Answer the question in the picture attached, ignore 1, 4 and 6.

Use a = 8, b = 10 Let axyz - ay3 +xz? =bw be a homogenous polynomial in PS(x,y,z,w), describing an algebraic variety Vin P3. 1. Show the view of V in affine patches Ux, Uy, Uz, Uw. when x=1,y=1, z=1, w=1. 2. What is the dimension of V? 3. Is V an irreducible variety? 4. Find all singular points. 5. Give the ideal of V. Is it prime? Is your variety irreducible? Describe the ring k(V) = O(V) of polynomials (regular functions) on V. 6. Calculate curvature at (at least two) smooth points. Curvature surfaces definition.docx # See other file 7. Describe the symmetries of your surface V. Is it bounded or unbounded? 8. Can you find a line on your surface