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Algebraic Geometry

May 24, 2022 | Mathematics - Algebra

May 24, 2022 | Mathematics - Algebra

Algebraic Geometry

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.
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?

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