Review Problems

• Find a birational map from the curve y = x to some smooth curve by blowing up at the origin. • Parametrize the quadratic surface x + 2y + 3z = 1. • The parametrization of C : x − 2y = 1 defines a rational map PQ −→ C(Q). Show that the induced map K(C) −→ K(PQ) is an isomorphism. • Show that x = t + t, y = t + 1 parametrizes the conic C : x− 2xy + 2x + y − 3y + 2 = 0. Show that the inverse map is also polynomial, and that the induced map on the coordinate rings is an isomorphism. • A valuation on a ring R is a map v : R \ {0} −→ N such that v(rs) = v(r) + v(s) and v(r + s) ≥ min{v(r), v(s)}. For a prime p and nonzero a ∈ Z, define vp(a) = n if a = bp for some integer n not divisible by p. Show that vp is a valuation. • Understand what an exact sequence is. Show that the following sequences are exact: