Advanced Algebra II
Homework 8 due on May. 18, 2007
In this exercise, k always denote an algebraically closed field.
(1) * Complete the exercises and incomplete proofs in the note.
(2) * Let (R, m) be a Noetherian local domain. Show that dim R ≤ dimkm/m2.
(3) Prove the associative law for the group structure on non-singular cubics.
(4) Determine the singularities and its embedded dimension of V(x2+ y3+ z5) ⊂ A3.
(5) Determine the singularities and its embedded dimension of V(xz−
y2) ⊂ A3.
(6) Prove Pascal’s theorem.
(7) Let F be a homogeneous polynomial in k[x, y, z] of degree d.
Show that dF = xFx+ yFy + zFz.
(8) Consider F = V(y2z − x3 + xz2) and G = V(xy − z2) in P2. Determine the intersection multiplicities at each intersection point and verify Bezout’s theorem.
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