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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?