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.
Exercises:
- 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);
- 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.