MATH 584 FINAL EXAM Fall 2020

Your
final paper and
poster are due Dec 17th 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.)

The points colored in purple are to be turned in as a draft on Dec 9th for the review and input.
They will be graded as midterm work and revised to be included in your final
paper.

Below
is the structure of the paper. The
poster should summarize your work in a form that can be presented at a
conference next semester (see the attached template).

Start
your paper with points 1 and 2 below, and then follow the other requirements
(you can change the order).

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 Ux,
Uy, Uz, Uw. . 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 = Uw,
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} = (mP , rl,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 a goes to +infinity, -
infinity? Show appropriate images.

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

See the example below (note:
only names of Journals and title of books are in italics):

[1] A. Bremner, A. CHoudhry, M. Ulas,
Constructions of diagonal quartic and sextic surfaces with infinitely many
rational points. International J of Number Theory, 2014.

[2] F. Catanese, G. Ceresa.
Constructing Sextic Surfaces with a Given Number of Nodes.J. Pure Appl.
Algebra 23, 1-12, 1982.

[3] W. Barth. Two projective surfaces
with many nodes, admitting the symmetries of the icosahedron, Journal of
Algebraic Geometry 5 (1): 173–186, 1996.

[4] D. Jaffe, D. Ruberman. A sextic surface cannot have 66 nodes, Journal
of Algebraic Geometry 6 (1): 151–168, 1997.

[5] S.
Endraß. Surfer. http://surf.sourceforge.net. A project of the Mathematisches
Forschungsinstitut Oberwolfach and the Technical University Kaiserslautern
(2008).

[6] J. Harris.
Algebraic Geometry: a first course. Springer, GTM 133, 1992.

[7] G.
Salmon. A treatise on the analytic geometry of three dimensions.
Dublin: Hodges, Smith, & Co., 1865.

[8]https://csuci.blackboard.com/bbcswebdav/courses/2172_MATH_584_1_2576/elliptic_curve_addition%281%29.pdf
accessed on Nov 29, 2020.

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