The Properties of Your Cubic Surface

The Properties of Your Cubic Surface

describing the properties of your cubic surface and related curves including appropriate computer-generated images. Please make your title interesting and include an abstract at the beginning of the paper. Submit as pdfs and electronic files in CANVAS (i.e. pdf+ source MSword, Latex files, (as text) , ppt, etc. Include your computer codes if you use them for calculations.

 

 

SURFACES

1. Title, name, abstract – (write this part at the end highlighting your results).
2. Give basic definitions of P³ (x,y,z,w), an algebraic variety in P³, an irreducible variety, singular points, a dimension a variety.
3. Give the homogenous equation f of your surface V in P³ (x,y,z,w). What is the space classifying all degree 3 surfaces in P³? What is the dimension of this classifying space? Show the calculations.
4. Give definitions of U_x, U_y, U_z, U_w. . Graph your surface for w=1 , x=1, y=1, z=1 (use Surfer or any other graphing software).

 

5. Find singular points of V in P³ or prove there are none. Find the inflection points on your surface or prove there are none.

 

6. Pick a non-singular point and write the equation of a tangent space. What dimension is your variety at non-singular points?
7. Calculate Gaussian curvature of your surface V given by the equation F(x,y,z) = 0 in R³ = U_w,  at least two non-singular points. Analyze the changes of the curvature on your surface? Is it always positive or negative? Is there a curve when the curvature equals always to 0?
8. Describe the ideal I(V). Is it prime? Is your variety irreducible in P³ (justify)?
9. Describe the ring O(V) of regular functions on your surface. Describe the field of rational functions K(V). Is your V birational to P²?
10. Describe symmetries of your V, Aut(V) – give generators or  matrices if possible.

You can consult these:

Example: Aut (Sphere} = (m_P , r_l,a id, where P is any plane passing through, the center, l is any line passing through the center, and a is any angle }  is infinite, and can be described by the group of orthogonal 3 x 3  matrices  O(3).

http://www-groups.mcs.st-andrews.ac.uk/~john/geometry/Lectures/L10.html

 

https://en.wikipedia.org/wiki/Crystallographic_point_group

 

 

11. Find lines on your variety by solving the equations in variables s and t (you may use computers) or justify that there are none.
12. Consider curves (divisors) on V given by V intersected with a plane x=0, y=0, z=0, w= 0. Calculate genus of each curve, if possible.

(The genus formula for a smooth curve on a plane is  g= (d-1)(d-2)/2 ,     where d is a degree of the polynomial defining the curve).

13. What can you say about the family of curves given by equations x=a (degree, irreducibility, singularities, genus,  etc.)? What can you say about the family of curves given by equations y=b?    What can you say about the family of curves given by equations z=c? What can you say about the family of curves given by equations w=d?
14. Are there any other interesting curves that lie on your variety V (i.e. not plane sections)? For example, a twisted cubic or an elliptic curve (g=1) may lie on your surface.
15. Define a family of (interesting) deformations of V parameterized  by a in R¹.  What happens to irreducibility, singularities, symmetries, lines, etc.? What happens when the parameter agoes to +infinity, – infinity? Show appropriate images.

Bibliography – cite all sources you have used, including our textbook and a calc book