Let v be a variety and p isin v show that there is a

Question

Math 176: Algebraic Geometry, Fall 2014- Assignment 5

1. (a) Let V be a variety and P ∈ V. Show that there is a one-to-one corespondence between prime ideals in O_P(V) and the subvarieties of V that pass through P. (Hint: If I is a prime ideal in O_P(V), show that I ∩ Γ(V) is a prime ideal in Γ(V), and I is generated by I ∩ Γ(V).)

(b) Why is the previous problem important?

2. Let V ⊂ Aⁿ and W ⊂ A^m be varieties. Let φ: V → W be an isomorphism of varieties. Prove O_φ(P) (W) ≅ O_P(V).

3. (a) Show that the map φ: V(x² + y² - 1) → A¹ given by φ(x, y) = x/1 - y is a dominant rational map.

(b) Show that ψ: A¹ → V(x² + y² - 1) given by ψ(t) = ((2t/t² + 1), (t² - 1/t² + 1)), is a dominant rational map.

(c) Deduce that V(x² + y² - 1) ∼ A¹. Draw a picture to explain what's going on.

4. A conic is a variety of the form

V(ax² + by² + cxy + dx + ey + f)

where a, b, c, d, e, f ∈ k. It is well known that any irreducible conic can be transformed into either the conic V(y - x²), V(xy - 1) or the conic V(x² + y² - 1) via an affine transformation (that is, via a linear transformation followed by a translation). Deduce that any irreducible conic is birationally equivalent to A¹.