# 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

- Title, name, abstract – (write this part at the end highlighting your results).
- Give basic definitions of
**P**(^{3}*x,y,z,w*), an algebraic variety in**P**, an irreducible variety, singular points, a dimension a variety.^{3} - Give the homogenous equation
*f*of your surface**V in P**(^{3}*x,y,z,w*). What is the space classifying all degree 3 surfaces in**P**? What is the dimension of this classifying space? Show the calculations.^{3} - Give definitions of
*U*. Graph your surface for w=1 , x=1, y=1, z=1 (use Surfer or any other graphing software)._{x}, U_{y}, U_{z}, U_{w}.

- Find singular points of
**V**in**P**or prove there are none. Find the inflection points on your surface or prove there are none.^{3 }

- Pick a non-singular point and write the equation of a tangent space. What dimension is your variety at non-singular points?
- Calculate Gaussian curvature of your surface
**V**given by the equation*F(x,y,z)*= 0 in**R**^{3 }=*U*, 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?_{w} - Describe the ideal I(
**V**). Is it prime? Is your variety irreducible in**P**(justify)?^{3} - Describe the ring
*O*(**V**) of regular functions on your surface. Describe the field of rational functions*K*(**V**). Is your V birational to**P**^{2}? - 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

- Find lines on your variety by solving the equations in variables
*s*and*t*(you may use computers) or justify that there are none. - 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).

- 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*? - 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. - Define a family of (interesting) deformations of
**V**parameterized by*a*in**R**. What happens to irreducibility, singularities, symmetries, lines, etc.? What happens when the parameter^{1}*a*goes to +infinity, – infinity? Show appropriate images.

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