Algebraic Geometry

a = 6, b = 2 are the values

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.

Curvature_surfaces _definition.docx

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

8. Can you find a line on your surface?