 6.10, giving the relation between dimension of V and transcendence degree of the function field k(V).

Exercises: see here (they are not from the book).

4. February 27.
Section 6.11 was finished, with some more detail than in the book on the criterion for nonsingularity in homogeneous coordinates. Then 6.12 was treated, also with the global blow-up of A2 in P1xA2. Then the statement of the 27 lines on a smooth cubic surface in P3 was given, and some examples of lines the Fermat cubic surface.

Exercises: 6.6 and 7.6 (graded), where in 7.6 one should also show that the automorphism group of S permutes the 27 lines transitively, and determine the number of other lines that a given line intersects.

5. March 6.
We started with Liu's book. Chapter 1 on commutative algebra was skipped (we will treat parts of it when necessary). Section 2.1.1, about the spectrum of a ring A as a topological space, was treated completely.

Exercises: (now from Liu's book) 1.1, 1.2, 1.5 (graded), and 1.8.

6. March 13.
Section 2.1.2 has been treated, rather fast because we had already seen such things in Reid's book. We then treated Section 2.2.1 about sheaves up to Corollary 2.13.

Exercises: 1.6, 1.7 (graded; just describe the prime ideals), 2.2, 2.7, 2.8.

7. March 20.
We finished section 2.1.2 but skipped the notion of sheaf associated to a presheaf, and (hence) also the notion of inverse image of a sheaf under a continuous map. An interesting example (the only one, in fact) of a 3-point space has been given. Section 2.2.2 has been treated up to Example 2.21.

Exercises: 2.4, 2.5, 2.9 and 2.14 (graded!).

8. April 3.
Definition 2.22 will be given. Then we start with section 2.3. We skip Def. 3.11 and Prop. 3.12

Exercises: Prove the assertions in Example 3.6, and show that the same happens for every closed point in Spec(Z[T]) (graded!), prove the assertions in Examples 3.15 and 3.16. And read all necessary material until Example 3.16.

9. April 10.
We start with section 2.3.2. We skip 3.19 and 3.20, but we do treat 3.21--3.24. Then we skip until Lemma 3.33, which we treat, as well as Example 3.34 (construction of projective space over a ring).

Exercises: 3.10 (graded!), 3.19, assuming Proposition 3.25, show that the affine plane A2k over a field k, minus the origin, is not an affine scheme.

10. April 20 (Thursday, and not Monday as usual!).
Room: 312. Time: 13:45-17:30.

Example 3.3.8 (affine line with double origin) was given. The next goal is the dictionary between function fields of transcendence degree 1 over a field k and projective integral normal curves over k. Def. 3.47 (affine varieties/k, algebraic variety/k, projective variety/k and morphisms) has been given. The decomposition of an algebraic variety/k in its irreducible components has been proved, and the fact that each irreducible component has a unique generic point (section 2.4.2). Reduced and integral schemes have been discussed (Def. 4.1 and Prop. 4.2 (a) and (b), Def. 4.16 and Prop. 4.17).

No exercises this week.

11. April 24.
We start with Prop. 4.18. Then we start the construction of a projective integral normal curve associated to a function field over a field k. Then we make a jump to Section 4 of Chapter 3. Definition 1.1, Lemma 1.4, the first statement of Proposition 1.5, and Example 1.6 are treated. Then the construction and some properties of the normalisation of an integral scheme X in an algebraic extension K-->L of its function field are given: Definition 1.24, Proposition 1.25, Lemma 1.26 and Propositio 1.27. The example of hyperelliptic curves over fields k of characteristic different from 2 is given. More precisely: we construct the normalisation of P1 with function field k(x) in the extension given by y2-f, for f in k[x] of degree 2d>0 such that gcd(f,f')=1.


  1. describe the normalisation as above but now for f of degree 2d+1 and describe in all cases the fibre over the point at infinity on P1 (graded);
  2. Let p be a prime number not equal to 2, k a field of characteristic p, and a an element of k that is not a pth power. Show that the affine algebraic curve over k given by the equation y2 = xp-a is integral and normal, but that the curve defined by the same equation but then over an algebraic closure of k is singular.
12. May 1.
We continue. We will try to prove the anti-equivalence between function fields of transcendence degree one and curves that are normal, integral and admit a finite morphism to the projective line over k.

Bas Edixhoven <>
Last modified: Mon Apr 24 17:31:52 MEST 2006