Some of the topics covered will include
Syllabus:
We skip around a bit especially for the first few weeks.
Algebraic Varieties in Kn. Projective space KPn.(pp.3-5)
Projective hyperplanes, linear subspaces, and varieties. (pp.3-8).
A statement of Bezout's Theorem. (p.173)
Counting tangent lines to a fixed conic from a fixed point.
Conics simultaneously containing 5 points (p.12) or tangent to 5 lines.
Gauss map of a planar curve. (p.188)
Ideals of varieties (p.48), Hilbert Basis theorem, K[x1,
... , xn] is Noetherian.
Zariski topology (pp.17-18). Finite sets as varieties (pp. 6-7).
Twisted Cubic(pp. 9-10)
Complete and set-theoretic intersections (pp.136-137)
Hilbert's Nullstellensatz and consequences (pp. 48-62)
Regular Maps (pp. 21-25)
Cones, projections, and products (pp.31-40)
Dimension definitions (pp.133-135)
Regularity of Zariski open dense subset of complex hypersurface
Oct.25- Rational functions and maps (pp.72-81)
Incidence and secant varieties (pp.142-143)
Smoothness and Zariski tangent space (pp.174-177)
Stratification of complex varieties
Projective tangent spaces (181-183)
Degree and Bezout's theorem (224-229), (235-237)
Remarks on lectures:
Friday, Sept.3: There was a siign error in deriving the equation for the set of conics tangent to the line y=0 . A point on the intersection of the general conic ax2 +by2 +cz2+dxy+eyz+fxz = 0 and the line y=0 satisfies ax2 + cz2 + fxz = 0 . In the affine coordinate system z=1, the tangency condition is obtained by setting d/dx( ax2 + c + fx) equal to 0 , that is, 2ax + f = 0 . Substituting x = -2a/f , gives the correct quadratic relation 4ac - f2 = 0 . Alternately one can work in the other affine coordinate system x=1, where one uses d/dz(a + cz2 + fz) = 0 to substitute into a + cz2 + fz = 0 and again obtain the same relation 4ac - f2 = 0 .
Wed., Oct.19: As in Harris, our field K will always be algebraically closed.
Mon., Nov. 30: To give rigorous definitions of the projection
and intersection multiplicities we now assume that K = C ,
the complex field, and use the classical topology to define distinguished
neighborhoods.
Exercises:
1. Suppose H is a nonempty subset of
n-dimensional projective space KPn .
Show that H is a projective hyperplane
iff each coordinate neighborhood image Fj(H
intersect Uj) , if nonempty, is an n dimensional affine
hyperplane in Kn
iff {(0,...,0)}union{ (a0,...,an+1) in
Kn+1 : [a0,...,an+1]
in H } is an n dimensional linear subspace of
Kn+1 .
2. Suppose p(x0,x1,x2) is a nonconstant homogeneous polynomial of degree 2 . Find all the possible topological types of the corresponding projective real algebraic variety V = { [x0,x1,x2] in RP2 : p(x0,x1,x2) = 0 }.
3. Exercise 1.3 on page 6 of Harris. (See the hint in the back of the book.)
4. Exercise 1.21. There is a unique planar conic
passing through 5 given points (in general position) in
CP2.
Hint: The condition that the conic ax2+by2+cz2+dxy+eyz+fxz
= 0 pass through the points p1=(x1,y1,z1),
... , p5=(x5,y5,z5)
is a system of 5 homogeneous linear equations in the 6 unknowns
a,b,c,d,e,f . One must show that the rank of the
5 x 6 coefficient matrix is 5 for generic choice of
p1 , ..., p5 in (C3)5
= C15 . By showing this condition is a Zariski open set
in C15 one needs only check that it is nonempty (by finding
one specific good choice of p1 , ..., p5
) to conclude that it is both open and dense.
5. If A is a Noetherian ring and F : A -> B is a surjective ring homomorphism, then B is Noetherian.
6. If A is Noetherian and S is a multiplicative subset of A , then S-1A = { a/s : a in A, s in S } is Noetherian.
7. Show that a set of 7 points in general in the projective plane CP2 may be described as the set-theoretic intersection of two curves. How about 8 points?
8. Exercise 5.5.
9. Exercise 5.9.
10. Exercise 2.2.
11. Show that the field C of complex numbers has infinite transcendency degree over the field Q of rational numbers.
12. Show that subvarieties V and W of Kn are isomorphic if and only if their corresponding coordinate rings A(V) and A(W) are isoorphic as K algebras.
13. Show that a regular function on CPn is a constant.
14. Show that if V1 , ... ,
Vj are the irreducible components of a projective
variety V , then
dim V = supi dim Vi .
15. Show that the field of rational functions K(KPn) = K(Kn) = K(x1,...,xn) = { p/q | p,q in K[x1,...,xn] , q not identically 0} .
16. Show that if f : X --> Y is rational and f* : K(Y) -> K(X) is an isomorphism, then f | U is an isomorphism for some dense open subset U of X .
17. Show that dim LkV = n! / k!(n-k)! when dim V = n ..
18. Exercise 15.15
19. Exercise 16.14
20. Exercise 18.18
Some Pictures : (contributions welcome)
This page is maintained by Robert Hardt ( email
)
Last edited 12/9/99.