Algebraic Geometry

syeduru06

Please find the below questions and let me know if anyone can solve it. I am attaching the textbook for your reference.

Value of a =6, b=2

Let axyz - ay^3 +xz^2 =bw^3 be a homogenous polynomial in P3(x,y,z,w), describing an algebraic variety V in 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.

7. Describe the symmetries of your surface V. Is it bounded or unbounded?

8. Can you find a line on your surface?

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