1. Let k be an arbitrary field. When is A

3de Bachelor Wiskunde Academiejaar 2017-2018 1ste semester, 31 januari 2018

Oefeningen Algebraische Meetkunde

1. Let k be an arbitrary field. When is A

(k) \ {(0, 0)} an algebraic subset of A

(k)?

2. Let k be an arbitrary field, and let V ⊂ A

(k) be the reducible curve whose components are the 3 coordinate axes.

(a) Determine I = I(V ).

(b) Let J = (XY + XZ, Y Z). Show that V (XY + XZ, Y Z) = V .

(c) Show that XY 6∈ J , and X

Y

∈ J. What is the relation between I and J?

(d) Show that I cannot be generated by two elements.

3. Let F = X

Y − Z

, G = Y

− Z

be projective curves in P

(C). Compute the intersection points and their multiplicities.

4. Let F and G be two projective plane curves in P

(R).

(a) Give an example of F and G such that V (F, G) is the empty set.

Viewing F and G as polynomials in R[X, Y, Z] ⊂ C[X, Y, Z], F and G determine also projective plane curves V

(F ) and V

(G) in P

(C). Now assume that F and G have odd degree.

(b) Show that if P = (a, b, c) ∈ V

(F ) then also the complex conjugate ¯ P = (¯ a, ¯ b, ¯ c) ∈ V

(F ).

(c) Assume that F and G have no common component. Show that V (F, G) is nonempty.

Hint: look at Bezout’s formula.

(d) Is the assumption that F and G have no common component in (c) necessary?

Tijd: 3 uur; Cursus en nota’s mogen gebruikt worden.